c©Copyright 2016
Maxwell W. Libbrecht
Understanding human genome regulation through entropic
graph-based regularization and submodular optimization
Maxwell W. Libbrecht
A dissertation
submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
University of Washington
2016
Reading Committee:
William Stafford Noble, Chair
Jeffrey A Bilmes
Philip Green
Program Authorized to Offer Degree:
Computer Science and Engineering
University of Washington
Abstract
Understanding human genome regulation through entropic graph-based regularization and
submodular optimization
Maxwell W. Libbrecht
Chair of the Supervisory Committee:
Professor William Stafford Noble
Genome Sciences
I am interested in developing computational methods to improve understanding of human genome
regulation. This thesis is organized around two novel machine learning methods. First, I present a
new method in the field of posterior regularization. The genomic neighborhood of a gene influences
its activity, a behavior that is attributable in part to domain-scale regulation, in which regions of
hundreds or thousands of kilobases known as domains are regulated as a unit. I developed a method
called entropic graph-based posterior regularization that makes it possible to jointly model all
available genomic data sets, including chromatin state information from ChIP-seq and chromatin
conformation information from 3C-based assays. Using this approach, I produced a comprehensive
model of chromatin domains in eight human cell types, thereby revealing the relationships among
known domain types and identifying a new category of domain which I term “specific expression
domains.” Second, I present new methods from the field of submodular optimization. Due to the
high cost of sequencing-based genomics assays such as ChIP-seq and DNase-seq, a panel of at
most 3-10 assays is usually performed on each cell type. I present submodular selection of assays
(SSA), a method for choosing a diverse panel of genomic assays that leverages methods from
the field of submodular optimization. SSA performs better than alternative strategies in practice,
is computationally efficient and extremely flexible, and is theoretically optimal under certain
assumptions. This application may also serve as a model for how submodular optimization can be
applied to other discrete problems in biology. I applied a similar technique to remove redundancy in
protein sequence data sets. This method applies submodular optimization to choose representative
sets of protein sequences, achieving better results both in terms of reduction in redundancy and
functional diversity. These methods are widely applicable to computational problems in genome
biology and provide opportunities for the further development of methods.
TABLE OF CONTENTS
Page
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Genomics assays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Semi-automated genome annotation . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 2: Machine learning in genetics and genomics . . . . . . . . . . . . . . . . . . 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Machine learning application areas . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Scope of this review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Supervised versus unsupervised learning . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Semi-supervised learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Which type of method to use . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Generative versus discriminative modeling . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Incorporating prior knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Implicit prior knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 Probabilistic priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.3 Prior information in non-probabilistic models . . . . . . . . . . . . . . . . 20
2.5 Handling heterogeneous data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Feature selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Imbalanced class sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 Handling missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Modeling dependence among examples . . . . . . . . . . . . . . . . . . . . . . . 28
2.10 Future of machine learning in genomics . . . . . . . . . . . . . . . . . . . . . . . 29
i
Chapter 3: Entropic graph-based posterior regularization . . . . . . . . . . . . . . . . . 34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 EGPR for Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 Optimizing q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 Optimizing rM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Updating θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 EGPR for Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6.1 Learning a Gaussian Mixture on Poorly-Separated Data . . . . . . . . . . 52
3.6.2 Comparison with Related Inference Methods . . . . . . . . . . . . . . . . 53
3.7 Application: Genome Annotation Using Physical Interaction Information . . . . . 55
3.7.1 Synthetic Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7.2 Real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.9 Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.10 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Chapter 4: Joint annotation of chromatin state and chromatin conformation reveals
relationships among domain types and identifies domains of cell type-specific
expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Chromatin domains co-localize with domains of similar activity . . . . . . 67
4.2.2 Graph-based regularization expresses a pairwise prior in a SAGA method . 67
4.2.3 Using GBR to integrate 3D structure information improves prediction of
replication and topological domains . . . . . . . . . . . . . . . . . . . . . 68
4.2.4 Joint domain annotation of chromatin state and chromatin conformation
captures previously-described domain types . . . . . . . . . . . . . . . . . 69
4.2.5 Repressive domains are divided into constitutive, facultative and quiescent
heterochromatin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.6 Active domains are divided between broad and specific gene expression . . 73
4.2.7 Lamina association is driven by a complex structure of domains . . . . . . 74
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4.2.8 Developmentally-consistent domain boundaries are marked by identifiable
sequence elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.9 Using GBR to transfer information between cell types improves accuracy
of predicting functional elements . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.1 Histone modification, open chromatin and replication timing signal data . . 86
4.4.2 Hi-C data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.3 Segway model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.4 Graph-based regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4.5 GBR optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.6 GBR graph for incorporating Hi-C data . . . . . . . . . . . . . . . . . . . 93
4.4.7 GBR graph for annotation of multiple cell types . . . . . . . . . . . . . . . 93
4.4.8 Circular permutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.9 Topological domain agreement . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.10 Signal variance explained . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.11 Genomic element prediction . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.12 Developmental replication domains . . . . . . . . . . . . . . . . . . . . . 97
4.4.13 Consistent domain boundaries . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5 Data access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6 Supplementary Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6.1 Review of existing SAGA methods for using data from multiple cell types . 98
4.6.2 Related optimization methods . . . . . . . . . . . . . . . . . . . . . . . . 99
4.7 Supplementary Figures/Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Chapter 5: Choosing panels of genomics assays using submodular optimization . . . . . 110
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.1 Submodular selection of assays identifies diverse panels of genomics assays 114
5.2.2 Three metrics evaluate the quality of a set of genomics assays. . . . . . . . 118
5.2.3 Panels chosen by SSA outperform alternative methods according to three
evaluation metrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.4 SSA can select a subset of performed assays as input to an expensive
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4.1 Genomics data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.3 Submodular optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.4 Facility location function . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.5 Assay type similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.6 Evaluation cross-validation strategy . . . . . . . . . . . . . . . . . . . . . 132
5.4.7 Assay imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.4.8 Functional element prediction . . . . . . . . . . . . . . . . . . . . . . . . 133
5.4.9 Annotation-based evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5 Source code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.6 Author contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.7 Competing interests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.8 Supplementary Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.9 Supplementary Note 1: Comparison of potential objective functions . . . . . . . . 137
5.10 Supplementary Note 2: Comparison among various similarity measures . . . . . . 138
5.11 Supplementary Note 3: Comparison of different supports for similarity computation 142
5.12 Supplementary Note 4: Comparison of different correlation metrics as the similarity
measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.13 Supplementary Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Chapter 6: Eliminating redundancy among protein sequences using submodular opti-
mization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.1 Submodularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.2 Similarity functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.3 Subset quality measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2.4 Optimization algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.2.5 Comparison to threshold algorithm (CD-HIT) . . . . . . . . . . . . . . . . 160
6.2.6 Protein sequence data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
iv
6.3.1 Submodular optimization effectively eliminates redundancy in sequence
data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3.2 Representative subsets chosen by submodular optimization exhibit high
structural diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3.3 A mixture of submodular objective functions outperforms any single com-
ponent objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3.4 Representative subset selection methods perform better than repurposed
clustering methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Chapter 7: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.1 Genome annotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.1.1 Annotation with multiple length scales . . . . . . . . . . . . . . . . . . . 178
7.1.2 Hierarchical label complexity . . . . . . . . . . . . . . . . . . . . . . . . 179
7.1.3 Multiple cell types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.2 Statistical evaluation of differences between genomics assays . . . . . . . . . . . . 181
7.3 Improved submodular objective for choosing panels of genomics assays . . . . . . 183
7.4 Future problems in computational genomics . . . . . . . . . . . . . . . . . . . . . 184
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
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LIST OF FIGURES
Figure Number Page
2.1 Machine learning. A canonical example of a machine learning application. A
training set of DNA sequences is provided as input to a learning procedure, along
with binary labels indicating whether each sequence is centered on a TSS or not.
The learning algorithm produces a model which can then be subsequently used, in
conjunction with a prediction algorithm, to assign predicted labels to unlabeled test
sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 A gene finding model. A simplified gene finding model, which captures the basic
properties of a protein-coding gene. The model takes as input the DNA sequence of
a chromosome or a portion thereof and produces as output detailed gene annotations.
Note that this simplified model is incapable of identifying overlapping genes or
multiple isoforms of the same gene. . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Two models of TF binding. (A) A position-specific frequency matrix model, in
which the entry in row i and column j represents the frequency of the ith base
occurring at position j in the training set. (B) A linear support vector machine
model of TF binding. Labeled positive and negative training examples are provided
as input, and a learning procedure adjusts the weights on the edges to predict the
given label. (C) The figure plots the mean accuracy (± 95% confidence intervals),
on a set of 500 simulated test sets, of predicting TF binding as a function of the
number of training examples. The two series correspond to a PSFM model or an
SVM. (D) Schematic of generative versus discriminative modeling. The generative
model characterizes both classes completely, whereas the discriminative model
focuses on the boundary between the classes. . . . . . . . . . . . . . . . . . . . . 30
2.4 Incorporating a prior into a PSFM. A simple, principled method for putting a
probabilistic prior on a PSFM involves augmenting the observed nucleotide counts
with “pseudocounts” and then computing frequencies with respect to the sum. The
magnitude of the pseudocount corresponds to the weight assigned to the prior. . . 31
vi
2.5 Three ways to accommodate heterogeneous data in machine learning. The
task of predicting gene function labels requires methods that take as input gene
expression data, protein sequences, protein-protein interaction networks, etc. These
diverse data types can be encoded into fixed-length features, represented using
pairwise similarities (kernels), or directly accommodated by a probability model. . 32
2.6 Inferring network structure. Methods that infer each relationship in a network
separately, such as computing the correlation between each pair, can be confounded
by indirect relationships. Methods that infer the network as a whole can identify
only direct relationships. Inferring the direction of causality inherent in networks is
generally more challenging than inferring the network structure [Pearl, 2000], so
many network inference methods, such as Gaussian graphical model learning, infer
only the network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Illustration of optimization algorithm. Solid ovals denote closed-form update steps.
Dashed ovals with dotted expansion lines denote updates that are implemented by
alternating optimization. Pairs of opposing arrows indicate alternating optimization
implemented by iterating each update to convergence. . . . . . . . . . . . . . . . 49
3.2 Using EGPR to learn ambiguous clusters. Shape denotes true class, color denotes
predicted class, and colored arrows denote cluster means as they evolve between
iterations of EM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Comparison of EGPR with related inference methods. The X axis shows σ, a
hyperparameter controlling the difficulty of inference. The Y axis shows the average
accuracy over 200 simulations of MAP inference on the model in question (95%
Wilcoxon test confidence intervals). . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Synthetic model of genome spatial interactions. Color and labeled division lines
indicate learned labels along the hypothesized 501 bp genome. Large filled circles
indicate observed positions. Dotted lines indicate EGPR edges. (A) Without EGPR.
(B) With EGPR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 (A) Strategy for utilizing physical interaction information. (B) Improvement in
RMSE over chain model for EGPR and SQGPR for 29 experiments. (C) Relative
improvement in RMSE between EGPR and SQGPR for each of 29 experiments. . . 59
vii
4.1 (A) Strategy for incorporating Hi-C data using GBR. (B) Effect of GBR hyperpa-
rameters on topological domain agreement. The X axis indicates the value of the
graph weight hyperparameter λG. Y axis indicates the average over 10 annotations
(one for each input histone modification data set) of the fraction improvement in
topological domain label agreement by adding GBR over Segway without GBR.
(C) Same as (B), but Y axis indicates improvement in replication timing standard
deviation explained and average is over 29 annotations. Annotations used four
labels, 10kb resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Statistics of domain annotation made with Segway augmented with GBR. (A)
Heatmap of association of IMR90 domain labels to IMR90 data sets used in train-
ing. (B) Fraction of genome covered by each domain type. (C) Segment length
distribution for each domain type. (D) Label-specific histograms of values for five
notable data sets in IMR90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 (Caption on following page.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Enrichment of each domain type with respect to lamina-associating domain bound-
aries. X axis indicates position with respect to lamina-associating domains, with
each domain stretched or shortened to the median length of 0.8 Mb. Y axis indicates
label enrichment or depletion (log(obs/expected)). . . . . . . . . . . . . . . . . . 81
4.5 (A) Example of consistent domains. (B) Distribution of number of overlapping
boundaries compared to a permutation control (binomial test 95% confidence inter-
val error bars). Vertical dashed line denotes consistent boundary threshold. (C,D)
Cumulative density of distances from consistent domain boundaries to the nearest
(C) promoter and (D) distal CTCF motif. Distal CTCF motifs are defined as all
CTCF motifs more than 5 kb from a promoter. . . . . . . . . . . . . . . . . . . . 82
4.6 (A) Strategy for using GBR to transfer information between cell types. (B,C)
Performance of predicting the locations of GM12878 (B) EP300 and (C) CTCF
ChIP-seq peaks with and without using GBR to integrate information from K562.
Each plot shows the fraction of elements detected as a function of the number of
bases predicted (Methods). Results are shown on the test set (10 Mbp). Model
training and label ordering was performed on the training set (10 Mbp). The X
axis is plotted up to 1 kbp times the total number of elements. These plots can
be interpreted, for example, in the context of an enhancer validation experiment,
in which case it shows how many sequences would need to be tested in order to
discover a certain number of enhancers. . . . . . . . . . . . . . . . . . . . . . . . 83
4.7 Models of (A) domain types and (B) co-regulated regions. . . . . . . . . . . . . . 85
4.8 GBR model. Squares and circles denote discrete and continuous random variables
respectively. Filled-in and unfilled shapes denote observed and unobserved variables,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
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4.1 (A) Average squared difference between replication timing values at left and right
sides of significant contacts, as a function of genomic distance, relative to a permu-
tation control (t-test 95% confidence interval grey error regions). (B) Confusion
matrix of Segway annotation labels at left and right sides of significant contacts
(without GBR). Color depicts log2(obs/expected) relative to a permutation control
(Methods). Pairs of annotation labels at significantly interacting positions match
more often than expected by chance (binomial test p < 10−16). . . . . . . . . . . . 104
4.2 Fraction of annotation changed by GBR. Y axis depicts the fraction of positions
that receive a different label between an annotation without GBR and an annotation
with GBR using a certain set of hyperparameters. X axis and color depict the λG
and λR1 hyperparameters respectively. . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Correlation of Hi-C contact strength between IMR90 and H1-hESC. X and Y axes
are log p-values of association of a given pair of positions. Color indicates density
of points. Black lines indicate density contours in 0.1% bins. Spearman r = 0.57. 105
4.4 Distribution of domain labels across eight cell types. Y axis indicates log2(bases
covered by label ` in celltype A / (bases covered by label ` / 8)). . . . . . . . . . . 106
4.5 Visualization of GO term enrichment for genes in IMR90 (A) BRD domains and
(B) SPC domains using REVIGO [Supek et al., 2011]. Each bubble represents a
cluster of related enriched GO terms. X and Y axes are projected semantic axes
defined using multidimensional scaling on the semantic similarity of each pair of
terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.6 Enrichment of consistent Segway boundaries for consistent replication domain
boundaries. (A) Fraction of consistent replication domain boundaries overlapping
consistent Segway domain boundaries as a function of the overlap distance. (B)
Same as (A), but fraction of Segway domain boundaries. We used replication
domain boundaries called by Pope et al. [2014]. We defined replication boundaries
occurring in more than 10 out of 18 cell types as consistent. . . . . . . . . . . . . 108
4.7 Genomic distance distribution of significant IMR90 Hi-C contacts (q < 0.05). . . 108
4.8 Schematic of the three formulations of the objective and the alternating maximiza-
tion strategy. Edges in this figure indicate KL terms, labeled according to their
weight in the objective. Boxed formulae are update steps. We perform two refor-
mulations, first splitting q into q and rM linked by a KL term of weight λR1, then
splitting rM into rM and sM , linked by a KL term of weight λR2. . . . . . . . . . . 109
ix
4.9 Comparison between inference methods on synthetic data. The X axis shows
σ, a hyperparameter controlling the difficulty of inference. The Y axis shows
the average accuracy over 200 simulations of MAP inference on the model in
question (95% Wilcoxon test confidence intervals). Bars correspond to five different
inference methods: 1) inference on each position independently, with no chain
model (Independent); 2) inference on the chain alone, without using W (Chain);
3) loopy belief propagation on the chain plus extra factors of Pr(Xi = Xj) =
sigmoid(λwij), where λ controls the strength of these factors; (Loopy BP) 4) a
variant of GBR using the regularization graph W and a squared-error penalties as
described in [He et al., 2013] (SQ-GBR); and 5) GBR using the regularization graph
W (our method, GBR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1 Schematic of the selection process performed by submodular selection of as-
says (SSA). The method takes as input all available existing genomics assays, where
each assay is represented as a real-valued track over the genome. In the SSA-past
mode, SSA selects a panel of already-performed assays to use as input to an ex-
pensive computational analysis. In the SSA-future mode, SSA chooses a panel of
assay types to be performed in a new cell type. In both cases, the resulting data
sets are provided as input to downstream analyses, which may include imputing
assays that weren’t performed, predicting the locations of functional elements, or
semi-automated genome annotation. . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 (Caption on following page) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Evaluation strategies. Schematics of the three evaluation metrics: (a) assay impu-
tation, (b) functional element prediction, and (c) annotation-based evaluation, as
well as (d) the overall cross-validation evaluation strategy. . . . . . . . . . . . . . 119
5.4 Relationship between the facility location objective function and evaluation
metrics. Each dot corresponds to one of 40 randomly-chosen panels. Pink triangles
indicate results from maximizing the SSA-future facility location function; red dia-
monds indicate the panel of most-frequently performed assay types (Supplementary
Table 5.5). These results were computed in GM12878, using panels of four assay
types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
x
5.5 Performance of panel selection strategies. (a) Boxplots show the distribution
of evaluation metrics over 40 random panels on data from cell type GM12878.
The panels of most-frequently performed assays are composed of the top k most
frequent assay types available in our data set, where k is the size of the panel. Each
evaluation metric is normalized to lie within [0, 1] by subtracting the lowest value
and dividing by the highest. (b) Scatter plot between the performance of SSA-past
and SSA-future across cell types K562, GM12878, and H1-hESC. Each dot in the
plot corresponds to the two variants of SSA for a panel size evaluated using a metric
in a cell type. The performance is measured as the fraction of panels that perform
worse than the SSA-chosen panel, estimated by comparing to 40 randomly-selected
panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.6 Comparison among various submodular functions in terms of the Spearman correla-
tion between function valuation and the performance metrics. . . . . . . . . . . . . 139
5.7 Comparison among various similarity aggregation strategies in terms of the Spear-
man correlation between the facility location function valuation instantiated by the
similarity measure and the performance metrics. . . . . . . . . . . . . . . . . . . 141
5.8 Performance of the Dnase peaks-based SSA relative to an estimate of the perfor-
mance on all possible panels. The vertical axis shows the fraction (%) of panels
that perform worse than the SSA-chosen panel for a given setting, estimated by
comparing to 40 randomly-selected panels. . . . . . . . . . . . . . . . . . . . . . 143
5.9 Performance of the random genomics positions-based SSA relative to an estimate
of the performance on all possible panels. . . . . . . . . . . . . . . . . . . . . . . 144
5.10 Scatter plot between the two variants of SSA with different genomic support for
similarity computation. Each dot in the plot corresponds to the performance (again
measured as relative to an estimate of the performance on all possible panels) of the
two variants of SSA for a selection budget evaluated using a metric in a cell type
(i.e., one bar in Figure 5.8 and its counterpart in Figure 5.5). Its x- and y-values
are the performance measure for the DNase peaks-based SSA and random genomic
positions-based SSA, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.11 Performance of the absolute Spearman correlation-based SSA relative to an estimate
of the performance on all possible panels. . . . . . . . . . . . . . . . . . . . . . . 146
5.12 Scatter plot between the two variants of SSA with different correlation metrics as
the similarity measure. Each dot in the plot corresponds to the performance (again
measured as relative to an estimate of the performance on all possible panels) of the
two variants of SSA for a selection budget evaluated using a metric in a cell type
(i.e., one bar in Figure 5.11 and its counterpart in Figure 5.5). Its x- and y-values are
the performance measure for the Spearman correlation-based SSA and the Pearson
correlation-based SSA, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 147
xi
6.1 Protein sequence representative set selection. Points represent protein sequences,
projected into 2D based on their pairwise similarities according to BLAST using
t-SNE [Van der Maaten and Hinton, 2008]. All sequences are depicted from the
SCOPe Class “Membrane and cell surface proteins and peptides.” Color with text
labels indicates SCOPe fold, and italicized text indicates SCOPe family within
a given fold. Black circles indicate the top ten representative sequences chosen
by the greedy algorithm on a mixture of facility-location (Rankprop sim; 0.5
weight) and sum-redundancy (percent ID sim; 0.5 weight). Numbers labeling these
circles indicate choice order. Choice #9 is colored red because it is the second
chosen sequence within the same fold. To produce the 2D embedding, we first
computed a n × n pairwise distance matrix (for a set of n sequences) with the
formula: distance(i, j) = 1 − percent id(i, j)/100. We used multidimensional
scaling (MDS) to project this distance matrix to a 30×n feature matrix . This MDS
preprocessing step is similar to the standard preprocessing step used by tSNE that
applies principle component analysis (PCA) to project a high-dimensional feature
matrix to a 30 × n feature matrix [Van der Maaten and Hinton, 2008]. We used
tSNE to project this 30×n matrix into a 2×n matrix, which is plotted. The diffuse
points in the center are a common feature of tSNE plots, and are a consequence of
the algorithm imperfectly recapitulating the n-dimensional similarity matrix in two
dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 (A) Average redundancy in representative sets. The vertical axis is calculated
for a representative set A as (1/|A|)∑a1,a2∈A s(a1, a2), where s(a1, a2) is percent
identity. Vertical lines correspond to the sizes of subsets selected by CD-HIT using
40% and 90% sequence identity thresholds. (B) Histogram of pairwise similarities
within representative sets. We produced subsets of size 12,614 sequences (the size
reported by CD-HIT with a 40% identity threshold) for all methods by finding the
smallest subset larger than the target size and randomly removing sequences to
reach this size. Vertical axis indicates the number of pairs of sequences with a given
percent identity within each size-12,614 representative set, in 5% bins. Vertical line
indicates 40% identity threshold used by CD-HIT. . . . . . . . . . . . . . . . . . 162
xii
6.3 Coverage of SCOPe families. (A) The vertical axis indicates is the number of
families with at least one representative, normalized to the range [0, 1] using the
formula f ′(x) = f(x)−E(x)
M(x)−E(x) , where x is a subset size, f(x) is the value for the
method in question on that subset size, E(x) is expected (random mean) and
M(x) is maximum possible (defined as M(x) = x if x < F , and M(x) = F
otherwise, where F = 4,994 is the total number of SCOPe families). Vertical lines
correspond to the sizes of subsets selected by CD-HIT using 40% and 90% sequence
identity thresholds. (B) Area under the curve comparison. Bars indicate selection
algorithms, ordered by their area under the curve. Horizontal axis indicates area
under the “family” coverage curve, calculated as follows. Let V be the full set of
sequences, i ∈ 1 . . . |V | be a subset size, m be a particular method, A(m)i be the
subset of size i chosen by m, and f(A) be the number of SCOPe Families covered
by A. Area under the curve = 1|V |
∑|V |
i=1 f(A
(m)
i ). For subset sizes for which we did
not compute a subset, f() is imputed with linear interpolation. . . . . . . . . . . . 163
6.4 (A) Same as 6.3A, but including mixture objectives. (B) Comparison of mixture
weights. Each mixture objective is a combination of facility-location and sum-
redundancy objectives, each with either a percent ID or Rankprop similarity function.
Horizontal axis indicates weight on facility-location (weight on sum-redundancy
equals one minus the weight on facility-location). Line color indicates mixture
objective. Vertical axis indicates area under the curve for SCOPe family coverage
(see Supplementary Figure 6.10). To avoid overfitting, the statistics in (A) were
computed on the subset of sequences with the SCOPe Class “All beta proteins”. . . 165
6.5 Performance of clustering methods repurposed for representative set selection. Plot
is same as Figure 6.3, but includes clustering-based methods instead of submodular
optimization-based. Results are computed on our development set, composed of the
subset of the data (∼21%) with the SCOPe Class “All beta proteins”. . . . . . . . 166
6.6 Comparison of the CD-HIT source and our implementation of the same algorithm.
Horizontal axis indicates number of sequences in a given subset, focused on the
region between 40% and 90% percent ID thresholds (horizontal black lines indicates
these thresholds). Line color indicates quality measure; vertical axis indicates
quality value. All quality measures are linearly transformed to fall between zero
and one. Solid lines with triangles indicate performance with our implementation;
dotted lines with circles indicate the CD-HIT source. . . . . . . . . . . . . . . . . 168
xiii
6.7 Optimization performance on each submodular objective: (A) facility-location with
percent ID sim; (B) facility-location with Rankprop sim; (C) sum-redundancy with
percent ID sim. Horizontal axis indicates size of representative set. Vertical axis
indicates facility-location objective value (higher is better). Color and point type
indicates choice method. The highly similar lines in panel (B) result from the fact
that the Rankprop similarity function has a small dynamic range (∼0.9-1.0). . . . 171
6.8 Coverage of SCOPe categories: (A) Families; (B) Superfamilies, and (C) Folds.
Horizontal axis indicates size of representative set. Vertical axis indicates number
of SCOPe categories with at least one representative. Color and point type indicates
choice method. Black curve indicates maximum possible performance. Vertical
black lines indicate subset sizes reported by CD-HIT for thresholds of 40% and 90%. 172
6.9 Same as Supplementary Figure 6.8, but vertical axis is normalized as in Figure 2. . 173
6.10 Area under the curve for SCOPe Family coverage. Bars indicate selection algo-
rithms, ordered by their area under the curve. Horizontal axis indicates area under
the curve, calculated as follows. Let V be the full set of sequences, i ∈ 1 . . . |V | be
a subset size, m be a particular method, A(m)i be the subset of size i chosen by m,
and f(A) be the number of SCOPe Families covered by A. Area under the curve
= 1|V |
∑|V |
i=1 f(A
(m)
i ). For subset sizes for which we did not compute a subset, f()
is imputed with linear interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.11 Area under the curve for SCOPe Superfamily coverage. Plot is same as Supple-
mentary Figure 6.10, except that horizontal axis indicates Superfamily coverage.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.12 Area under the curve for SCOPe Fold coverage. Plot is same as Supplementary
Figure 6.10, except that horizontal axis indicates Fold coverage. . . . . . . . . . . 176
6.13 Area under the curve for SCOPe Family coverage on development set. Plot is same
as Supplementary Figure 6.10, except that results are computed on our development
set, composed of the subset of the data (∼21%) with the SCOPe Class “All beta
proteins”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
xiv
LIST OF TABLES
Table Number Page
2.1 Selected applications of machine learning in genetics and genomics. . . . . . . 7
2.2 Glossary of machine learning terminology. . . . . . . . . . . . . . . . . . . . . 10
2.3 Selected machine learning methods. . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Measuring performance of a classifier. Different applications for a machine
learning classifier necessitate different performance measures. . . . . . . . . . . . 25
4.1 Known types of domains. Note that the lengths of domains depend greatly on the
method used to define them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Summary of learned domain types. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Parameters of all genome annotations . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3 GO terms enriched for genes in BRD domains . . . . . . . . . . . . . . . . . . . . 102
4.4 GO terms enriched for genes in SPC domains. . . . . . . . . . . . . . . . . . . . . 103
5.1 The choice order of SSA on all assay types. . . . . . . . . . . . . . . . . . . . . . 144
5.2 Performing SSA after one assay has already been performed. We ran SSA
while restricting the output panel to include each assay type in turn, by initializing
the submodular greedy algorithm with the assay type in question and then running
the algorithm as normal. As expected, restricting the panel to include a particular
assay type de-prioritizes assay types that measure similar types of activity. (Top)
Choice order of histone modification assays by SSA if one assay type is required to
be included. (Bottom) Choice order of transcription factors by SSA if one assay type
is required to be included. In both tables, each assay type in the table is encoded
with a different color to allow easy comparison to different columns. . . . . . . . . 148
xv
5.3 Incorporating modular weighting to encourage certain assay types to be cho-
sen. We ran SSA with a variant that weights certain assay types more highly in the
selection process. In this experiment, we examine the choice order of assay types
by changing the weights on histone modification assays in a selection experiment
involving both histone modification and transcription factor assays. We implement
this experiment by first defining a combined objective hα(A) = f(A) + αm(A),
where f is the facility location objective and m is the modular function, with each
modular score capturing the importance of the item. Note that a set function m is
modular if m(A) =
∑
a∈Am(a) holds for all A ⊆ V . We define a hyperparameter
α > 0 that controls the trade-off between f and m. We define the modular function
m by assigning the score m(a) for each histone mod assay as 1 and for other assay
types as 0. Given a choice of α, we use the greedy algorithm to optimize hα leading
to an ordering of the assays. The table displays the resulting order for various
choices of α. All histone modification assay types are color encoded for ease of
comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.4 The choice order of cell types using SSA methodology. . . . . . . . . . . . . . . . 150
5.5 The 10 most frequently performed assay types. . . . . . . . . . . . . . . . . . . . 150
5.6 The choice order of SSA-past on common assay types shared among the six EN-
CODE tier 1+2 cell types: K562, GM12878, H1-hESC, A549, HepG2, and HeLa-S3.150
xvi
1Chapter 1
INTRODUCTION
More than a decade after the sequencing of the human genome, much is still unknown about how
the genome functions. A primary function of the genome is the production of RNA transcripts of
genes, which go on to produce the proteins that carry out most cellular activity. Gene regulation
occurs through a number of biochemical processes. In the cell, most DNA is wrapped around
histone proteins, forming a DNA-protein complex called chromatin. DNA-binding proteins (tran-
scription factors) act on chromatin, changing its biochemical activity by recruiting other factors,
covalently altering histone proteins, moving and removing histones to reveal the underlying DNA,
and ultimately expressing or silencing a given gene. This complicated process is a key determinant
of human traits, evolution and disease. I am interested in developing computational methods to
improve our understanding of gene regulation.
1.1 Genomics assays
A class of experimental methods known as genomics assays are a powerful tool for understanding
gene regulation. These experimental assays leverage DNA sequencing technology to assay various
types of biochemical properties. These assays have become possible recently due to the dramatic
increase in efficiency of high-throughput DNA sequencing technology. Due to this technological
improvement, in the last decade, genomics assays have revolutionized the study of genome biology.
A representative genomics assay is chromatin immunoprecipitation followed by sequencing
(ChIP-seq). This assay aims to identify the binding sites of a given transcription factor (i.e. DNA-
binding protein)—call that factor X—in the genome of a population of human cells. These cells
can be human cells grown in culture (cell lines), a sample of human primary tissue or any other
source of cells in a homogenous state. In a ChIP-seq experiment, the researcher applies a series
2of biochemical reactions to break up the genome into small fragments of DNA, then applies an
antibody to extract the fragments that are bound to X (the term “immunoprecipitation” derives
from this antibody step). The researcher then uses DNA sequencing technology to read the DNA
sequence of each fragment. This sequence is known as a “read”. If that sequence occurs exactly
once in the human reference genome, the read is taken as evidence that the transcription factor
binds to the matching position. A typical ChIP-seq experiment costs on the order of $500 and
measures 107–108 such reads, so each binding event is measured hundreds of times to give statistical
confidence. In that way, a ChIP-seq assay measures genome-wide binding of a given transcription
factor in a population of cells relatively cheaply.
Many different types of genomics assays have been developed to query a wide variety of types
of genome activity, including transcription factor binding, DNA accessibility, transcription, 3D
genome architecture of the nucleus, and others. These assays generally proceed in similar ways
as ChIP-seq, but replace the antibody step with a different step that extracts DNA fragments that
exhibit some other activity of interest. In the last decade, several efforts have aimed to systematically
apply genomics assays to a variety of human cells. The two largest such efforts are the ENCODE
and Roadmap Epigenomics consortia. These two consortia have performed thousands of genomics
assays on hundreds of human cell types. These rich data sets provide the opportunity to greatly
improve our understanding of human genome biology. However, their scale and complexity
necessitate the development of computational methods for their analysis.
1.2 Semi-automated genome annotation
A class of computational methods called semi-automated genome annotation (SAGA) algorithms
are widely used to analyze genomics data sets. These algorithms take as input a collection of
genomics data sets performed in a given cell type represented as real-valued tracks over the genome.
They simultaneously partition the genome and label each segment with an integer label such that
positions with the same label have similar patterns of activity. Each label is taken to represent a
given type of genome element, and a researcher interprets what biological phenomenon each integer
label corresponds to (hence the name “semi-automated”). These methods therefore both (1) identify
3new types of genomic elements and (2) identify the function of each position in the genome. They
have been shown [Ernst and Kellis, 2010, Hoffman et al., 2012] to re-discover many known types of
elements, including genes, promoters, enhancers and heterochromatic (inactive) regions. I apply
and extend SAGA methods throughout the work presented in this thesis.
1.3 This thesis
This thesis begins with a review of how machine learning is currently applied to genetic research
(Chapter 2). The subsequent chapters are organized around novel machine learning methods in two
fields, posterior regularization and submodular optimization. Chapter 3 presents a machine learning
method called entropic graph-based posterior regularization. This method integrates prior work on
unsupervised models such as SAGA with methods for semi-supervised learning, allowing new types
of information to be efficiently incorporated into structured probabilistic models. In Chapter 4, I
apply this method to investigate genomic domain regulation, in which regions of up to millions of
base pairs are regulated as a unit. Chapters 5 and 6 present new methods in computational biology
that apply techniques from submodular optimization. In Chapter 5, I present Submodular Selection
of Assays (SSA), a method for choosing which genomics assays to perform on a new cell type. In
Chapter 6, I present a submodular optimization-based method for removing redundancy in protein
sequence data sets. In Chapter 7, I suggest several directions for future work in this field.
4Chapter 2
MACHINE LEARNING IN GENETICS AND GENOMICS
Abstract
The field of machine learning promises to enable computers to assist humans in making sense of
large, complex data sets. In this review, we outline some of the main applications of machine learning
to genetic and genomic data. In the process, we identify some recurrent challenges associated with
this type of analysis and provide general guidelines to assist in the practical application of machine
learning to real genetic and genomic data.
Summary
• The field of machine learning is concerned with the development and application of computer
algorithms that improve with experience.
• Machine learning methods can be divided into supervised, semisupervised and unsupervised
methods. Supervised methods are trained on examples with labels (e.g. “gene” or “not gene”)
and then are used to predict these labels on other examples, whereas unsupervised methods
find patterns in data sets without the use of labels, and semisupervised methods leverage
patterns in unlabeled data to improve power at predicting labels.
• An application requires different machine learning methods depending on whether one is
interested in interpreting the output model or one is simply concerned with predictive power.
Generative models, which posit a probabilistic distribution over input data, are generally
best for interpretability, whereas discriminative models, which seek only to model labels, are
generally best for predictive power.
5• Prior information can be added to a model in order to train the model more effectively given
limited data, limit the complexity of the model, or incorporate data not used by the model
directly. Prior information can be incorporated explicitly in a probabilistic model or implicitly
through the choice of features or similarity measures.
• Choosing an appropriate performance measure depends strongly upon the application task.
Machine learning methods are most effective when they optimize an appropriate performance
measure.
• Network estimation methods are appropriate when the data contains complex dependencies
among examples. These methods work best when they take into account the confounding
effects of indirect relationships.
2.1 Introduction
The field of machine learning is concerned with the development and application of computer
algorithms that improve with experience [Mitchell, 1997]. Thus, for example, in genomics machine
learning can be used to “learn” how to recognize the locations of transcription start sites (TSSs) in a
genome sequence [Ohler et al., 2002]. The process typically proceeds in three stages (Figure 2.1).
First, a machine learning researcher develops an algorithm that they believe will lead to successful
learning. Second, the algorithm is provided with a large collection of TSS sequences as well as,
optionally, a list of sequences that are known not to be TSSs. The annotation indicating whether a
sequence is a TSS or not is known as the label. The algorithm processes these labeled sequences
and stores a model. Third, novel, unlabeled sequences are given to the algorithm, and it uses the
model to predict labels (“TSS” or “not TSS”) for each sequence. If the learning was successful,
then all or most of the predicted labels will be correct. If the labels associated with test set examples
are known—i.e., if these examples were held out of the training set on purpose for use in testing
the performance of the learning system—then the performance of the machine learning algorithm
can be assessed immediately. Otherwise, in a prospective validation setting, the TSS predictions
6Model
Machine!
learning!
algorithm
Predicted!
labels
TSS
TSS
TSS
TSS
Not TSS
Not TSS
Not TSS
Not TSS
2.
3.
6.
8.
4.
1.
5.
7.Testing set
2.
3.
6.
8.
4.
1.
5.
7.
Prediction!
algorithm
Training set
TSS
TSS
TSS
TSS
Not TSS
Not TSS
Not TSS
Not TSS
Labels Data!
(“features”)
Figure 2.1: Machine learning. A canonical example of a machine learning application. A training
set of DNA sequences is provided as input to a learning procedure, along with binary labels
indicating whether each sequence is centered on a TSS or not. The learning algorithm produces a
model which can then be subsequently used, in conjunction with a prediction algorithm, to assign
predicted labels to unlabeled test sequences.
7Input data Task
DNA sequence Identify transcription start sites, splice sites, exons, etc.
DNA sequence Identify TF binding sites
DNA sequence Identify genes
Gene expression Predict regulatory relationships
Gene expression data Identify biomarkers for a disease
Histone and TF ChIP-seq data Partition and label the genome with chromatin state annotation
DNA sequence + gene expression + . . . Predict gene function
DNA sequence + histone mods + . . . Predict gene expression
DNA sequence + histone mods + . . . Predict variant deleteriousness
Sequence variants + gene expression + . . . Predict disease phenotype or prognosis
Table 2.1: Selected applications of machine learning in genetics and genomics.
produced by the machine learning system must be tested independently in the lab. Note that this is
an example of a subtype of machine learning called “supervised learning,” which is described in
more detail below (Section 2.2).
This process of algorithm design, learning, and testing is simultaneously analogous to the
scientific method on two different levels. First, the design-learn-test provides a principled way to
test a hypothesis about machine learning: “I hypothesize that algorithm X can successfully learn to
recognize TSSs.” Second, the algorithm itself can be used as a hypothesis generator: “I hypothesize
that sequence Y is a TSS.” In this second setting, the resulting scientific theory is instantiated in
the model produced by the learning algorithm. In this case, a key question, which we return to in
Section 2.3, is whether and how easily a human can make sense of this model.
2.1.1 Machine learning application areas
Machine learning methods have been applied to a huge variety of problems in genomics and
genetics (Table 2.1). Perhaps most significantly, machine learning has been used to annotate a wide
variety of genomic sequence elements. In addition to TSSs, algorithms can be trained to identify
splice sites [Degroeve et al., 2002], promoters [Bucher, 1990], enhancers [Heintzman et al., 2007],
positioned nucleosomes [Segal et al., 2006], etc. In general, if you can compile a list of sequence
elements of a given type, then you can probably train a machine learning method to recognize those
8elements. Furthermore, models that each recognize an individual type of of genomic elements can
be combined, along with (learned) logic about their relative locations, to build machine learning
systems capable of annotating genes, including their full UTR/intron/exon structure, along entire
eukaryotic chromosomes [Picardi and Pesole, 2010].
In addition to learning to recognize patterns in DNA sequences, machine learning can take as
input data generated by other genomic assays, such as microarray or RNA-seq expression data,
chromatin accessibility assays such as DNase-seq, MNase-seq, and FAIRE, or histone modification,
transcription factor (TF) binding ChIP-seq data, etc. Gene expression data can be used to learn to
distinguish between different disease phenotypes and, in the process, to identify potentially valuable
disease biomarkers (Section 2.6). Chromatin data can be used, for example, to annotate the genome
in an “unsupervised” fashion, thereby potentially allowing for the identification of novel classes of
functional elements (Section 2.2).
Machine learning has also been used extensively to assign functional annotations to genes. Such
annotations most frequently take the form of Gene Ontology term assignments [Gene Ontology
Consortium, 2000b]. Predictive algorithms can take as input any one or more of a wide variety of
data types, including the genomic sequence, gene expression profiles across various experimental
conditions or phenotypes, protein-protein interactions, synthetic lethality data, open chromatin data,
histone modification or TF binding ChIP-seq data, etc. As an alternative to GO term prediction,
some predictors instead identify co-functional relationships; i.e., the machine learning method
outputs a network in which genes are nodes and an edge between genes A and B indicates that the
two genes share a common function [Fraser and Marcotte, 2004].
Finally, a wide variety of machine learning methods have been developed to help understand the
mechanisms underlying gene expression. Some techniques aim to predict the expression of a gene
based solely on the DNA sequence [Beer and Tavazoie, 2004], whereas others take into account
ChIP-seq histone modification [Karlic et al., 2010] or TF binding [Ouyang et al., 2009] profiles at
the gene promoter region. More sophisticated methods attempt to jointly model the expression of all
genes in a cell by training a network model [Friedman, 2004]. Like a co-functional network, each
node in a gene expression network is a gene; however, edges in this case represent, for example,
9regulatory relationships between TFs and their targets.
Many of the problems listed above can also be solved using techniques drawn from the field
of statistics. Indeed, the line between machine learning and statistics is at best blurry, with some
preferring the term “statistical learning” over “machine learning” [Hastie et al., 2001]. Historically,
the field of machine learning grew out of the artificial intelligence community, where the term
“machine learning” became popular in the late 90s. In general, machine learning researchers have
tended to focus on a subset of problems within statistics, emphasizing in particular the analysis of
large, heterogeneous data sets. Accordingly, many core statistical concepts, such as calibration of
likelihood estimates, statistical confidence estimation and power calculations, are essentially absent
from the machine learning literature.
2.1.2 Scope of this review
This review provides an overview of machine learning as it is applied to problems in genomics. We
discuss the main categories of machine learning methods and the key considerations that must be
made when applying these methods to genomics problems. We do not not attempt to catalogue all
machine learning methods or all reported applications of machine learning to genomics, nor do
we discuss any particular method in great detail. Instead, the review begins by explaining several
key distinctions in machine learning and then outlining some of the major challenges researchers
face in applying machine learning methods to practical problems in genomics. We hope that the
reader will take away from this review an idea of what problems machine learning approaches
might be applicable to and which types of methods are likely to be effective for these problems.
For a more detailed treatment of machine learning applied to particular subfields of genetics and
genomics, the reader may consult reviews of these specific topics [Hamelryck, 2009, Swan et al.,
2013, Upstill-Goddard et al., 2013, Yip et al., 2013].
The review will cover the following topics. Section 1 describes the major classes of machine
learning problems: supervised, unsupervised and semisupervised. Section 2 concerns the tradeoff
between maximizing a model’s performance and interpretability. Sections 3-5 describe strategies a
researcher can use to guide a machine learning model, through prior knowledge, means of integrating
10
Example One data instance used in a machine learning task.
Feature A single measurement or descriptor of an example used in a ma-
chine learning task.
Label The target of a prediction task. In classification, the label is discrete
(e.g., “expressed” or “not expressed”); in regression, the label is
real-valued (e.g., a gene expression value).
Supervised algorithm Machine learning algorithm that is trained on labeled examples and
used to predict the label of unlabeled examples.
Unsupervised algorithm Machine learning algorithm that does not require labels, such as a
clustering algorithm.
Input space Set of features chosen as input to a machine learning method.
Kernel methods A class of machine learning methods (e.g. SVM) that employ a
type of similarity measure between feature vectors called a kernel.
Bayesian network A representation of a probability distribution that specifies the
structure of dependencies between variables as a network.
Overfitting A common pitfall in machine learning analysis where a complex
model is trained on too little data and becomes specific to the
training data, resulting in poor performance on other data.
Curse of dimensionality The observation that analysis can sometimes become more difficult
as the number of features increases, particularly because overfitting
becomes more likely.
Feature selection The process of choosing a smaller set of features from a larger
set, either before applying a machine learning method or as part of
training.
Class skew A case where the two classes in a supervised learning problem are
present with differing frequencies. Usually necessitates careful
choice of performance measure.
Table 2.2: Glossary of machine learning terminology.
heterogeneous data sets and feature selection. Sections 6-9 describe several important pitfalls and
special cases of machine learning: imbalance between classes in supervised problems, missing data
and networks of dependencies between examples. Finally, in Section 10, we describe the outlook
for machine learning in genomics in coming years.
2.2 Supervised versus unsupervised learning
Machine learning methods can usefully be segregated into two primary categories: supervised
versus unsupervised methods. To illustrate the difference between the two, consider again the gene
11
5’	
  UTR	
   Ini*al	
  exon	
  
Internal	
  
intron	
  
Internal	
  
exon	
  
Final	
  
exon	
   3’	
  UTR	
  
Non-­‐
coding	
  
Input:	
  genomic	
  sequence	
  
GCCTGGGAAAACCCTCAACTTT...	
  
Output:	
  gene	
  annota2on	
  
Gene	
  model	
  
Figure 2.2: A gene finding model. A simplified gene finding model, which captures the basic
properties of a protein-coding gene. The model takes as input the DNA sequence of a chromosome
or a portion thereof and produces as output detailed gene annotations. Note that this simplified
model is incapable of identifying overlapping genes or multiple isoforms of the same gene.
12
Method Su
pe
rvi
sed
Un
su
pe
rvi
sed
Ge
ne
rat
ive
Di
scr
im
ina
tiv
e
Linear regression X X
Artificial Neural Network (ANN) X X X
Support Vector Machine (SVM) X X
k-means clustering X X
Hierarchical clustering X X
Hidden Markov Model (HMM) X X X
Table 2.3: Selected machine learning methods.
finding problem: given the DNA sequence of a chromosome, the goal of a gene finding algorithm
is to predict the locations and detailed intron/exon structure of all of the protein-coding genes on
the chromosome (Figure 2.2). The most straightforward solution to this problem leverages what
we already know about the genome to help us build a predictive model. In particular, a supervised
learning algorithm for gene finding requires as input a training set of labeled DNA sequences, where
the labels specify the locations of the start and end of the gene (TSS and TTS) as well as all of the
splice sites in between. The model then uses this training data to learn general properties of genes,
such as what DNA sequence pattern typically occurs near a donor or acceptor splice site, the fact
that in-frame stop codons should not occur within coding exons, the expected length distributions
of 5’ and 3’ untranslated regions (UTRs) and of initial, internal and final introns, etc. The trained
model can then use these learned properties to identify novel genes that resemble the genes in the
training set.
On the other hand, supervised learning only makes sense when a labeled training set is available.
Consider, for example, interpreting a heterogeneous collection of epigenomic data sets, such as
those generated by the ENCODE and Epigenomics Roadmap consortia. A priori, we expect that
the patterns of chromatin accessibility, histone modifications and TF binding along the genome
should be able to provide us with a detailed picture of the biochemical and functional activity of the
genome. We may also expect that these activities could be accurately summarized using a relatively
small set of labels. If we are interested in discovering what types of labels best explain the data
13
rather than imposing a pre-determined set of labels on the data, then we must use unsupervised
rather than supervised learning. In this type of approach, the machine learning algorithm takes
as input only the unlabeled data and the desired number of different labels to assign [Day et al.,
2007, Ernst and Kellis, 2012, Hoffman et al., 2012]. The algorithm then automatically partitions the
genome into segments and assigns a label to each segment, with the goal of assigning the same label
to segments that have similar data. Because the method is unsupervised, a human must subsequently
assign semantics to each label, a step which is not required for a supervised gene finding algorithm.
The benefit of the unsupervised approach, however, is the ability to train when labeled examples are
unavailable and the ability to identify potentially novel types of genomic elements. The latter is
fundamentally impossible with a supervised method, which can only identify more examples of the
types of labels it learns from.
2.2.1 Semi-supervised learning
Intermediate between supervised and unsupervised learning is semi-supervised learning [Chapelle
et al., 2006]. In supervised learning, the algorithm receives as input a collection of data points,
each with an associated label, whereas in unsupervised learning, the algorithm receives the data
but no labels. The semi-supervised setting is a mixture of these two: the algorithm receives a
collection of data points, but only a subset of those points have associated labels. In practice, gene
finding systems are often trained using a semi-supervised approach, where the input is a collection
of annotated genes plus a complete (unlabeled) genome sequence. The learning procedure begins
by constructing an initial gene finding model based solely on the labeled subset of the training data.
Then the model is used to scan the genome, and tentative labels are assigned throughout the genome.
These tentative labels can then be used to improve the learned model, and the procedure iterates
until no new genes are found. The semi-supervised approach can work much better than a fully
supervised approach because the model is able to learn from a much larger set of genes: all genes in
the genome, rather than only the subset of genes that have been identified with high confidence.
14
2.2.2 Which type of method to use
When faced with a new machine learning task, the first question is often whether to use a supervised,
unsupervised or semi-supervised approach. In some cases, the answer to this question is obvious.
For example, if no labels are available, then you can only perform unsupervised learning. However,
when labels are available, it is not always the case that taking a supervised approach is a good
idea. This is because every supervised learning method rests upon the implicit assumption that the
distribution responsible for generating the training data set is the same as the distribution responsible
for generating the test data set. This assumption will be respected if, for example, you take a single
labeled data set and randomly subdivide it into a training set and a testing set. However, it is often the
case that you plan to train an algorithm on a training set that is generated differently from the testing
data to which the trained model will eventually be applied. A gene finder trained using a training set
of human genes will probably not work very well at finding genes in the mouse genome. Often,
the train/test divergence is less obvious. For example, a TSS data set generated by cap analysis of
gene expression (CAGE) data will not contain non-polyadenylated genes [Stamatoyannopoulos,
2010]. If such genes also exhibit differences around the TSS, then the resulting TSS predictor will
be biased. In general, supervised learning should be employed only when the training set and test
set are expected to exhibit similar statistical properties.
What about semi-supervised learning? When supervised learning is feasible, it is often the
case that additional, unlabeled data points are easy to obtain. How do you decide whether to use
a supervised or semi-supervised approach? In theory, semi-supervised learning requires making
certain assumptions about the data set [Chapelle et al., 2006]). In practice, however, assessing
these assumptions can be very difficult. Therefore, a good rule of thumb is to use semi-supervised
learning if two conditions hold: you don’t have very much labeled data and you have a very large
amount of unlabeled data.
15
2.3 Generative versus discriminative modeling
Applications of machine learning methods generally have one of two goals: prediction or interpreta-
tion. Consider the problem of predicting, on the basis of a ChIP-seq experiment, the locations at
which a given TF will bind genomic DNA. This task is analogous to the TSS prediction task (Fig-
ure 2.1), except that the labels are derived from ChIP-seq peaks. A researcher applying a machine
learning method to this problem may either want to understand what properties of a sequence are
most important for determining whether or not a TF will bind (interpretation) or simply predict
the locations of TF binding as accurately as possible (prediction). There are tradeoffs between
accomplishing these two goals—methods that optimize prediction accuracy often do so at the cost
of interpretability.
The distinction between generative and discriminative models plays a large role in the tradeoff
between interpretability and performance. Generative models build a full model of the distribution
of features in each of the two classes and then compares how those two distributions differ from
one another. In contrast, the discriminative approach focuses on accurately modeling just the
boundary between the two classes. From a probabilistic perspective, the discriminative approach
involves modeling just the conditional distribution of the label given the input feature data sets,
as opposed to the joint distribution of the labels and features. Schematically, if we imagine that
our task is to separate two groups of points in a two-dimensional space (Figure 2.3A), then the
generative approach builds a full model of the distribution of points in each of the two classes
and then compares how those two distributions differ from one another, while the discriminative
approach focuses only on separating the two classes.
A researcher adopting a generative approach to the TF binding problem begins by imagining
what procedure could be used to generate the observed data. A widely used, generative model of TF
binding employs a position-specific frequency matrix (PSFM, Figure 2.3B), in which a collection of
aligned binding sites of width w are summarized in a 4× w matrix M of frequencies, where the
entry at position Mi,j represents the empirical frequency of observing the ith DNA base at position
j. We can generate a random bound sequence according to this PSFM model by drawing w random
16
numbers, each in the range [0, 1〉. For the jth random number we select the corresponding DNA
base according to the frequencies in the jth column of M . Conversely, scoring a candidate binding
site using the model corresponds to computing the product of the corresponding frequencies from
the PSFM. This value is called the likelihood. Training a PSFM is simple—you simply compute the
empirical frequency of each nucleotide at each position.
A simple example of a discriminative algorithm is the support vector machine (SVM, Fig-
ure 2.3C) [Boser et al., 1992, Noble, 2006]. The goal of the SVM is to learn to output a “1”
whenever it is given a positive training example and a “-1” whenever it is given a negative training
example. In the TF binding prediction problem, the input sequence of length w is encoded as a
binary string of length 4w, where each bit corresponds to the presence or absence of a particular
nucleotide at a particular position.
This generative modeling approach offers several compelling benefits. First, the generative
description of the data implies that the model parameters have well-defined semantics relative to
the generative process. Accordingly, as shown in the example above, the model not only makes
predictions for where a given TF binds but also provides a story about why the TF binds there. If
we compare two different potential binding sites, we can see not only that the model prefers one site
over another but also that the preference comes from, e.g., the preference for an A rather than a T at
position 7 of the motif. Second, generative models are frequently stated in terms of probabilities,
and the probabilistic framework provides a principled way to handle problems like missing data.
For example, it is still possible for a PSFM to make a prediction for a binding site where one or
more of the bound residues is unknown. This is accomplished by probabilistically averaging over
the missing bases. The probabilistic framework is also helpful because the output has well-defined,
probabilistic semantics. This can be helpful when making downstream decisions about how much
to trust a given prediction.
In many cases, including the TF binding example, the training data set contains a mixture of
positive and negative examples. In a generative setting, these two groups of examples are modeled
separately, each with its own generative process. For example, for the PSFM model, the negative (or
background) model is often a single set B of nucleotide frequencies, representing the overall mean
17
frequency of each nucleotide in the negative training examples. To generate a length-w sequence
according to this model, we again generate w random numbers, but now each base is selected
according to the frequencies in B. To use the foreground PSFM model together with the background
model B, we compute a likelihood ratio, i.e., which is simply the ratio of the likelihoods computed
with respect to the PSFM and with respect to B.
The primary benefit of the discriminative approach is that it provably achieves better performance
with infinite training data [Ng and Jordan, 2002, Jordan, 1995]. In practice, analogous generative
and discriminative approaches often converge to the same solution, and generative approaches
can sometimes perform better with limited training data. However, when the amount of labeled
training data is reasonably large, then the discriminative approach will tend to find a better solution,
in the sense that it will predict the desired outcome more accurately when tested on previously
unseen data (assuming, as usual, that the data are drawn from the same underlying distribution as
the training data). To illustrate this phenomenon, we simulated data according to a simple PSFM
model and trained a PSFM and SVM, varying the number of training examples. To train a model
of width 19 nucleotides to discriminate between bound and non-bound sites with 90% accuracy
requires eight training examples for a PSFM model and only four examples for an SVM model
(Figure 2.3D). This improvement in performance is achieved because, by not bothering to accurately
characterize the “easy” parts of the 2D space, the discriminative model does a better job at solving
the discrimination task at hand. Thus, empirically, the discriminative approach will tend to give
more accurate predictions
The flipside of this accuracy, however, is that by solving a single problem well, the discriminative
approach fails to solve other problems at all. Specifically, because the internal parameters of a
generatively trained model have well-defined semantics, we can use the model to ask a variety of
related questions, e.g., not just “Does CTCF bind to this particular sequence?” but “Why does
CTCF bind to this sequence more tightly than to some other sequence?” The discriminative model,
in contrast, only allows us to answer the single question for which it was designed. Thus, choosing
between a generative and discriminative model involves a trade-off between predictive accuracy and
interpretability of the model.
18
While the distinction between generative and discriminative models plays a large role in deter-
mining the interpretability of a model, the model’s complexity—that is, the number of parameters it
has—can be just as imporant. Models of either type with a large number of parameters tend to be
difficult to interpret, but can generally achieve higher accuracy than models with few parameters,
given enough data. The complexity of a model can be limited either by choosing a simple model or
by using a feature selection strategy to restrict the complexity of the learned model.
2.4 Incorporating prior knowledge
In many applications, the success or failure of a machine learning method depends upon the extent
to which the method successfully encodes various types of prior knowledge about the problem at
hand. Indeed, much of the “art” of practical machine learning involves understanding the details
of a particular problem and then selecting an algorithmic approach that allows those details to be
accurately encoded. There is provably no optimal machine learning algorithm that works best for all
problems [Wolpert and Macready, 1997], so this selection of approach that matches the researcher’s
prior knowledge about the problem is crucial to the success of the analysis.
Often, the encoding of prior knowledge is implicit in the framing of the machine learning
problem. Consider, for example, the prediction of nucleosome positioning from primary DNA
sequence [Segal et al., 2006]. The “labels” for this prediction task can be derived, for example,
from an MNase sequencing assay. In this case, after appropriate processing, each base is assigned
an integer count of the number of nucleosome-sized fragments that cover the base. It might seem
natural, therefore, to frame the problem as a regression, where the input is, say, a sequence of length
201 bp and the output is the predicted coverage at the center of the sequence. However, in practice
we may be particularly interested in identifying nucleosome-free regions. Hence, rather than asking
the algorithm to solve the problem of predicting exact coverage at each base, we might instead opt
to predict whether each base occurs in a nucleosome-free region or not. Switching from regression
to classification makes the problem easier, but more importantly, this switch also encodes the prior
knowledge that regions of high MNase accessibility are of particular biological interest.
19
2.4.1 Implicit prior knowledge
In other cases, prior knowledge is encoded implicitly in the choice of the data that we provide
as input to the machine learning algorithm. For example, Yip et al. [Yip et al., 2012] trained a
collection of machine learning methods to distinguish among various types of genomic elements.
Using chromatin data—DNaseI accessibility and ChIP-seq profiles of histone modifications and
TF binding—one classifier distinguished between regulatory regions that are close to a gene versus
regulatory regions that are far away from any gene. In this context, design of the input space,
i.e., the features that are provided as input to the classifier, is of critical importance. In the Yip et
al. study, features were computed by averaging over a 100 bp window. The choice of a 100 bp
window likely reflects the prior knowledge that, at least for histone modification data, the data are
arguably only meaningful at approximately the scale of a single nucleosome (147 bp) or larger.
Moreover, replacing a single, averaged feature with 100 separate features may also be problematic
(see Section 2.6). In contrast, for DNase accessibility data, it is plausible that averaging over 100
bp may remove some useful signal. Alternatively, prior knowledge may be implicitly encoded in
the learning algorithm itself, where some types of solutions are preferred over others [Urbanowicz
et al., 2012a]. Therefore, in general, the choice of input data sets, their representations, and any
preprocessing must be guided by prior knowledge about data and application.
2.4.2 Probabilistic priors
In a probabilistic framework, some forms of prior knowledge can be represented explicitly by
specifying a prior distribution over the data. A common prior distribution is the uniform, or
“uninformative,” prior which, despite the name, can be quite useful in some contexts. Consider,
for example, a scenario in which we have gathered a collection of ten validated binding sites for a
particular transcription factor (Figure 2.4), and we observe that the sequences cluster around a clear
consensus motif. If we represent these sequences using a pure PSFM, then because our data set is
quite small, a significant number of the entries in the PSFM will be zero. Consequently, the model
will assign any sequence that contains one of these zero entries an overall probability of zero, even
20
if the sequence otherwise exactly matches the motif. This is counter-intuitive. The solution to this
problem is to encode the prior knowledge that every possible DNA sequence has the potential to be
bound by a given transcription factor. The result is that even sequences containing nucleotides that
we have never observed in a given position can still be assigned a non-zero probability by the model.
In many probability models, much more sophisticated priors have been developed to capture
more complex prior knowledge. A particularly successful example is the use of Dirichlet mixture
priors in protein modeling [Brown et al., 1993]. Here, the idea is that if we are examining an
alignment of protein sequences, and if we see many aligned leucines in a particular column, then
because we know that leucine and valine are biochemically similar, we may want to assign a high
probability to sequences that contain a valine in that same column, even if we have never seen
a valine there in our training data. The name “Dirichlet mixture” refers to the fact that the prior
distribution is represented as a Dirichlet distribution, and that the distribution is a mixture in which
each component corresponds to a group of biochemically similar amino acids. Such priors lead to
significantly improved performance in modeling evolutionarily related families of proteins [Brown
et al., 1993] and in discovering protein motifs [Bailey and Elkan, 1995].
2.4.3 Prior information in non-probabilistic models
On the other hand, incorporation of prior knowledge into non-probabilistic methods can be more
challenging. For example, discriminative classifiers like artificial neural networks (ANNs) or random
forests do not provide any explicit mechanism for representing prior knowledge. The topology of a
multi-layer ANN can represent information about dependencies between input features in the input
space, but more general priors cannot be represented.
One class of discriminative methods does provide a more general mechanism for representing
prior knowledge, when that knowledge can be encoded into a generalized notion of similarity.
Kernel methods are algorithms that substitute in place of a simple similarity function (specifically,
the cosine of the angle between two input vectors) a general class of mathematical functions called
kernels [Scho¨lkopf and Smola, 2002]. The flagship kernel method is the SVM classifier, which has
been widely used in many fields including biological applications ranging from DNA and protein
21
sequence classification to mass spectrometry analysis [Noble, 2006]. Other methods that can employ
kernels include support vector regression as well as classical algorithms like k-means clustering,
principal components analysis, and hierarchical clustering. Prior knowledge can be provided to
a kernel method by selecting or designing an appropriate kernel function. For example, a wide
variety of kernels can be defined between pairs of DNA sequences, where the similarity can be
based on shared k-mers, irrespective of their positions within the sequences [Leslie et al., 2002],
on nucleotides occuring at particular positions [Ra¨tsch and Sonnenburg, 2004], or on a mixture
of the two [Zien et al., 2000]. A DNA sequence can even encode a simple model of molecular
evolution, using algorithms similar to the Smith-Waterman alignment algorithm [Saigo et al., 2006].
Furthermore, the Fisher kernel provides a general framework for deriving a kernel function from
any probability model [Jaakkola and Haussler, 1998]. In this way, formal probabilistic priors can
be used in conjunction with any kernel method. Kernel methods have a rich literature, which is
reviewed in more detail in [Shawe-Taylor and Cristianini, 2004].
2.5 Handling heterogeneous data
Another common challenge in learning from real biological data is that the data themselves are
heterogeneous. For example, consider the problem of learning to assign Gene Ontology terms to
genes. For a given term, such as “cytoskeleton dependent intracellular transport,” a wide variety of
types of data might be relevant, including the amino acid sequence of the gene’s protein product,
that protein’s inferred evolutionary relationships to other proteins across a variety of species, the
microarray or RNAseq expression profile of the gene across a variety of phenotypic or environmental
conditions, the number and identity of neighboring proteins identified using yeast two-hybrid or
tandem affinity purification tagging experiments, GFP-tagged microscopy images, etc (Figure 2.5).
Such data sets are difficult to analyze jointly because of their heterogeneity: an expression profile
is a fixed-length vector of real values; protein-protein interaction information is a binary network,
and protein sequences are variable length strings drawn from a discrete alphabet. Many statistical
and machine learning methods for classification assume that all of the data can be represented as
fixed-length vectors of real numbers. Such methods cannot directly be applied to heterogeneous
22
data.
The most straightforward way to solve this problem is to transform each type of data into vector
format prior to processing (Figure 2.5A). This was the approach taken, for example, by Pen˜a-Castillo
et al. in a critical assessment of methods for predicting gene function in mice [Pen˜a-Castillo et al.,
2008]. Participating research groups were provided with separate matrices, each representing a single
type of data: gene expression, sequence patterns, protein interactions, phenotypes, conservation
profiles, and disease associations. All matrices shared a common set of rows, one per gene, but
the number and meanings of the columns differed from one matrix to the next. For example, gene
expression data were represented directly, whereas protein sequence data were represented indirectly
via annotations from repositories such as Pfam [Sonnhammer et al., 1997] and InterPro [Apweiler
et al., 2001], and protein-protein interactions were represented as square matrices in which entries
were shortest-path lengths between genes.
Alternatively, each type of data can be encoded using a kernel function (Figure 2.5B), with one
kernel for each data type. Mathematically, using kernels is formally equivalent to transforming each
data type into a fixed-length vector; however, in the kernel approach the vectors themselves are
not always represented explicitly. Instead, the similarity between two data elements—amino acid
sequences, nodes in a protein-protein interaction network, etc.—is encoded in the kernel function.
The kernel framework allows more complex kernels to be defined from combinations of simple
kernels. For example, a simple summation of all the kernels for a given pair of genes is itself a
kernel function. Furthermore, the kernels themselves can encode prior knowledge by, e.g., allowing
for statistical dependencies within a given data type but not between data types [Pavlidis et al.,
2002]. Kernels also provide a general framework for automatically learning the relative importance
of each type of data relative to a given classification task [Lanckriet et al., 2004].
Finally, probability models provide a very different method for handling heterogeneous data.
Rather than forcing all the data into a vector representation or requiring that all data be represented
using pairwise similarities, a probability model explicitly represents diverse data types in the model
itself (Figure 2.5C). An early example of this type of approach assigned gene functional labels to
yeast genes on the basis of gene expression profiles, physical associations from affinity purification,
23
two-hybrid and direct binding measurements, as well as genetic associations from synthetic lethality
experiments [Troyanskaya et al., 2003]. The authors used a Bayesian network, which is a formal
graphical language for representing the joint probability distribution over a set of random variables
[Pearl, 1998]. Querying the network with respect to the variable representing a particular Gene
Ontology label yields the probability that a given gene is assigned that label. In principle, the
probabilistic framework allows a Bayesian network to represent any arbitrary data type and perform
joint inference over heterogeneous data types using a single model.
In practice, such inference can be challenging because a joint probability model over hetero-
geneous data may contain a very large number of trainable parameters. Therefore, an alternative
method for handling heterogeneous data in a probability model is to “chain” the data types together.
This approach makes use of the general mechanism for handling prior knowledge by treating one
type of data as “prior” to another. For example, as we have discussed, the problem of predicting
where along the genome sequence a particular transcription factor will bind can be framed as a
classification problem and solved using a PSFM. However, in vivo, the binding of a TF depends
not only on the native affinity of the TF for a given DNA sequence, but also upon competitive
binding by other molecules. Accordingly, measurements of chromatin accessibility, as provided by
assays such as DNase-seq [Song and Crawford, 2010], can provide valuable information about the
overall accessibility of a given genomic locus to TF binding. A joint probability model can take this
accessibility into account [Wasson and Hartemink, 2009, Pique-Regi et al., 2011], but training such
a model can be challenging. The alternative approach uses the DNaseI data to create a probabilistic
prior and then applies this prior during the scanning of the PSFM [Cuellar-Partida et al., 2011].
2.6 Feature selection
In any application of machine learning methods, the researcher must decide which data to provide
as input to the algorithm. As noted above, this choice provides a method for incorporating prior
knowledge into the procedure, because the researcher can decide which features of the data are
likely to be relevant and which are probably irrelevant. On the other hand, choosing relevant features
can itself be a scientifically interesting question. Consider, for example, the problem of training a
24
multiclass classifier to distinguish, on the basis of gene expression measurements, among different
types of cancers [Ramaswamy et al., 2001]. Such a classifier could be valuable in two ways. First,
the classifier itself could help to establish accurate diagnoses in cases of atypical presentation
or histopathology. Second, the model produced during the learning phase could perform feature
selection, identifying subsets of genes whose expression patterns contribute specifically to different
types of cancers. In general, feature selection can be carried out within any supervised learning
algorithm, in which the algorithm is given a large set of features (or input variables) and then
automatically decides to ignore some or all of the features, focusing on the subset of features most
relevant to the task at hand.
In practice, it is important to distinguish among three distinct motivations for performing feature
selection with respect to a given task. First, in some cases, we want to identify a very small set of
features that yield the best possible classifier. For example, we may want to produce an inexpensive
way to identify a disease phenotype on the basis of the measured expression levels of a handful of
genes. Such a classifier, if it is accurate enough, might serve as the basis for an inexpensive clinical
assay. Second, we may want to use the classifier to understand the underlying biology [Glaab
et al., 2012, Tibshirani, 1996, Urbanowicz et al., 2012b]. In this case, we want the feature selection
procedure to identify all and only the genes whose expression is actually relevant to the task at hand,
in the hopes that the corresponding functional annotations or biological pathways might provide
insight into the etiology of disease. Third, we often simply want to train the most accurate possible
classifier [Tikhonov, 1943]. In this case, we hope that the feature selection enables the classifier to
identify and eliminate noisy or redundant features. Researchers are often disappointed to find that
feature selection cannot optimally perform more than one of these three tasks simultaneously.
Feature selection is especially important in the third case because analysis of high-dimensional
data sets, including genomic, epigenomic, proteomic or metabolomic data, suffers from the curse of
dimensionality [Keogh and Mueen, 2011]. The “curse” refers to the general observation that many
types of analysis become more difficult as the number of input dimensions (i.e., data measurements)
grows very large. For example, as the number of data features input to a machine learning classifier
grows, it is increasingly likely that, by chance, one feature perfectly separates the training examples
25
Scenario Example performance measure
All predictions are equally important. Accuracy (fraction of all predictions that are correct)
Some number of follow-up tests are planned Precision (fraction of positives that are correct) among
a fixed number of predicted enhancers
Positive predictions will be published Sensitivity (number of enhancers that are identified)
among predictions with a pre-determined precision
(e.g. 95%).
Table 2.4: Measuring performance of a classifier. Different applications for a machine learning
classifier necessitate different performance measures.
into positive and negative classes. This phenomenon leads to good performance on the training data
but poor generalization to data not used in training due to the model being overfit to the training data.
Feature selection methods and dimensionality reduction techniques, such as principal components
analysis or multidimensional scaling, aim to solve this problem by projecting the data from higher
to lower dimensions.
2.7 Imbalanced class sizes
A common stumbling block in many applications of machine learning to genomics is the large
imbalance (or skew) in the relative sizes of the groups being classified. For example, suppose you
are trying to use a discriminative machine learning method to predict the locations of enhancers
in the genome. Starting with a set of 641 known enhancers, you break the genome up into 1000
bp segments and assign each segment a label (“enhancer” or “not enhancer”) based on whether or
not it overlaps a known enhancer. This procedure produces 1,711 positive examples and roughly
3,000,000 negative examples—2,000 times as many negative examples as positive. Unfortunately,
most software cannot handle 3,000,000 examples.
The most straightforward solution to this problem is to select a random, smaller subset of the
data. However, in the case of enhancer prediction, selecting a random 50,000 examples results in
49,971 negative examples and just 28 positive examples. This many positive examples is far too
small to train an accurate classifier. To demonstrate this problem, we simulated 93 noisy genomics
26
assays using a Gaussian model for enhancers and background positions based on data produced by
the ENCODE Consortium [ENCODE Project Consortium, 2012]. We trained a logistic regression
classifier to distinguish between enhancers and non-enhancers on the basis of these data. The overall
accuracy of the predictions, i.e., the percentage of predictions that were correct, was 99.9%. This
sounds reasonably good until you consider that a null classifier that simply predicts everything to be
non-enhancer achieves nearly the same accuracy.
In this context, it is more appropriate to separately evaluate sensitivity, defined as the fraction
of enhancers detected, and the precision, defined as the percentage of predicted enhancers that are
truly enhancers. Our balanced classifier has a high precision but a very low sensitivity, detecting
only 0.5% of enhancers. The behavior of the classifier can be improved by instead using all the
enhancers to train and then picking a random set of 49,000 non-enhancer positions as negative
training examples. However, balancing the classes in this way results in the classifier learning to
reproduce this artificially balanced ratio. The resulting classifier achieves much higher sensitivity
(81%) but very poor precision (40%). This classifier is thus not useful for finding enhancers that can
be validated experimentally.
It is possible to trade off sensitivity and precision while retaining the training power of a balanced
training set by placing weights on the training examples. In the enhancer prediction case, we use
the balanced training set, but during training we weight each negative example 36 times as much as
a positive example. Doing so results in an excellent sensitivity of 53% with a precision of 95%.
In general, which performance measure is most appropriate depends on the intended application
of the classifier (Table 2.4). For problems such as identifying which tissue a given cell comes
from, where it may be equally important to identify rare and abundant tissues, the overall number
of correct predictions may be the most informative measure of performance. In others, such
as enhancer detection, predictions in one class may be more important than the other. A wide
variety of performance measures are used in practice, including the F1 measure, the receiver
operating characteristic (ROC) or precision-recall (PR) curve [Manning and Schu¨tze, 1999, Davis
and Goadrich, 2006], and others [Cohen, 1968]. Machine learning classifiers perform best when
they optimize for a realistic performance measure.
27
2.8 Handling missing data
Machine learning analysis can often be complicated by missing data values. Missing values can
come from a variety of sources, such as defective cells in a gene expression microarray, unmappable
genome positions in a functional genomics assay, or measurements that are unreliable because
they saturate the detection limits of an instrument. Missing data values can be divided into two
types: (1) values that are missing at random or for reasons unrelated to the task at hand, such as
defective microarray cells, and (2) values whose absences provides information about the task at
hand, such as saturated detectors. The presence or absence of values of the second type are usually
best incorporated directly into the model.
The simplest way to deal with data that is missing at random is to impute the missing values
[Luengo et al., 2012]. This can be done either with a very simple strategy, such as replacing all
missing values with zero, or a more sophisticated strategy. For example, Troyanskaya et al. used the
correlations between data values to impute missing microarray values [Troyanskaya et al., 2001].
For each target gene expression profile, the authors found the 10 most similar other expression
profiles and replaced each missing value in the target profile with an average of the values in the
similar profiles. Imputing data points this way can be used as a preprocessing step for any other
analysis, but downstream analyses are blind to this information and cannot make use of either the
fact that a data point is missing or the added uncertainty resulting from missing data values.
Another method for dealing with missing data is to include in the model informaton about
the missingness of each data point. For example, Kircher et al. aim to predict the deleteriousness
of mutations based on functional genomics data [Kircher et al., 2014]. The functional genomics
data provided a feature vector associated with each mutation, but some of these features were
missing. Therefore, for each feature, the authors added a Boolean feature indicating whether the
corresponding feature value was present. The missing values themselves were then replaced with
zeroes. A sufficiently sophisticated model will be able to learn the pattern that determines the
relationship between the feature and presence or absence indicator. An advantage of this approach
to handling missing data is that it is applicable whether or not the absence of a data point is
28
significant—if it is not, the model will learn to ignore the absence indicator.
Finally, probability models can explicitly model missing data by considering all the potential
missing values. For example, this approach is used by Hoffman et al. to analyze functional genomics
data, which contain missing values due to mappability issues from short read sequencing data
[Hoffman et al., 2012]. The probability model annotates the genome by assigning a label to each
position and modeling the probability of observing a certain value given the label. Missing data
points are handled by summing over all possibilities for that random variable in the model. This
approach, called marginalization, represents the case where a particular variable is unobserved.
However, marginalization is only appropriate when data points are missing for reasons unrelated
to the task at hand. When the presence or absence of a data point is likely to be correlated with
the values themselves, then incorporating presence or absence explicitly into the model is more
appropriate.
2.9 Modeling dependence among examples
So far we have focused on machine learning tasks that involve sets of independent instances of a
common pattern. However, in some domains, individual entities, such as genes, are not independent,
and the relationships among them are important. By inferring such relationships, it is possible to
integrate many examples into a meaningful network. Such a network might represent physical
interactions among proteins, regulatory interactions between genes, or symbiosis between microbes.
Networks are useful both for understanding the biological relationships between entities and as
input into a following analysis that makes use of these relationships.
The most straightforward way to infer the relationships among examples is to consider each pair
independently. In this case, the problem of network learning reduces to a normal machine learning
problem, defined on pairs of individuals rather than individual examples. Qiu et al. used an SVM
classifier to predict whether a given pair of proteins physically interact based on data such as their
sequences and cellular localization [Qiu and Noble, 2008].
A downside of any approach that considers each relationship independently is that such methods
cannot take into account the confounding effects of indirect relationships (Figure 2.6). For example,
29
in the case of gene regulation, an independent model cannot infer whether a pair of genes directly
regulates each other or they are both regulated by a third gene. Such spurious inferred relationships,
called transitive relationships, can be removed by methods which infer the graph as a whole. For
example, Friedman et al. inferred a Bayesian network on gene expression data that models which
genes regulate each other [Friedman et al., 2000]. Such a network includes only direct effects,
and models indirect correlations through multiple-hop paths in the network. Methods that infer a
network as a whole are therefore more biologically interpretable, because they remove these indirect
correlations. A large number of such methods have been described, as reviewed in [Bacardit and
Llora`, 2013, Koski and Noble, 2012].
2.10 Future of machine learning in genomics
Machine learning is most effective in situations where one would like to make sense of large, complex
data sets. Accordingly, machine learning methods are likely to become ever more important to
genomics as more large sets become available, through projects such as the 1000 Genomes Project
or the NIH’s 4D Nucleome initiative. On the other hand, even in the presence of massive amounts
of data, machine learning techniques cannot generally be applied in a completely arbitrary fashion.
In practice, effective application of machine learning methods requires a certain amount of “art.”
In particular, theoretical and practical knowledge of both machine learning methodology and the
particular application area are often necessary to achieve good performance. As new technologies
for generating large genomic and proteomic data sets emerge, pushing beyond DNA sequencing to
mass spectrometry, flow cytometry and high-resolution imaging methods, demand will increase not
only for new machine learning methods but also for experts capable of applying and adapting them
to big data sets. In this sense, both machine learning itself and machine learning researchers are
likely to become increasingly important to genetics and genomics.
30
A
+
+
+
+
+
+++ +
-
-
-
-
-
--— -
+
+
+
+
+
+++ +
-
-
-
-
-
--— -
Generative Discriminative B
AAGTGT 
TAATGT 
AATTGT 
AATTGA 
ATCTGT 
AATTGT 
TGTTGT 
AAATGA A 0.75 0.75 0.25 0.00 0.25  C 0.00 0.00 0.13 0.00 0.00 
G 0.00 0.13 0.13 1.00 0.00 
T 0.25 0.13 0.50 0.00 0.75 
6
8
Pr(AAGTGT) = 0.75 * 0.75 * 0.13 * 1.00 * 1.00 * 0.75  
           = 0.05 
C
A C G T A C G T A C G TA C G T
?
Input:	
  A	
  A	
  G	
  T	
  G	
  T	
  
Output:	
  “bound”	
  or	
  “not	
  bound”	
   D
l
l
l
l
l
l
l
l
l
l
l
l
l l l l
0.8
0.9
1.0
10 100
Number of training examples
Ac
cu
ra
cy Method
l
l
PSFM
SVM
Figure 2.3: Two models of TF binding. (A) A position-specific frequency matrix model, in which
the entry in row i and column j represents the frequency of the ith base occurring at position j in
the training set. (B) A linear support vector machine model of TF binding. Labeled positive and
negative training examples are provided as input, and a learning procedure adjusts the weights on
the edges to predict the given label. (C) The figure plots the mean accuracy (± 95% confidence
intervals), on a set of 500 simulated test sets, of predicting TF binding as a function of the number
of training examples. The two series correspond to a PSFM model or an SVM. (D) Schematic
of generative versus discriminative modeling. The generative model characterizes both classes
completely, whereas the discriminative model focuses on the boundary between the classes.
31
AAGTGT 
TAATGT 
AATTGT 
AATTGA 
ATCTGT 
AATTGT 
TGTTGT 
AAATGA 
A 6 6 2 0 2 
C 0 0 1 0 0 
G 0 1 1 8 0 
T 2 1 4 0 6 
A 6.1 6.1 2.1 0.1 2.1 
C 0.1 0.1 1.1 0.1 0.1 
G 0.1 1.1 1.1 8.1 0.1 
T 2.1 1.1 4.1 0.1 6.1 
A 0.73 0.73 0.25 0.01 0.25 
C 0.01 0.01 0.13 0.01 0.01 
G 0.01 0.13 0.13 0.96 0.01 
T 0.25 0.13 0.49 0.01 0.73 
Input	
   Counts	
   Counts	
  +	
  pseudocounts	
   Frequencies	
  
Figure 2.4: Incorporating a prior into a PSFM. A simple, principled method for putting a proba-
bilistic prior on a PSFM involves augmenting the observed nucleotide counts with “pseudocounts”
and then computing frequencies with respect to the sum. The magnitude of the pseudocount
corresponds to the weight assigned to the prior.
32
Gene expression Genetic interactions Amino acid sequences
Gene
Condition MCDLLVISHSSL 
MLLLDRTSCSRI 
MKVLIDRVYDRD 
Ge
ne
Condition
Ge
ne
Gene
0 1 0 1 
1 0 0 1 
0 0 0 0 
0 1 1 0
Ge
ne
K-mers
Cytoskeleton !
dependent !
intracellular !
transport
Feature!
extraction Kernel!transformation Probability!model
Gene
Ge
ne
Gene
Ge
ne
Gene
Ge
ne
+
+
A B C
Gene expression
Sequences
Genetic interactions
Figure 2.5: Three ways to accommodate heterogeneous data in machine learning. The task of
predicting gene function labels requires methods that take as input gene expression data, protein
sequences, protein-protein interaction networks, etc. These diverse data types can be encoded into
fixed-length features, represented using pairwise similarities (kernels), or directly accommodated
by a probability model.
33
True regulatory !
network
Regulation
Correlation
Correlation!
network
Inferred!
network
Figure 2.6: Inferring network structure. Methods that infer each relationship in a network
separately, such as computing the correlation between each pair, can be confounded by indirect
relationships. Methods that infer the network as a whole can identify only direct relationships.
Inferring the direction of causality inherent in networks is generally more challenging than inferring
the network structure [Pearl, 2000], so many network inference methods, such as Gaussian graphical
model learning, infer only the network.
34
Chapter 3
ENTROPIC GRAPH-BASED POSTERIOR REGULARIZATION
Abstract
Graph smoothness objectives have achieved great success in semi-supervised learning but have
not yet been applied extensively to unsupervised generative models. We define a new class of
entropic graph-based posterior regularizers that augment a probabilistic model by encouraging
pairs of nearby variables in a regularization graph to have similar posterior distributions. We
present a three-way alternating optimization algorithm with closed-form updates for performing
inference on this joint model and learning its parameters. This method admits updates linear in
the degree of the regularization graph, exhibits monotone convergence, and is easily parallelizable.
We are motivated by applications in computational biology in which temporal models such as
hidden Markov models are used to learn a human-interpretable representation of genomic data.
On a synthetic problem, we show that our method outperforms existing methods for graph-based
regularization and a comparable strategy for incorporating long-range interactions using existing
methods for approximate inference. Using genome-scale functional genomics data, we integrate
genome 3D interaction data into existing models for genome annotation and demonstrate significant
improvements in predicting genomic activity.
3.1 Introduction
Graph-based methods have recently been successful in solving many types of semi-supervised
learning problems [Chapelle et al., 2006, Das and Smith, 2011, Joachims, 1999, Subramanya et al.,
2010, Subramanya and Bilmes, 2009, 2011, Zhu et al., 2004, Zhu and Ghahramani, 2002]. These
methods assume that data instances lie in a low-dimensional manifold that may be represented as
a graph. They optimize a graph smoothness criterion, which states that data instances nearby in
35
the graph should be more likely to receive the same label. In a semi-supervised learning setting,
optimizing this criterion has the effect of spreading labels from labeled to unlabeled instances.
Despite the success of graph-based methods for semi-supervised learning, there has not been as
much study of the use of graph smoothness objectives in an unsupervised setting. In unsupervised
problems, we do not have labels but instead have a generative model that is assumed to explain
the observed data given the latent labels. While some types of relationships between instances
(for example, the relationship between neighboring words in a sentence or neighboring bases in
a genome) can easily be incorporated into the generative model, it is often inappropriate to encode
a graph smoothness assumption into the model this way, for two reasons. First, in some cases, it is
not clear what probabilistic process generated the labels with respect to the graph. Some objectives
and distance measures that are successful for semi-supervised learning do not have probabilistic
analogues. Second, large models must obey factorization properties (e.g., a tree or chain as in
hidden Markov models) to facilitate the use of efficient dynamic programming algorithms such as
belief propagation. Graphs representing similarity between variables do not in general satisfy these
structure requirements because they tend to be densely clustered, leading to very high-order factors.
In this paper, therefore, we propose a new regularization approach for expressing a graph
smoothness objective over a probabilistic model. We employ the posterior regularization (PR)
framework of Ganchev et al. [2010], in which a probabilistic model is regularized through a term
defined on an auxiliary posterior distribution variable. We define a powerful posterior regularizer
which encourages pairs of variables to have similar posterior distributions by adding a penalty
based on their Kullback-Leibler (KL) divergence. The pairs of penalized variables are encoded in
a regularization graph which may be entirely different from the graphical model on which inference
is performed. This regularizer graph need not have low treewidth and admits efficient optimization
even when fully connected. We call our strategy of adding KL regularization penalties entropic
graph-based posterior regularization (EGPR).
We show that inference and learning using this regularizer can be performed efficiently using
a three-way alternating optimization algorithm with closed-form updates. This algorithm alternates
between (1) smoothing marginal posteriors according to a regularization similarity graph, (2) per-
36
forming probabilistic inference in a graphical model with the same dependence structure as the
unregularized model, and (3) updating model parameters. The updates are linear in the degree
of the regularization graph and are easily parallelizable [Bilmes and Subramanya, 2012], in our
experiments scaling to tens of millions of variables. We show that this procedure corresponds to
a generalization of the EM algorithm.
We apply this approach to improve existing methods for annotating the human genome [Day
et al., 2007, Hoffman et al., 2012, Ernst and Kellis, 2010]. Methods for genome annotation distill
genomic data into a human-interpretable form by simultaneously partitioning the genome into
non-overlapping segments and assigning labels to each segment. This type of analysis has recently
had great success in interpreting the function of the human genome and formed an integral part
of the analysis of the NIH-sponsored ENCODE project ([ENCODE Project Consortium, 2012,
Hoffman et al., 2013], http://www.nature.com/encode).
However, exiting annotation methods use temporal models such as hidden Markov models and
therefore cannot efficiently incorporate data on the genome’s 3D structure. This 3D structure has
been shown to play a key role in gene regulation and other genomic processes. In our experiments
on synthetic data, a model using EGPR outperforms comparable models using either other regu-
larization strategies (e.g., squared error) or loopy belief propagation. On ENCODE data, a model
using EGPR predicts genome activity much more accurately than the currently-used chain models
as well as other forms of regularizer. Thus EGPR provides a method for jointly modeling genome
activity and 3D structure.
3.2 Proposed Method
In an unsupervised learning problem, we are given a set of vertices V that index a set of n = |V |
random variables XV = {X1, . . . , Xn} and a conditional dependence graph G = (V,E). The
graphical model describes a probability distribution parameterized by θ that can be factorized as
pθ(xV ) =
1
Z
∏
C∈C φ
(C)
θ (xC) where each C ⊆ V is a fully connected clique in G. We denote
random variables with capital letters (e.g., XH) and instantiations of variables with lower-case (e.g.,
xH ∈ domain(XH)). We also use capitals to denote sets and lowercase to denote set elements (e.g.,
37
Xh for h ∈ H).
Training graphical models involves a set of observed data x¯O, where a subset of variablesO ⊆ V
is observed and the remainder H = V \ O are hidden. In general we may have many samples of
observed data, but for notational simplicity, in this work we assume the graphical model expresses
the independence between samples, so it is sufficient to assume just one sample.
In this work we require that every hidden variable have the same domain X , although our results
can easily be extended to a case where we have several sets of variables, each with a common
domain. When the probability distribution is governed by a set of parameters θ, penalized maximum
likelihood training corresponds to the optimization
maximizeθ J(θ) , L(θ) +R(θ) (3.1)
where L(θ) , log pθ(x¯O) = log
∑
xH
pθ(xH , x¯O), (3.2)
and where R(θ) is a regularizer that expresses prior knowledge about the parameters. Many reg-
ularizers are used in practice, such as the `2 or `1 norms, which encourage parameters to be small
or sparse, respectively.
Instead of placing a regularizer on the parameters themselves, it is often more natural to place a
regularizer on the posterior distribution, a technique called posterior regularization [Ganchev et al.,
2010]. This is done by introducing an auxiliary joint distribution q(XH), placing a regularizer on
q(XH), and encouraging q to be similar to pθ via a KL divergence penalty. The regularizer is
RPR(θ) , max
q
R′PR(θ, q) (3.3)
R′PR(θ, q) , −D(q(XH)‖pθ(XH |x¯O)) + PR(q), (3.4)
where D(·‖·) is the KL divergence D(p(X)‖q(X)) = ∑x p(x) log(p(x)/q(x)) and PR(q) is a
penalty term that expresses some prior knowledge about the posterior distribution. For notational
convenience, we also define J ′(θ, q) , L(θ) + R′(θ, q). Ganchev et al. [2010] showed how to
optimize this combined objective efficiently when PR(q) is a sum of terms over individual cliques
38
in the model. Such regularizers can be used for constraining the posterior of individual variables in
expectation, among other applications. However, graph smoothness objectives cannot be expressed
this way, because they involve arbitrary pairs of variables.
When we have a graph smoothness assumption, we are given a weighted, undirected regular-
ization graph over the hidden variables GR = (H,ER), where ER ⊆ H ×H is a set of edges with
non-negative similarity weights w : ER → R+, such that a large w(u, v) indicates that we have
strong belief that Xu and Xv should be similar. The regularization graph GR is entirely separate
from the conditional dependence graph G and, in particular, need not obey any decomposition or
factorization properties to admit efficient inference.
He et al. [2013] introduced a regularizer of the following form. Let λG be a hyperparameter
controlling the strength of regularization. The regularizer is
PRGPR(q) , −λG
∑
(u,v)∈ER
w(u, v)‖q(Xu)− q(Xv)‖22.
He et al. showed how to optimize this regularizer using an exponentiated gradient descent method.
Although this regularizer shows good results for some problems, the use of squared error—or
indeed any p-norm—to represent dissimilarity between probability distributions can be highly
suboptimal, as we demonstrate empirically in Sections 3.6 and 4.2.9. Squared error is based on
a Gaussian error model, which is not appropriate for probability values, and it under-penalizes
differences between small probability values. p-norms are defined over all real numbers, while
posteriors must lie within the range [0, 1] (and live in a simplex).
A more justified way to measure divergence between probability distributions is to employ the
KL divergence. The KL divergence measures the difference of exponents in the probability and
so evaluates differences between small and large probabilities more uniformly. Also, Pinsker’s
inequality [Csisza´r and Tusna´dy, 1984] combined with the relationship of `-norms implies that
D(p‖q) ≥ 1
2
‖p − q‖21 ≥ 12‖p − q‖2` , for all ` ≥ 1, where ‖ · ‖` is the `-norm. Hence, minimizing
KL divergence minimizes an upper bound on all `-norms.
As a concrete example, consider two pairs of probability distributions over two events: p1 =
39
[0.55, 0.45] vs. q1 = [0.45, 0.55], and the second pair p2 = [0.1, 0.9] vs. q2 = [10−10, 1 − 10−10].
The first pair of distributions (p1, q1) are fairly similar, with both events being roughly equally likely.
The second pair (p2, q2) is quite dissimilar, with the first event being reasonably likely in the first
case p2 and astronomically unlikely in the second q2. Squared error actually regards the first pair as
more dissimilar than the second pair, while KL divergence identifies the second pair as much more
dissimilar. Despite the advantages of KL divergence, although all posterior regularization objectives
include a KL term binding q to be similar to pθ, to our knowledge, no existing methods define the
posterior regularizer itself using KL. Results in Sections 3.6 and 4.2.9 will show, moreover, that
KL significantly improves over squared error across the board.
In this work, we propose such a posterior regularizer, which we term entropic graph-based
posterior regularization (EGPR). The posterior regularizer is
PREGPR(q) ,
− λG
∑
(u,v)∈ER
w(u, v)D(q(Xu)‖q(Xv)), (3.5)
and J ′EGPR(θ, q) and R′EGPR(θ, q) are defined according to Equations (4.3) and (4.5) respectively
using the corresponding regularizers. That is,
maximizeθ,q J ′EGPR(θ, q) , L(θ) +R′EGPR(θ, q),
R′EGPR(θ, q) , −D(q(XH)‖pθ(XH |x¯O)) + PREGPR(q).
The KL divergence in Equation 4.6 is symmetrized because GR is undirected—that is w(u, v) =
w(v, u)—so D(qu‖qv) and D(qv‖qu) appear with the same weight in the regularizer.
In the next section, we describe a novel alternating optimization algorithm for solving Equa-
tion (4.2) with an EGPR regularizer. Unlike other recent prominent examples of alternating
optimization in machine learning [Wang et al., 2008], each update of this method has a closed-form
solution. EGPR can be employed either as a regularizer for training the parameters or for inference
directly. In the training case, an EM-like algorithm described in Section 3.3.3 is used to compute
40
and output θ, which can then be used for inference either with or without EGPR. In the inference
case, q is computed and output as the posterior marginals, as described in Section 3.4.
3.3 EGPR for Training
We first describe how to compute argmaxq J ′EGPR(θ, q), then describe how this algorithm can be
used in combination with an EM-like algorithm for learning θ.
3.3.1 Optimizing q
The EGPR regularizerR′EGPR(θ, q) is convex in q; therefore, we could compute q using any convex
optimization algorithm. However, general-purpose convex optimization algorithms do not scale
to problems with millions or billions of variables such as those present in genomics. Therefore,
we instead propose a novel alternating optimization strategy for performing this optimization more
efficiently.
To enable closed form updates for this objective, we reformulate J ′EGPR(θ, q) by introducing a
new variable rM(XH). Like q, rM is a distribution over XH , but we require that rM be factorizable
as a product of marginals—that is rM(xH) =
∏
h r
M
h (xh). (In this manuscript, we use the notation
pM(XH) to indicate that p is a product of marginals.) We define the graph regularizer over rM and
add an additional term λR1D(q(XH)‖rM(XH)), which encourages q and rM to be similar. As we
show below, restricting rM in this way means that the reformulated objective is a lower bound on
the original rather than being equivalent. We maximize this lower bound as an approximation to
maximizing the original. The reformulated regularizer is
PR′EGPR-R1(q, rM) , −λR1D(q(XH)‖rM(XH))
+ fR1(r
M)
(3.6)
41
and
fR1(r
M) ,
− λG
∑
(u,v)∈FEGPR
w(u, v)D(rM(Xu)‖rM(Xv)), (3.7)
where J ′EGPR-R1(θ, q, r
M) andR′EGPR-R1(θ, q, rM) are defined according to Equations (4.3) and (4.5)
respectively using the corresponding regularizers. That is,
maximizeθ,q,rM J ′EGPR-R1(θ, q, r
M) ,
L(θ) +R′EGPR-R1(θ, q, rM),
(3.8)
R′EGPR-R1(θ, q, rM) ,
−D(q(XH)‖pθ(XH |x¯O)) + PREGPR-R1(q, rM).
(3.9)
First, we show that rM ≈ q for large values of λR1, so optimizing the reformulated regularizer is
equivalent to optimizing a lower bound on the original.
Lemma 3.3.1. For distributions p ∈ P and q ∈ Q where P ∩ Q 6= ∅ and a continuous function
J(p, q), let J˜(p, q;λ) = J(p, q) − λD(p‖q), and p∗λ, q∗λ ∈ argmaxp∈P,q∈Q J˜(p, q;λ). Then the
following hold:
lim
λ→∞
D(p∗λ‖q∗λ) = 0, (3.10)
lim
λ→∞
‖p∗λ − q∗λ‖` = 0 (3.11)
for any `, where ‖ · ‖` is the `-norm, and
lim
λ→∞
max
p∈P,q∈Q
J˜(p, q;λ) ≤ max
p∈P
J(p, p). (3.12)
Proof. Consider any  > 0 and any p′ ∈ P , q′ ∈ Q such thatD(p′‖q′) > . Let pˆ ∈ argmaxp∈P∩Q J(p, p)
42
and consider any λ′ ≥ (1/)(J(p′‖q′)− J(pˆ‖pˆ)).
J˜(p′, q′;λ′) = J(p′, q′)− λD(p′‖q′) (3.13)
< J(p′, q′)− λ (3.14)
≤ J(p′, q′)− 
(1/)(J(p′‖q′)− J(pˆ‖pˆ)) (3.15)
= J(pˆ, pˆ) (3.16)
Therefore, D(p∗‖q∗) ≤  when λ ≥ λ′. This proves Proposition (3.10).
We have that
D(p‖q) ≥ 1
2
‖p− q‖21 ≥
1
2
‖p− q‖2` (3.17)
(3.18)
for any `-norm. The first inequality is Pinsker’s inequality and the second follows from the rela-
tionship of `-norms. Proposition (3.11) follows from this combined with Proposition (3.10).
Due to Proposition (3.11) and the continuity of J(p, q),
lim
λ→∞
max
p∈P,q∈Q
J˜(p, q;λ)− max
p∈P∩Q
J(p, p) = 0. (3.19)
Proposition (3.12) follows from this and the fact that P ∩Q ⊆ P .
Therefore, for sufficiently large λR1, optimizing Equation (3.7) is equivalent to optimizing a
lower bound on Equation (4.6). This form allows us to compute q efficiently, which is shown as
follows.
Theorem 3.3.2. Define q∗(XH) , argmaxq J ′EGPR-R1(θ, q, rM). Then,
q∗(xH) =
pθ(xH , x¯O)
1/(1+λR1)
∏
h∈H r
M
h (xh)
λR1/(1+λR1)∑
x′H
pθ(x′H , x¯O)1/(1+λR1)
∏
h∈H r
M
h (x
′
h)
λR1/(1+λR1)
.
(3.20)
43
Proof. For ease of notation, we group all terms that do not depend on q into one function K3.3.2(r).
Since we must respect the sum-to-one property of q, we form the Lagrangian by adding the term
λ3.3.2(1−
∑
xH
q(xH))
L3.3.2(q, λ3.3.2) = −D(q(XH)‖pθ(XH |XO))− λR1D(q(XH)‖rM(XH))− λ3.3.2(1−
∑
xH
q(xH)) +K3.3.2(r)
(3.21)
=
∑
xH
q(xH) log
pθ(xH |x¯O)rM(xH)λR1
q(xH)1+λR1
− λ3.3.2(1−
∑
xH
q(xH)) +K3.3.2(r)
(3.22)
=
∑
xH
q(xH) log
pθ(xH , x¯O)r
M(xH)
λR1
q(xH)1+λR1
− log pθ(x¯O)− λ3.3.2(1−
∑
xH
q(xH)) +K3.3.2(r)
(3.23)
0 =
∂L
∂q(xH)
= − log pθ(xH , x¯O)rM(xH)λR1 + log q(xH)1+λR1 + 1 + λR1 − λ3.3.2 (3.24)
=⇒ q(xH) ∝ pθ(xH , x¯O)
1
1+λR1 rM(xH)
λR1
1+λR1 (3.25)
This is identical to the original model pθ, but with one additional factor rMh (xh)
λR1/(1+λR1) over
each label. Critically, because rM is factorizable such that each factor involves just one variable
Xh, q∗(XH) is factorizable in the same way as the unregularized model pθ(XH , x¯O). For example,
if the original model was an HMM, q still factors as a chain. Therefore, the normalization constant
can be computed using any algorithm for exact or approximate probabilistic inference on factorized
models, such as belief propagation, with similar computational cost as the unregularized model.
3.3.2 Optimizing rM
While the last reformulation enabled closed form updates for q, the objective still does not admit
closed-form updates for rM . This is due to the fact that an objective of the form D(p‖q) +D(p‖r)
44
admits closed form updates for p, whileD(p‖q)+D(r‖p) does not. Therefore, we again reformulate
PREGPR-R1(θ, q, rM) by adding a new variable sM , where sM is also a distribution overXH restricted
to be factorizable as a product of marginals. As before, we add a term λR2D(sM(XH)‖rM(XH)),
which encourages sM ≈ rM . We define the graph regularizer KL divergence terms to have sM on
the left and rM on the right—that is, in the form D(sMu (Xu)‖rMv (Xv))—which will enable efficient
optimization for both variables.
PR′EGPR-R2(q, rM , sM) , −λR1D(q(XH)‖rM(XH))
+ max
sM
fR2(r
M , sM)
(3.26)
fR2(r
M , sM) , −λR2D(sM(XH)‖rM(XH))
− λG
∑
(u,v)∈FEGPR
w(u, v)D(sMu (Xu)‖rMv (Xv)).
J ′EGPR-R2(θ, q, r
M , sM) andR′EGPR-R2(θ, q, rM , sM) are defined according to Equations (4.3) and (4.5)
respectively using the corresponding regularizers. That is,
maximizeθ,q,rM ,sM J ′EGPR-R2(θ, q, r
M , sM) ,
L(θ) +R′EGPR-R2(θ, q, rM , sM),
(3.27)
R′EGPR-R1(θ, q, rM , sM) ,
−D(q(XH)‖pθ(XH |x¯O)) + PREGPR-R2(q, rM , sM).
(3.28)
By Lemma 3.3.1, optimizingREGPR-R2(q) is equivalent to optimizingREGPR-R1(q) for large values of
λR2. This regularizer can be optimized in rM and sM using closed-form updates, shown as follows.
Theorem 3.3.3. For notational simplicity, define a new regularization graph with self-edges of
weight λR2/λG, E ′EGPR , ER ∪ {(h, h) | h ∈ H}, and w′(u, v) , w(u, v) + δ(u = v)λR2/λG. Let
rM
∗
(XH) ∈ argmaxrM J ′EGPR-R2(θ, q, rM , sM) and sM ∗(XH) ∈ argmaxsM J ′EGPR-R2(θ, q, rM , sM).
45
Then,
rMv (xv) =
λR1q
M
v (xv) + λG
∑
(u,v)∈E′EGPR w
′(u, v)sMu (xv)
λR1 + λG
∑
(u,v)∈E′EGPR w
′(u, v)
,
(3.29)
sMu
∗
(xu) =
exp
∑
(u,v)∈E′EGPR
w′(u,v) log rMv (xu)∑
(u,v)∈E′EGPR
w′(u,v)∑
x′u
exp
∑
(u,v)∈E′EGPR
w′(u,v) log rMv (x′u)∑
(u,v)∈E′EGPR
w′(u,v)
.
(3.30)
Proof. In its current form, PREGPR-R2(q) involves a sum over all values of q(XH). However, the
following lemma shows how the factorizability of rM facilitates expressing the objective in a form
that involves only sum over values of each variable Xh.
Lemma 3.3.4. For distribution p(XV ) and factorizable distribution qM(XV ) =
∏
v∈V q
M
v (Xv),
define pMv (Xv) ,
∑
XV \v
p(XV ).
D(p‖qM) =
∑
v∈V
D(p(Xv)‖q(Xv))−H(p) +
∑
v∈V
H(q(Xv)). (3.31)
46
Proof.
D(p‖qM) = −H(p)−
∑
xV
p(xV ) log
(∏
v∈V
qMv (Xv)
)
(3.32)
= −H(p)−
∑
xV
p(xV )
∑
v∈V
log qMv (Xv) (3.33)
= −H(p)−
∑
v∈V
∑
xv
∑
xV 6=v
p(xV ) log q
M
v (Xv) (3.34)
= −H(p)−
∑
v∈V
∑
xv
(
log qMv (Xv)
) ∑
xV 6=v
p(xV ) (3.35)
= −H(p)−
∑
v∈V
∑
xv
(
log qMv (Xv)
)
pMv (xv) (3.36)
=
∑
v∈V
D(p(Xv)‖q(Xv))−H(p) +
∑
v∈V
H(q(Xv)) (3.37)
Define qMh to be the marginal distribution of q over Xh, q
M
h (Xh) ,
∑
XH\h
q(XH). Using
Lemma 3.3.4,
PREGPR-R2(q) = max
rM
(
−λR1
∑
h∈H
D(qMh (Xh)‖rMh (Xh)) +H(q)−
∑
h∈H
H(qMh (Xh)) + fR2(r
M)
)
.
(3.38)
We now proceed to derive the update steps. We first derive the update for rM . The Lagrangian
47
for the optimization of rMv is
L3.3.3–1(r
M
v , λ3.3.3–1) (3.39)
= λR1D(q
M
v (Xv)‖rMv (Xv)) + λG
∑
(u,v)∈E′EGPR
w′(u, v)D(sMu (Xu)‖rMv (Xu)) (3.40)
+ λ3.3.3–1(1−
∑
xv
rMv (Xv)) +K3.3.3–1(q, s
M , rMH\v) (3.41)
0 =
∂L
∂rMv (xv)
= −
λR1qMv (xv) + λG ∑
(u,v)∈E′EGPR
w′(u, v)sMu (xv)
 1
rMv (xv)
+ λ3.3.3–1 (3.42)
=⇒ rMv (xv) ∝ λR1qMv (xv) + λG
∑
(u,v)∈E′EGPR
w′(u, v)sMu (xv) (3.43)
∑
xv
λR1qMv (xv) + λG ∑
(u,v)∈E′EGPR
w′(u, v)sMu (xv)
 (3.44)
= λR1


*
1∑
xv
qMv (xv) + λG
∑
(u,v)∈E′EGPR
w′(u, v)


*
1∑
xv
sMu (xv) (3.45)
=⇒ rMv (xv) =
λR1q
M
v (xv) + λG
∑
(u,v)∈E′EGPR w
′(u, v)sMu (xv)
λR1 + λG
∑
(u,v)∈E′EGPR w
′(u, v)
(3.46)
48
We next derive the update for sM . The Lagrangian for the optimization of sMu is
L3.3.3–2(s
M
u , λ3.3.3–2) (3.47)
= λG
∑
(u,v)∈E′EGPR
w′(u, v)D(sMu (Xu)‖rMv (Xu)) + λ3.3.3–2(1−
∑
xu
sMu (Xu)) +K3.3.3–2(q, r
M , sMH\u)
(3.48)
= λG
∑
xu
sMu (xu)
∑
(u,v)∈E′EGPR
w′(u, v) log
sMu (xu)
rMv (xu)
+ λ3.3.3–2(1−
∑
xu
sMu (Xu)) +K3.3.3–2(q, r
M , sMH\u)
(3.49)
0 =
∂L
∂sMu (xu)
= λG
∑
(u,v)∈E′EGPR
w′(u, v) log
1
rMv (xu)
+
λG ∑
(u,v)∈E′EGPR
w′(u, v)
 (1 + log sMu (xu))− λ3.3.3–2
(3.50)
=⇒ sMu (xu) ∝ exp
∑
(u,v)∈E′EGPR w
′(u, v) log rMv (xu)∑
(u,v)∈E′EGPR w
′(u, v)
(3.51)
3.3.3 Updating θ
The preceding section described an algorithm for computing argmaxqREGPR-R2. This algorithm can
be combined with an EM-like algorithm in order to learn a θ that (locally) optimizes JEGPR-R2, as we
describe in this section. We use an alternating EM-like algorithm to compute θ.
E-step:
q(t+1) ∈ argmaxq,rM ,sM J ′EGPR-R2(θ(t), q, rM , sM)
M-step: θ(t+1) ∈ argmaxθ J ′EGPR(θ, q(t+1))
49
Update ✓  ✓⇤(q)Update 
Update Update 
Update 
q  q⇤(✓, rM )
rM , sM  (rM , sM )⇤(q)q  q⇤(✓, rM )
Update rM , sM  (rM , sM )⇤(q)r , sM  (rM , sM )⇤(q)
Figure 3.1: Illustration of optimization algorithm. Solid ovals denote closed-form update steps.
Dashed ovals with dotted expansion lines denote updates that are implemented by alternating
optimization. Pairs of opposing arrows indicate alternating optimization implemented by iterating
each update to convergence.
The preceding section showed how to perform the E-step. To compute the M-step,
argmaxθ J
′
EGPR(θ, q
(t+1))
= argmaxθ Eq(t+1)(XH) [log pθ(XH , x¯O))]
(3.52)
The M-step takes the same form as the EM algorithm presented in [Neal and Hinton, 1999]. The
update for θ depends on the particular factorization and parameterization properties of the model. Be-
cause the posterior distribution q(XH) obeys the same factorization properties as the unregularized
model pθ(XH , XO), the same closed-form updates for θ can be used.
Therefore, the upper bound on the EGPR objective, JEGPR-R2, can be minimized using a three-
way alternating optimization algorithm, which proceeds by alternating closed-form updates to rM
and rM to convergence, alternating this whole update of rM/sM with closed-form updates to q
until convergence, then finally alternating updates to q and θ until convergence. A schematic of the
algorithm is shown in Figure 3.1. The full algorithm in pseudocode is shown in Algorithm 1.
50
Theorem 3.3.5. The modified EM algorithm monotonically increases the relaxed EGPR objective:
JEGPR-R2(θ
(t)) ≤ JEGPR-R2(θ(t+1)). (3.53)
Proof. Function q∗(·) of Algorithm 1 implements coordinate descent on q, rM and sM . D(p‖q) and
D(p‖p) are jointly strictly convex in p and q and bounded below by 0. Thus, J ′EGPR-R2 is bounded
below and jointly strictly convex in q, rM and sM . Convergence to the global optimum of J ′EGPR-R2
in q, rM and sM follows from its strict convexity [Warga, 1963].
JEGPR-R2(θ
(t)) = J ′EGPR-R2(θ
(t), q(t+1), rM
(t+1)
, sM
(t+1)
)
≤ J ′EGPR-R2(θ(t+1), q(t+1), rM (t+1), sM (t+1))
≤ JEGPR-R2(θ(t+1))
The first equality follows from the global optimality of q(t+1), rM (t+1) and sM (t+1). The second
inequality follows from the fact that θ(t+1) is chosen to maximize J ′EGPR-R2. The third inequality
follows from the fact that J ′EGPR-R2(θ, q, r
M , sM) is a lower bound on JEGPR-R2(θ).
3.4 EGPR for Inference
In addition to its use for regularized learning, EGPR can be used directly as an inference algorithm.
To do this, we compute q∗ ← argmaxq J ′EGPR-R2(θ, q) and use this distribution as the posterior. As
discussed above, q∗ obeys the same factorization properties as pθ, so algorithms such as Viterbi
and belief propagation can be used to compute the MAP solution and marginal distributions of q
respectively. Using EGPR as an inference algorithm results in posteriors which are smooth with
respect to the graph GR.
3.5 Related work
The most straightforward way to express similarity information in an unsupervised model is to
encode it in the graphical model. For example, one might add a factor between Xu and Xv as in
51
φ(xu, xv) = λ1(xu = xv). This form of interaction is quite different from EGPR, because adding
factors changes the “implementation” of the model, while EGPR regularizes what the model does.
Moreover, there are two problems with this approach. First, it is not always clear what form of
interaction is most appropriate. In particular, although KL-based penalties have been very successful
in graph-based semi-supervised learning, they cannot be converted to an equivalent set of probability
factors. Second, adding similarity edges to the probabilistic model results in a model that does not,
in general, have low tree-width, so efficient exact inference algorithms such as belief propagation
cannot be used, and one must resort to approximate inference. As we show in Section 4.7, EGPR
performs better than the loopy belief propagation (LBP) approximate inference algorithm on an
augmented graph.
Three methods take a similar approach to ours, by augmenting a probabilistic model with a
graph regularizer. First, Altun et al. [2005] describe a graph regularization for max-margin models
applied to pitch-accent prediction and optical character recognition. However, this method involves
a matrix inversion step, and thus it cannot scale to large models. Second, Subramanya et al. [2010]
combine a temporal conditional random field with a regularizer that expresses pairwise squared-error
penalties derived from unlabeled data. They apply this method to the part-of-speech tagging task
[Subramanya et al., 2010] and later to related problems in natural language [Das and Petrov, 2011,
Das and Smith, 2011]. That work, however, resorts to a purely heuristic update step and lacks any
optimality guarantees.
Third, He et al. [2013] present an approach based on an exponentiated gradient descent algorithm.
Like our approach, He’s approach exhibits monotone convergence. Although He’s work has many
similarities with our approach, He’s work differs from ours in three important ways. First, He’s
method uses a squared-error penalty, which, as argued above, is less appropriate for probability
distributions than Kullback-Leibler divergence [Bishop, 1995, p. 226]. As shown in Section 4.7,
using squared error also results in worse performance in practice. Second, the exponentiated gradient
descent method is applied to semi-supervised handwriting recognition and part-of-speech tagging,
while we apply EGPR to an unsupervised genome annotation problem. Third, He et al. [He et al.,
2013] use an exponentiated gradient descent strategy, while we use alternating optimization.
52
Our alternating optimization approach has several benefits over the exponentiated gradient
descent method of He et al. [He et al., 2013]. First, the He et al. algorithm involves gradient
calculations over each clique in the conditional dependence graph, with order O(k|C|) for a variable
of dimension k involved in cliques of size |C|. Our alternating minimization algorithm, by contrast,
has closed-form updates of order O(k) for q, rM and sM . (Updates for θ can still involve O(k|C|)
calculations, but these are generally very fast). Therefore, the alternating optimization algorithm
is more appropriate for extremely large models or models with large cliques or high-order factors.
Second, empirical studies of KL-based graph smoothness objectives have shown that alternating op-
timization algorithms perform better than gradient-based methods for these objectives [Subramanya
and Bilmes, 2011, 2009]. Finally, the alternating optimization strategy is parallelizable [Bilmes and
Subramanya, 2012] and extremely simple to implement because the graph regularization step has
simple, closed-form updates with no learning rate hyperparameters, and the posterior calculation
can be performed using any probabilistic inference method on a model with the same conditional
dependence graph as the unregularized model.
3.6 Simulations
All implementations in the below use the GMTK, the graphical models toolkit [Bilmes, 2000], and
its use of virtual evidence factors, for HMM and dynamic graphical model inference
3.6.1 Learning a Gaussian Mixture on Poorly-Separated Data
First, we consider a simple example that demonstrates the utility of EGPR (Figure 3.2). Suppose
we wish to learn a mixture of four Gaussians on data lying in a circle in 2D, using EM training to
find cluster centers. Clearly, the training problem is underspecified in this case, because any set of
centers at 90 degrees from one another relative to the circle’s center represents an optimum of the
model likelihood. However, suppose we additionally have pairwise information that certain slices
of the circle should form clusters. To represent this additional information, we form a regularization
graph that connects all pairs of positions within the same cluster and run EM with EGPR using
53
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
X
Y
Without EGPR
l
l
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l
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l
l
l
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l
l
l
l
l
l
l
l
l
X
Y
True
class
l 1
2
3
4
Predicted
class
l
l
l
l
1
2
3
4
With EGPR
Figure 3.2: Using EGPR to learn ambiguous clusters. Shape denotes true class, color denotes
predicted class, and colored arrows denote cluster means as they evolve between iterations of EM.
this graph. EGPR finds the true cluster centers and recovers the true labels with 100% accuracy
compared to 74% accuracy with EM alone. This example demonstrates how pairwise information
can be integrated in the training process to produce a trained model that implicitly incorporates this
information.
3.6.2 Comparison with Related Inference Methods
To evaluate the efficacy of EGPR, we compared EGPR to two related methods: 1) approximate
inference on a graphical model with the same dependence structure, and 2) GPR using squared-error
penalties. We compared to the approximate inference method loopy belief propagation (LBP)
because it is one of the most widely used approximate inference methods. While we would have
preferred to perform this comparison using our genomics data sets (see Section 4.2.9), due to the
large size of the models and dense connectivity of the regularization graphs, it appeared that even
our implementations of these methods would take months to converge. Therefore, we instead
performed this comparison using synthetic data. We generated a chain of length n = 200, with
(XH , XO) = (Z1:200, Y1:200), whereZ1:200 ∈ {0, 1}n and Y1:200 ∈ Rn. We defined an HMM over this
54
0.0
0.1
0.2
0.3
0.4
0.5 1 1.5 2
Noise level (sigma)
Er
ro
r
Method
Independent
Chain
Loopy BP
SQGPR
EGPR
Figure 3.3: Comparison of EGPR with related inference methods. The X axis shows σ, a hyperpa-
rameter controlling the difficulty of inference. The Y axis shows the average accuracy over 200
simulations of MAP inference on the model in question (95% Wilcoxon test confidence intervals).
chain with transition probabilities Pr(Zi = Zi+1) = 0.9 and emission probabilities Yi ∼ N(Zi, σ),
where we vary σ to control the difficulty of the problem—higher σ results in more challenging
inference. We generated a graph W ∈ Rn×n over the vertices of the chain by setting wij = 1 with
probability 0.4 if Zi = Zj , wij = 1 with probability 0.1 if Zi 6= Zj , and wij = 0 otherwise. This
model is meant to simulate the task of labeling a chain (such as a genomic sequence) where we
have noisy information about which pairs of positions have the same label.
We compared five methods of inference: 1) inference on each position independently, with no
chain model; 2) inference on the chain alone, without usingW ; 3) LBP on the chain plus extra factors
of Pr(Xi = Xj) = sigmoid(λwij), where λ controls the strength of these factors; 4) GPR using the
regularization graph W and a squared-error penalties as described in [He et al., 2013] (SQGPR);
and 5) EGPR using the regularization graph W . We chose hyperparameters for each model (λG,
λR1 and λR2 for GPR and λ for LBP) using a training set of 200 simulations. We evaluated results
according to the average accuracy over 200 simulations of MAP inference on the model in question.
EGPR significantly outperforms all other models for all experiments (Figure 3.3, providing
nearly as much improvement in accuracy as does the chain model itself. The pattern of accuracy is
55
instructive in understanding the properties of each model. LBP performs very well when there is little
noise, but becomes easily stuck in local optima on harder problems. GPR with squared error provides
a modest improvement over the chain model, but has poor performance relative to KL penalties,
consistent with previous work on semi-supervised methods [Subramanya and Bilmes, 2011, 2009].
3.7 Application: Genome Annotation Using Physical Interaction Information
Recently, many methods have been described that partition and label the human genome on the basis
of a number of genome-wide real-valued signal tracks, generally employing temporal models such as
HMMs [Day et al., 2007, Hoffman et al., 2012, Ernst and Kellis, 2010, Filion et al., 2010, Thurman
et al., 2007, Lian et al., 2008]. Formally, these methods aim to learn a labeling XH1:n ∈ {1..L}n
which associates each position in the genome with one of L integer labels, such that positions that
receive the same label exhibit similar patterns in the signal data. The input is comprised of a feature
vector XOi ∈ RF at each position that represents the output of biological experiments that measure
local properties of the DNA, including its interaction with binding proteins, its local structure, and
various types of chemical modifications. The process is “semi-automated” because a human assigns
a semantic interpretation of the integer labels subsequent to the unsupervised learning phase.
However, existing genome annotation methods cannot incorporate the genome’s 3D confor-
mation. The 3D arrangement of the genome in the nucleus plays a central role in gene regulation,
chromatin state and replication timing [Misteli, 2007, Dekker et al., 2002, Ryba et al., 2010, Dixon
et al., 2012]. Genome conformation can be investigated using chromatin conformation capture
experiments such as Hi-C [Lieberman-Aiden et al., 2009]. A Hi-C experiment outputs a matrix
of contact counts, where the number of contact counts of a pair of genomic positions is inversely
proportional to the positions’ 3D distance in the nucleus [Lieberman-Aiden et al., 2009, Ay et al.,
2014b]. Existing genome annotation methods can incorporate any data set that can be represented
as a vector defined linearly across the genome, but they cannot incorporate inherently pairwise Hi-C
data without resorting to simplifying transformations such as principle component analysis.
56
Algorithm 1 Efficient and scalable algorithm to optimize J ′EGPR-R2
1: function rM ∗(q)
2: for h ∈ H do
3: qMh (xh)←
∑
xH 6=h q(xH) (belief propagation)
4: end for
5: Initialize rM (0), sM (0) arbitrarily.
6: t1 ← 1
7: while not converged do
8: for v ∈ H do
9: rM
(t1)
v (xv)←
λR1q
M
v (xv)+λG
∑
(u,v)∈E′EGPR
w′(u,v)sM (t1−1)u (xv)
λR1+λG
∑
(u,v)∈E′EGPR
w′(u,v)
10: end for
11: for u ∈ H do
12: sMu
(t1)(xu)←
exp
∑
(u,v)∈E′EGPR
w′(u,v) log rM (t1−1)v (xu)∑
(u,v)∈E′EGPR
w′(u,v)
∑
x′u exp
∑
(u,v)∈E′EGPR
w′(u,v) log rM (t1−1)v (x′u)∑
(u,v)∈E′EGPR
w′(u,v)
13: end for
14: t1 ← t1 + 1
15: end while
16: return rM (t1)
17: end function
18:
19: function q∗(θ)
20: t2 ← 1
21: Initialize rM (0) arbitrarily.
22: while not converged do
23: q(t2)(xH)← pθ(xH ,x¯O)
1/(1+λR1)
∏
h∈H r
M (t2−1)
h (xh)
λR1/(1+λR1)∑
x′
H
pθ(x
′
H ,x¯O)
1/(1+λR1)
∏
h∈H rM
(t2−1)
h (x
′
h)
λR1/(1+λR1)
(belief propagation)
24: rM
(t2) ← rM ∗(q(t2))
25: t2 ← t2 + 1
26: end while
27: end function
28:
29: Initialize θ(0) arbitrarily.
30: t3 ← 1
31: while not converged do
32: q(t3) ← q∗(θ(t3−1))
33: θ(t3) ← argmaxθ Eq(t3)(XH) [log pθ(XH , x¯O))] (EM update)
34: end while
35: Output θ(t3)
57
We therefore present a novel strategy for integrating 3D conformation information using EGPR
in which we encourage pairs of positions which interact in 3D to receive the same label in the
annotation by connecting these positions with edges in an EGPR graph (Figure 3.4). While the
assumption that positions close in 3D space have similar regulatory state is not necessarily true at
a small scale (∼1 thousand base pair elements), it does generally hold at a large scale (∼1 million
base pair elements) [Lieberman-Aiden et al., 2009].
3.7.1 Synthetic Example
To motivate this approach, we first consider a simple synthetic model with 501 nodes and six
EGPR edges. Let Yi ∈ {0, 1} for i = 0 . . . 500. We assign Y1 = 0 and Y500 = 1. Let P (Y0:500) =∏499
i=0 0.9
1(Yi=Yi+1)0.11(Yi 6=Yi+1). In other words, the model places higher probability on neighboring
positions taking the same label but provides no other information. We construct an EGPR graph with
w0,200 = w0,400 = w200,400 = w100,300 = w100,500 = w300,500 = 1, corresponding to a hypothetical
2D arrangement of the chain. Without EGPR, the model learns a trivial labeling of the chain,
whereas with EGPR, the model learns a labeling corresponding to the 2D arrangement (Figure 3.4).
3.7.2 Real data
Next, to evaluate the efficacy of our approach, we performed genome annotation of the human
fibroblast cell line IMR90. We compared three models: (1) the chain model described in [Hoffman
et al., 2012], without 3D structure data, (2) the chain model augmented with 3D structure data
expressed with squared-error GPR (SQGPR), and (3) the chain model augmented with 3D structure
data expressed with EGPR. We would have liked to compare against loopy belief propagation as
well, but as mentioned previously, it appeared that our fastest implementation of this method would
take months to converge on this large data set. In order to evaluate our performance in a variety
of conditions, we ran each model separately once for each of the 29 available data sets in IMR90.
For SQGPR and EGPR, we used a GBR graph based on 3D structure data. To generate the
GBR graph representation of 3D structure used by EGPR and SQGPR, we used the Hi-C data set
58
A B
Figure 3.4: Synthetic model of genome spatial interactions. Color and labeled division lines
indicate learned labels along the hypothesized 501 bp genome. Large filled circles indicate observed
positions. Dotted lines indicate EGPR edges. (A) Without EGPR. (B) With EGPR.
of [Dixon et al., 2012] and processed the Hi-C data into a matrix of pairwise p-values using the
Fit-Hi-C method [Ay et al., 2014a]. To remove noise and decrease the degree of the graph, we
removed all contacts with uncorrected p-value p > 10−6 and multiplied the remaining p-values by
106, similar to a Bonferroni correction. We generated the GPR graph by setting the weight between
positions i and j, w(i, j), to w(i, j) , max(0,− loge(p(i, j)/106)), where p(i, j) is the p-value of
interaction between positions i and j. All annotations used four labels and binned the genome at
10,000 base pair resolution, comparable to the resolution of 40,000 bp used in [Dixon et al., 2012].
We chose the best-performing hyperparameters for each GPR model using a validation set.
To evaluate these annotations, we used the time during the cell cycle at which the DNA is
replicated as a gold standard [Woodfine et al., 2004]. Replication time is highly correlated with gene
expression and chromatin state and therefore is a good proxy for domain type [Lieberman-Aiden
et al., 2009]. To evaluate the accuracy with which an annotation predicts replication time, we
compute a prediction as follows. Let ai ∈ {1 . . . 4} and xi be the annotation label and replication
time at position i, respectively. We compute the replication time mean over the positions assigned a
59
A
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
A
B
C
D
A B C D
Chromatin 3D 
structure
A
B
C
D
A B C D
Hi-C contact 
matrixEGPR!
graph
Signal!
tracks
Hi-C
B C
l
0.00
0.01
0.02
0.03
EGPR SQGPR
Method
Ch
ai
n 
RM
SE
 −
 G
PR
 R
M
SE
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
0.00
0.01
0.02
0.03
0.000 0.005 0.010 0.015 0.020
Chain RMSE − SQGPR RMSE
Ch
ai
n 
RM
SE
 −
 E
G
PR
 R
M
SE
Figure 3.5: (A) Strategy for utilizing physical interaction information. (B) Improvement in RMSE
over chain model for EGPR and SQGPR for 29 experiments. (C) Relative improvement in RMSE
between EGPR and SQGPR for each of 29 experiments.
given label ` as
µ` ,
∑n
i=1 1(ai = `)xi∑n
i=1 1(ai = `)
for ` ∈ {1 . . . 4}. (3.54)
60
We define a predicted replication time vector xpi = µai and compute the root mean squared error
(RMSE) of this prediction as RMSE =
√∑
i(xi − xpi )2. EGPR consistently outperforms both the
chain model alone and SQGPR (Figure 3.5).
3.8 Discussion
We have defined entropic graph-based posterior regularization (EGPR), a method to encourage a
model’s posteriors to be smooth according to a regularization graph. This method is motivated by
graph-based methods for semi-supervised learning, which have had great success in that setting
but have not been studied thoroughly in an unsupervised setting. We showed that EGPR greatly
outperforms both the approximate inference method loopy belief propagation and previous methods
for graph-based regularization. We used EGPR to incorporate 3D structure data for semi-automated
genome annotation and showed that EGPR greatly improved the quality of the resulting annotations.
This method will thereby enable these methods to distill complicated pairwise contact matrices into
human-interpretable genome annotations. Moreover, because EGPR can use any graphical model
and similarity graph, it will likely have diverse applications in other fields such as natural language
and time-series analysis.
3.9 Frequently Asked Questions
This section collects questions we have received when presenting this work.
1. Why is KL divergence better than squared error for probability distributions? Squared
error is based on a Gaussian error model, which is not appropriate for probability values, and
it under-penalizes differences between small probability values. p-norms are defined over all
real numbers, while posteriors must lie within the range [0, 1] (and live in a simplex). A more
justified way to measure divergence between probability distributions is to employ the KL
divergence. The KL divergence measures the difference of exponents in the probability and so
evaluates differences between small and large probabilities more uniformly. Also, Pinsker’s
inequality [Csisza´r and Tusna´dy, 1984] combined with the relationship of `-norms implies
61
that D(p‖q) ≥ 1
2
‖p − q‖21 ≥ 12‖p − q‖2` , for all ` ≥ 1, where ‖ · ‖` is the `-norm. Hence,
minimizing KL divergence minimizes an upper bound on all `-norms.
As a concrete example, consider two pairs of probability distributions over two possible
events: [0.55, 0.45] vs. [0.45, 0.55], and [0.1, 0.9] vs. [10−10, 1 − 10−10]. The first pair of
distributions are very similar, with both events being roughly equally likely. The second pair is
quite dissimilar, with the first event being reasonably likely in the first case and astronomically
unlikely in the second. Squared error actually regards the first pair as more dissimilar, while
KL divergence correctly identifies the second pair as much more dissimilar. Despite the
advantages of KL divergence, although all posterior regularization objectives include a KL
term binding q to be similar to pθ, to our knowledge, no existing methods define the posterior
regularizer itself using KL. This is also discussed in Section 3.2.
In practice, although a regularizer based on squared error shows good results for some prob-
lems, we demonstrate the better performance of KL divergence empirically in Sections 3.6
and 4.2.9.
2. KL divergence is asymmetric in its two arguments. How does that affect the results?
As stated in Section 3.2, the KL divergence in Equation 4.6 is symmetrized because GR is
undirected—that is w(u, v) = w(v, u)—so D(qu‖qv) and D(qv‖qu) appear with the same
weight in the regularizer. That is,
∑
u,v w(u, v)D(qu‖qv) =
∑
u≤v 2w(u, v)(D(qu‖qv) +
D(qv‖qu)).
3. How does EGPR compare to other methods? This topic is discussed in detail in Sec-
tion 3.5.
(a) How does EGPR compare to approximate inference? Posterior regularization is so
named because it regularizes the model parameters indirectly via a penalty defined using
the posterior distribution. It is therefore fundamentally different from approximate (or
any) inference methods, which are used to perform probabilistic inference. Although
62
complementary, inference methods are quite a separate machine-learning concept from
parameter regularization strategies.
In addition, we demonstrate empirically in Section 3.6 that EGPR outperforms the
approximate inference algorithm loopy belief propagation (LBP) on a synthetic data
set. (Note that LBP can be understood either in the context of belief propagation or as
a variational method—see [Wainwright and Jordan, 2008].)
(b) How does EGPR compare to the first posterior regularization method of [Ganchev
et al., 2010]? The method of [Ganchev et al., 2010] requires each posterior regular-
ization term to involve variables in just one clique in the graphical model. Applying
this method to a graph-based posterior regularization objective would result in a high
tree-width model and intractable inference.
(c) How does EGPR compare to the graph-based posterior regularization method of
[He et al., 2013]? He et al. [2013] present an approach based on an exponentiated
gradient descent algorithm. Like our approach, He’s approach exhibits monotone con-
vergence. Although He’s work has many similarities with our approach, He’s work
differs from ours in three important ways. First, He’s method uses a squared-error
penalty, which, as argued above, is less appropriate for probability distributions than
Kullback-Leibler divergence [Bishop, 1995, p. 226]. As shown in Section 4.7, using
squared error also results in worse performance in practice. Second, the exponentiated
gradient descent method is applied to semi-supervised handwriting recognition and
part-of-speech tagging, while we apply EGPR to an unsupervised genome annotation
problem. Third, He et al. [He et al., 2013] use an exponentiated gradient descent strategy,
while we use alternating optimization.
Our alternating optimization approach has several benefits over the exponentiated gradi-
ent descent method of He et al. [He et al., 2013]. First, the He et al. algorithm involves
gradient calculations over each clique in the conditional dependence graph, with order
O(k|C|) for a variable of dimension k involved in cliques of size |C|. Our alternating
63
minimization algorithm, by contrast, has closed-form updates of order O(k) for q, rM
and sM . (Updates for θ can still involve O(k|C|) calculations, but these are generally
very fast). Therefore, the alternating optimization algorithm is more appropriate for
extremely large models or models with large cliques or high-order factors. Second,
empirical studies of KL-based graph smoothness objectives have shown that alternating
optimization algorithms perform better than gradient-based methods for these objec-
tives [Subramanya and Bilmes, 2011]. Finally, the alternating optimization strategy
is parallelizable and extremely simple to implement because the graph regularization
step has simple, closed-form updates with no learning rate hyperparameters, and the
posterior calculation can be performed using any probabilistic inference method on a
model with the same conditional dependence graph as the unregularized model.
3.10 Acknowledgments
This work was supported by NIH awards U41HG007000, R01ES024917, and R01GM103544 and
by the Princess Margaret Cancer Foundation.
64
Chapter 4
JOINT ANNOTATION OF CHROMATIN STATE AND CHROMATIN
CONFORMATION REVEALS RELATIONSHIPS AMONG DOMAIN
TYPES AND IDENTIFIES DOMAINS OF CELL TYPE-SPECIFIC
EXPRESSION
Abstract
The genomic neighborhood of a gene influences its activity, a behavior that is attributable in
part to domain-scale regulation, in which regions of hundreds or thousands of kilobases known
as domains are regulated as a unit. Previous studies using genomics assays such as chromatin
immunoprecipitation (ChIP)-seq and chromatin conformation capture (3C)-based assays have
identified many types of regulatory domains. However, due to the difficulty of integrating genomics
data sets, the relationships among these domain types are poorly understood. Semi-automated
genome annotation (SAGA) algorithms facilitate human interpretation of heterogeneous collections
of genomics data by simultaneously partitioning the human genome and assigning labels to the
resulting genomic segments. However, existing SAGA methods can incorporate only data sets
that can be expressed as a one-dimensional vector over the genome and therefore cannot integrate
inherently pairwise chromatin conformation data. We developed a new computational method,
called graph-based regularization (GBR), for expressing a pairwise prior that encourages certain
pairs of genomic loci to receive the same label in a genome annotation. We used GBR to exploit
chromatin conformation information during genome annotation by encouraging positions that are
close in 3D to occupy the same type of domain. Using this approach, we produced a comprehensive
model of chromatin domains in eight human cell types, thereby revealing the relationships among
known domain types. Through this model, we identified clusters of tightly-regulated genes expressed
in only a small number of cell types, which we term “specific expression domains.” We additionally
65
found that a subset of domain boundaries marked by promoters and CTCF motifs are consistent
between cell types even when domain activity changes. Finally, we showed that GBR can be used
for the seemingly unrelated task of transferring information from well-studied cell types to less
well characterized cell types during genome annotation, making it possible to produce high-quality
annotations of the hundreds of cell types with limited available data.
4.1 Introduction
Although the mechanism of regulation of a gene by a promoter directly upstream of its transcription
start site is well understood, this type of local regulation does not explain the large effect of
genomic neighborhood on gene regulation. The neighborhood effect is in part the consequence of
domain-scale regulation, in which regions of hundreds or thousands of kilobases known as domains
are regulated as a unit [Bickmore and van Steensel, 2013, Chakalova et al., 2005, Akhtar et al.,
2013]. Current understanding of domain-scale regulation is based on a number of domain types,
each defined based on a different type of data, such as histone modification ChIP-seq, replication
timing, or measures of chromatin conformation. However, as a result of the difficulty of integrating
genomics data sets, the relationships among these domain types are poorly understood. Therefore,
a principled method for jointly modeling all available types of data is needed to improve our
understanding of domain-scale regulation.
A class of methods we term semi-automated genome annotation (SAGA) algorithms are widely
used to jointly model diverse genomics data sets. These algorithms take as input a collection of
genomics data sets and simultaneously partition the genome and label each segment with an integer
such that positions with the same label have similar patterns of activity. These algorithms are “semi-
automated” because a human performs a functional interpretation of the labels after the annotation
process. Examples of SAGA algorithms include HMMSeg [Day et al., 2007], ChromHMM [Ernst
and Kellis, 2010], Segway [Hoffman et al., 2012] and others [Thurman et al., 2007, Lian et al., 2008,
Filion et al., 2010]. These genome annotation algorithms have had great success in interpreting
genomics data and have been shown to recapitulate known functional elements including genes,
promoters and enhancers.
66
However, existing SAGA methods cannot model chromatin conformation information. The
3D arrangement of chromatin in the nucleus plays a central role in gene regulation, chromatin
state and replication timing [Misteli, 2007, Dekker, 2008, Ryba et al., 2010, Dixon et al., 2012].
Chromatin architecture can be investigated using chromatin conformation capture (3C) assays,
including the genome-wide conformation capture assay, Hi-C. A Hi-C experiment outputs a matrix
of contact counts, where the contact frequency of a pair of positions is inversely proportional to the
positions’ 3D distance in the nucleus [Lieberman-Aiden et al., 2009, Ay et al., 2014b]. Existing
SAGA methods can incorporate any data set that can be represented as a vector defined linearly
across the genome, but they cannot incorporate inherently pairwise Hi-C data without resorting to
simplifying transformations such as principle component analysis.
We present a method for integrating chromatin architecture information into a genome annotation
method. Motivated by the observation that pairs of loci close in 3D tend to occupy the same type of
domain, we encourage these pairs to be assigned the same label in a genome annotation through
a pairwise prior. We developed a novel computational method, called graph-based regularization
(GBR), which performs inference in the presence of such a pairwise prior, and we extended the
existing SAGA algorithm Segway [Hoffman et al., 2012] to implement this method.
GBR can also be used for the seemingly-unrelated task of transferring information from well-
studied cell types for the annotation of cell types with limited available data. Consortia such as
ENCODE have characterized a small set of cell types in great detail using hundreds of genomics
assays. However, due to the high cost of genomics experiments, it is feasible to perform only a few
assays on any additional cell type of interest. For example, ENCODE and Roadmap Epigenomics
have each performed 2-10 experiments in more than 100 cell types, and it is common for an
individual lab to perform a small number of experiments on a particular cell type or perturbation of
interest. In such settings, it is crucial to leverage information garnered from well-studied cell types to
allow accurate annotation of other cell types using just a few experiments. We transfer information
from well-studied cell types with GBR by using the pairwise prior that loci that were assigned the
same label in many well-studied cell types should be more likely to receive the same label in a
cell type of interest. Therefore, GBR makes it possible to produce high-quality annotations of the
67
hundreds of cell types with limited available data.
4.2 Results
4.2.1 Chromatin domains co-localize with domains of similar activity
Previous research has shown that large chromatin domains (∼1 Mb) tend to co-localize with domains
of similar activity in 3D [Ryba et al., 2010, Lieberman-Aiden et al., 2009]. To further explore
this trend, we compared the 1D genomics data at pairs of statistically significantly interacting loci
(Supplementary Figure 4.1). As expected, we found that chromatin signals were highly consistent
at pairs of positions nearby in 3D. First, histone modification and replication signal values at pairs
of significantly interacting loci are more highly correlated than for a rotational permutation control,
which controls for the 1D pattern of the signal (Supplementary Figure 4.1A). Second, Segway labels
generated without using GBR from an annotation of the genome were assigned the same label
much more often than a rotational permutation control (Supplementary Figure 4.1B). Note that this
pattern is in stark contrast to small elements (∼100bp) such as promoters and enhancers, which do
not, in general, cluster with elements of the same type. These observations suggest that chromatin
conformation data might best be incorporated using a pairwise prior stating that a pair of positions
should be more likely to receive the same label if the positions are close in 3D. Therefore, we sought
to develop new methods for leveraging chromatin conformation data using this co-localization
pattern.
4.2.2 Graph-based regularization expresses a pairwise prior in a SAGA method
Existing SAGA algorithms use dynamic programming algorithms to perform inference in a chain-
structured Bayesian network such as a hidden Markov model. Dynamic programming algorithms
such as the forward-backward algorithm can be used to perform inference efficiently in models with
chain-structured dependencies; however, applying these methods in the presence of a pairwise prior
that connects arbitrary pairs of positions results in inference costs which grow exponentially in the
number of genomic positions. Therefore, these methods cannot be applied to genome annotation
68
problems with millions or billions of variables. We propose a novel convex optimization framework
which allows for efficient inference in this case.
The method takes as input a set of genomics data sets and weighted graph over the genomic
positions, where a large weight on a given pair of positions indicates that we have a strong prior
belief that this pair should receive the same label. It outputs a probability distribution over the
integer labels at each position. The method encourages pairs of positions connected by edges in the
graph to be assigned the same label by minimizing a measure of dissimilarity between their output
probability distributions called the Kullback-Leibler (KL) divergence.
We call this strategy of using a graph to incorporating a pairwise prior graph-based regularization
(GBR) (Methods). Note that in this manuscript we use the word “prior” in the non-technical sense of
“prior information,” not in the sense of a prior distribution for a Bayesian model. We have developed
an efficient, novel alternating minimization algorithm that optimizes this objective (Methods). Using
synthetic data, we determined that graph-based regularization outperforms alternative methods based
on approximate inference as well as existing methods for graph-based regularization (Chapter 3).
We then extended the SAGA method Segway to implement this algorithm [Hoffman et al., 2012]
(Section 4.4.3).
4.2.3 Using GBR to integrate 3D structure information improves prediction of replication and
topological domains
We used graph-based regularization to integrate chromatin conformation information using the
pairwise prior that positions close in 3D should be more likely to be identified as the same domain
type (Figure 4.1A). To do this, we construct a GBR graph that connects each pair of positions with
weight proportional to our statistical confidence that the positions physically interact [Ay et al.,
2014a] (Methods). This measure of statistical confidence controls for the bias of Hi-C for positions
close in 1D as well as biases for sequence features such as GC content and restriction site density.
Incorporating Hi-C data using GBR has the effect of both aligning domains to regions of
self-interacting chromatin and helping to determine the label of each segment. We evaluated
the first effect, as a sanity check, by computing the accuracy with which our annotation predicts
69
self-interacting regions of chromatin of size ∼1 Mb called topological domains [Dixon et al., 2012,
Filippova et al., 2014]. We evaluated the second effect by comparing to replication time, which
is highly correlated with gene expression and chromatin state and therefore is a good proxy for
domain type. In order to evaluate our performance in a variety of conditions, we ran Segway
augmented with GBR separately once for each of the 29 histone modification sets available in
IMR90, in each case using as input a single histone modification data set and a GBR graph based
on IMR90 Hi-C data. We found that the annotation’s ability to identify both topological domains
(p < 10−16, t-test) and replication time (p < 10−16, t-test) was greatly improved by using GBR
(Figure 4.1B,C). To evaluate the degree to which an annotation matches topological domains, we
computed, for each topological domain, the fraction of positions in the topological domain receiving
the same label, controlling for the length and label distribution by comparing to a circularly permuted
annotation (Methods). We measure the degree to which an annotation predicts replication time
by the variance in replication timing explained by the annotation (Methods). The improvement
from adding Hi-C with GBR was greater than the improvement achieved by instead adding another
histone modification data set, for 26/30 and 27/30 histone modification data sets for topological
domains and replication time, respectively. Furthermore, this improvement was consistent for a
large range of hyperparameters (several order of magnitude around optimal, Figure 4.1B,C). These
results demonstrate that incorporating Hi-C data using GBR greatly improves the quality of the
resulting annotation, and moreover that Hi-C is more informative for determining domain identity
than most other data types.
4.2.4 Joint domain annotation of chromatin state and chromatin conformation captures previously-
described domain types
Having verified the utility of GBR for incorporating Hi-C data, we next sought to investigate domain-
scale genome regulation using this method. Current understanding of domain-scale regulation is
based on a number of domain types (seven, by our count), each defined based on a different
type of data, such as histone modification, replication timing or 3C-based assays (Table 4.1). For
example, ChIP-seq on the histone modification H3K27me3 has revealed repressive domains known
70
Name Label(s)
Typical
length
Relevant
data types References
Topological All 0.8 Mb Hi-C Dixon et al. [2012], Filippova et al. [2014]
Open compartment /
early replicating
BRD,
SPC,
FAC
10 Mb
Hi-C,
Repli-(chip/seq)
Lieberman-Aiden et al. [2009]
Ryba et al. [2010]
Closed compartment /
late replicating
QUI,
CON,
FAC
10 Mb
Hi-C,
Repli-(chip/seq)
Lieberman-Aiden et al. [2009]
Ryba et al. [2010]
Quiescent QUI 0.5 Mb None
Ernst and Kellis [2010]
Hoffman et al. [2012]
Constitutive
heterochromatin CON 0.5 Mb H3K9me3 (ChIP-seq) Lachner et al. [2003]
Facultative
heterochromatin FAC 0.5 Mb H3K27me3 (ChIP-seq)
Morey and Helin [2010]
Pauler et al. [2009]
Lamina See text 0.6 Mb lamin B1 (DamID)
Guelen et al. [2008]
Wen et al. [2009]
Broad activity BRD 0.5 Mb
Transcription (i.e. H3K36me3)
(ChIP-seq)
Novel in human
cells (see text)
Specific activity SPC 0.5 Mb
Regulation (i.e. H3K27ac)
(ChIP-seq)
Novel in human
cells (see text)
Table 4.1: Known types of domains. Note that the lengths of domains depend greatly on the method
used to define them.
as facultative heterochromatin [Morey and Helin, 2010, Pauler et al., 2009], and 3C-based assays
have revealed regions of self-interacting chromatin known as topological domains [Dixon et al.,
2012, Filippova et al., 2014]. Because all of these domain types are defined using different types of
data, until now it has been difficult to understand the relationships among these domain types. GBR
provides a principled method for integrating all types of data into a unified annotation of domains.
We therefore used Segway with GBR to create such a unified annotation, in order to understand
what types of domains exist and their interrelationships.
We annotated the cell type IMR90 using all 30 signal data sets we had available in IMR90 and a
GBR graph derived from IMR90 Hi-C data, resulting in an annotation with median segment length
of 0.4 Mb (Figure 4.2A,B,C, Methods, Supplementary Tables 4.1,4.2). Including Hi-C into this
annotation using GBR changed the label of 6% of positions relative to an annotation without GBR,
meaning Hi-C has a slightly larger influence than the 1/31 = 3% difference one would expect from
adding one additional data set (Supplementary Figure 4.2). Because determining the optimal number
of labels for an annotation remains an open problem, we specified a somewhat larger number of
71
Data type Figure QUI CON FAC BRD SPC
Median
segment length 4.2C 0.6 Mb 1.4 Mb 0.4 Mb 0.6 Mb 0.2 Mb
Histone
modifications 4.2A,D None H3K9me3 H3K27me3
Transcription
(i.e. H3K36me3)
Regulation
(i.e. H3K27ac)
Replication timing 4.2A,D
Switching
late
Constitutively
late Mixed
Constitutively
early
Switching
early
GC content 4.3D Low Low High Mid High
Conservation 4.3E Conserved Nonconserved
Conserved or
accelerated Conserved Conserved
Hi-C eigenvalue
compartment 4.3F Closed Closed Mixed Open Open
Gene density 4.3A Low Low High High High
Gene expression 4.3B N/A N/A Repressed Average Increased
Lamin 4.4 Core Core Boundaries Flanks Flanks
Table 4.2: Summary of learned domain types.
labels than we expected to be supported in the data (eight), then manually merged labels that we
deemed to be redundant.
We compared our annotation to eight types of features (Table 4.2). On the basis of these analyses,
we merged labels that appeared redundant and assigned names to each integer label (or group of
labels) that best match our interpretation of their function. This procedure yielded five types of
domains: (1) broad expression (BRD), (2) specific expression (SPC), (3) facultative heterochromatin
(FAC), (4) constitutive heterochromatin (CON), and (5) quiescent (QUI). We describe the analyses
that led us to these names in the following sections.
In order to understand how domains change state between cell types, we additionally annotated
eight cell types using twelve data sets present in all eight types and a GBR graph representing
common 3D contacts generated by combining IMR90 and H1-hESC Hi-C data sets (Methods,
Supplementary Table 4.2). This strategy of combining Hi-C data sets is motivated by the consistency
of Hi-C across cell types (Supplementary Figure 4.3). Again, we used eight labels and merged
redundant labels, to which we assigned the same five names. We investigated the properties of these
five domain types.
72
4.2.5 Repressive domains are divided into constitutive, facultative and quiescent heterochromatin
Previous studies have reported two types of repressive domains. The first type, best known as
“constitutive heterochromatin” but sometimes referred to simply as “heterochromatin”, is regulated
by the HP1 complex and associated with the histone modification H3K9me3 [Lachner et al., 2003].
Constitutive heterochromatin is thought to repress permanently silent regions such as centromeres
and telomeres. As expected, one output domain type “CON” exhibits all the known properties
of constitutive heterochromatin. CON domains are associated with H3K9me3 (Figure 4.2A,D),
are extremely depleted for genes (Figure 4.3A), are associated with low GC content and lack of
evolutionary conservation (Figure 4.3D,E), appear within the Hi-C eigenvector closed compartment
(Figure 4.3F, Methods), and cover regions which are constitutively late replicating in all cell types
(Figure 4.3G). CON domains are depleted both for transcription factor motifs and for transcription
factor binding at motifs (Figure 4.3H,I).
The second known type of repressive domain is best known as “facultative heterochromatin” but
is also sometimes referred to as BLOCs or Polycomb-repressed chromatin [Morey and Helin, 2010,
Pauler et al., 2009]. Facultative heterochromatin is regulated by the Polycomb complex and is asso-
ciated with the histone modification H3K27me3. Facultative heterochromatin is thought to repress
tissue-specific genes in cells where they are inactive. As expected, one output domain type “FAC”
has all the known properties of facultative heterochromatin. FAC domains are marked by H3K27me3
(Figure 4.2A,D), and they are enriched for genes (Figure 4.3A), GC content (Figure 4.3D) and
conservation (Figure 4.3E), but strongly depleted for gene expression relative to an average across
cell types (Figure 4.3B), indicating that FAC domains have a direct repressive effect. FAC domains
are mixed between the open and closed compartments, indicating that facultative repression is
independent of compartment-driven repression (Figure 4.3F). However, FAC domains are almost
completely absent from the annotation of the embryonic stem cell line H1-hESC, consistent with
previous observations that H3K27me3 does not form domains in embryonic stem cells but rather
occurs only at so-called poised or bivalent promoters (Supplementary Figure 4.4) [Bernstein et al.,
2006].
73
Other semi-automated genome annotation analyses have reported a third type of repressive
domain, characterized by a lack of signal from any mark, termed “quiescent domains” [Hoffman
et al., 2012, Ernst and Kellis, 2010, Filion et al., 2010, Julienne et al., 2013]. We identified this
domain type as the QUI label (Figure 4.2A). Note that Segway marginalizes over missing data
rather than setting the values to zero (Section 4.4.3), so the QUI label is not simply an artifact of
unmappable regions. QUI domains are highly depleted for genes (Figure 4.3A) and occur in the
closed compartment (Figure 4.3F). QUI domains are depleted for transcription factor motifs but,
unlike FAC and CON domains, are not depleted for transcription factor binding at motifs, indicating
that QUI chromatin does not have a direct repressive effect (Figure 4.3H,I). The mechanism behind
the activity of QUI domains is unknown, but these results are consistent with a model in which QUI
domains lack any activating signals but are not directly repressed.
4.2.6 Active domains are divided between broad and specific gene expression
Previous studies of human domains have focused on various types of repressive domains but have
assigned all active chromatin to one domain category [Julienne et al., 2013, Wen et al., 2009, Pauler
et al., 2009]. However, studies in other organisms have reported multiple types of active domains
[Filion et al., 2010, Liu et al., 2011]. We therefore investigated whether our IMR90 annotation can
be used to identify types of human active domains. We found that active domains in IMR90 can be
split into BRD (“broad expression”) domains, characterized by transcription-associated marks such
as H3K36me3, and SPC (“specific expression”) domains, characterized by regulatory marks such as
H3K27ac. Both domain types are highly enriched for genes (Figure 4.3A). However, while genes in
BRD domains are mostly expressed across all cell types, a much larger fraction of active genes in
SPC domains are expressed only in a small number of cell types (Figure 4.3C). Furthermore, when
a gene is in a SPC domain, that gene is expressed at a much higher level than that gene’s average
across cell types, suggesting that SPC domains are highly activating (Figure 4.3B). In contrast, while
genes in BRD domains are highly expressed, this high expression generally occurs consistently
across cell types, indicating that BRD domains do not necessarily directly promote expression
(Figure 4.3B). Moreover, while both BRD and SPC domains are generally early-replicating in
74
IMR90, regions covered by SPC domains typically switch replication time between cell types, while
regions covered by BRD domains are typically early replicating in all cell types (Figure 4.3G).
These results suggest a model in which genes performing housekeeping functions such as DNA
repair have strong promoters but little other regulation, whereas genes specific to a given tissue
are regulated by a complex web of regulatory elements, allowing the genome to specify precise
conditions under which the gene is active.
To test this hypothesis, we computed the enrichment of Gene Ontology (GO) terms for genes
in BRD and SPC domains respectively [Gene Ontology Consortium, 2000a, Boyle et al., 2004].
We found that genes in BRD domains were enriched for housekeeping functions such as cell cycle
and DNA repair, while genes in SPC domains were enriched for IMR90-specific developmental
functions such as vasculature development and stimulus response (Supplementary Tables 4.3–
4.4, Supplementary Figure 4.5). In order to avoid hindsight bias, before looking at these GO
term enrichments, we mixed the enriched terms with an equal number of decoy terms matched
according to the number of genes associated with each term, and manually labeled which terms
matched our hypothesized functions for each domain (housekeeping for BRD, IMR90-specific for
SPC). We correctly identified 21/32 BRD enrichments (p = 0.055) and 54/64 SPC enrichments
(p = 1.4 × 10−6). This demonstrates that active regions can be divided into domains of broadly-
expressed housekeeping genes and domains of specifically-expressed developmental genes. To our
knowledge, this is the first time a split between domains of broad and specific expression has been
reported in human cells.
4.2.7 Lamina association is driven by a complex structure of domains
Previous work has shown that some repressive domains are marked with the histone modification
H3K9me2, associate with the factor lamin B1 and localize to the nuclear lamina [Guelen et al., 2008,
Wen et al., 2009]. We found that comparing lamina association to domain annotations based on
many data sets reveals a much more complex interaction than comparing to each mark individually
(Figure 4.4). As expected, repressive domains (QUI and FAC) are enriched inside lamina-associating
chromatin domains, while active domains are depleted. However, this analysis also reveals that
75
CON domains are depleted immediately inside lamina-associating domain boundaries while being
comparatively enriched at their centers. In contrast, FAC domains are highly enriched at lamina-
associating domain boundaries while being comparatively depleted at their centers. In addition,
while active domains (SPC and BRD) are depleted inside lamina-associating domains, they are
highly enriched directly outside their boundaries. These observations suggest that lamina-associating
domains form around a core of repressed chromatin and spread until they hit a strong active element.
4.2.8 Developmentally-consistent domain boundaries are marked by identifiable sequence ele-
ments
Previous research has shown that domain boundaries tend to be consistent between cell types even
when the state of the domain changes. For example, when a region’s replication time is perturbed
by leukemia, the boundaries of the resulting replication domain tend to occur at the same positions
as developmental replication timing domain boundaries [Ryba et al., 2012]. However, the cause of
these consistent domain boundaries remains unclear. We investigated the consistency of domain
boundaries using our domain annotations. As expected, domain boundaries frequently occurred
at consistent positions across cell types, even when the domains’ state changed (Figure 4.5A). To
identify these consistent domain boundaries, we combined all boundaries occurring in at least one
cell type and merged boundaries within 50 kb. We defined groups of five or more boundaries
as consistent (Methods) (Figure 4.5B). As expected, these consistent boundaries are enriched for
replication domain boundaries, but many consistent domain boundaries do not overlap a replication
domain boundary (Supplementary Figure 4.6). We additionally found that consistent domain
boundaries are highly enriched for promoters and CTCF motifs, suggesting that these elements may
drive domain boundary formation (Figure 4.5C,D).
76
4.2.9 Using GBR to transfer information between cell types improves accuracy of predicting
functional elements
Graph-based regularization can also be used for the seemingly-unrelated task of transferring in-
formation from well-studied cell types for the annotation of cell types with limited available data
(Figure 4.6A). Existing SAGA methods work well on data from a single cell type, but integrating
information between cell types remains an open problem. Existing methods for using data from
multiple cell types for genome annotation fail to effectively address this problem (Supplementary
Note 4.6.1). We propose a novel strategy for leveraging information from well-studied cell types
using the pairwise prior that if two positions received the same label in many well-studied cell types,
then they should be more likely to receive the same label in the target cell type (Figure 4.6A). To ex-
press this pairwise prior, we first perform a Segway annotation (without GBR) of each well-studied
cell type and create a GBR graph which connects each pair of positions with weight proportional
to the number of cell types in which the pair receive the same label, placing higher weight on cell
types similar to the cell type of interest (Methods). We then use this graph in combination with the
data sets available in the target cell type to produce an annotation of this cell type. Note that this
GBR graph represents an entirely different type of information from the graphs used to represent
Hi-C data in the previous sections, despite the fact that both types of data are represented as a graph.
To demonstrate the efficacy of this approach, we evaluated whether GBR improves an annota-
tion’s ability to predict enhancers and insulators. We simulated the case where the lymphoblastoid
cell type GM12878 has only eight histone modifications available, a panel of data types similar to
that assayed by Roadmap Epigenomics on hundreds of human tissues (Supplementary Note 4.2).
Because there are enough well-studied cell types to ensure that at least one reference is reasonably
closely related to any cell type of interest, we used the related leukemia cell type K562 as reference.
We annotated GM12878 using these eight histone modifications and a GBR graph derived from an
annotation of K562 (Methods). Incorporating information from K562 this way greatly improved
the accuracy with which the annotation detected enhancers and insulators (Figure 4.6B,C). We
evaluated the performance with which the GM12878 annotation predicts a certain type of functional
77
element by ordering the labels by their enrichment for the element on a training set and evaluating
the recall as more labels are added (Methods). The GBR annotation detects one third of EP300
binding sites (a proxy for enhancers [Visel et al., 2009]) by predicting just 25 kb as EP300-binding,
while the annotation produced without GBR predicts 43 kb before it detects this many sites (Fig-
ure 4.6B). Likewise, the GBR annotation detects one third of CTCF binding sites (a proxy for
insulators [Burgess-Beusse et al., 2002]) by predicting 124 kb, compared to 241 kb without GBR
(Figure 4.6C). Because the algorithm was not given any knowledge of enhancers or insulators as
input, it is reasonable to expect that the annotations achieve similar performance at detecting other
types of functional elements, for which we do not have gold-standard examples and therefore cannot
evaluate against. These results demonstrate that GBR effectively leverages information from a
reference cell type and therefore provides a method for producing high-quality annotations of the
hundreds of cell types with limited available data.
78
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
A
B
C
D
A B C D
Chromatin 3D 
structure
A
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A B C D
Hi-C contact 
matrixGBR!
graph
Signal!
tracks
Hi-C
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Graph weight
0.1 10. 000
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0.01 10 1e6 e
Av
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re
lat
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to
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log
ica
l d
om
ain
 a
gr
ee
m
en
t
Av
er
ag
e 
re
lat
ive
 st
de
v e
xp
lai
ne
d
Figure 4.1: (A) Strategy for incorporating Hi-C data using GBR. (B) Effect of GBR hyperparameters
on topological domain agreement. The X axis indicates the value of the graph weight hyperparameter
λG. Y axis indicates the average over 10 annotations (one for each input histone modification data
set) of the fraction improvement in topological domain label agreement by adding GBR over
Segway without GBR. (C) Same as (B), but Y axis indicates improvement in replication timing
standard deviation explained and average is over 29 annotations. Annotations used four labels, 10kb
resolution.
79
0.0
0.1
0.2
0.3
QU
I
CO
N
FA
C
BR
D
SP
C
label
Fr
ac
tio
n 
of
ge
no
m
e
BRD
CON
FAC
QUI
SPC
La
be
l
Segment length (MB)
0.1 1 10
Si
gn
al 
va
lue
SPC
BRD
FAC
CON
QUI
Do
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 ty
pe
1.0
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A
B C
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imr90_h3k27ac
fre
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q!
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m
e1
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9m
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as
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c!
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0a
c!
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K2
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c!
H4
K9
1a
c!
H2
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ac
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H2
BK
5a
c!
H3
K5
6a
c!
H3
K4
m
e1
!
H3
K9
ac
!
H3
K4
m
e2
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H3
K1
4a
c!
H3
K7
9m
e2
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H2
A.
Z!
H2
Ak
9a
c!
H3
K2
7m
e3
!
H3
K9
m
e3
!
RepliSeq H3K9me3 H3K36me3 H3K27me3 H3K27ac
t l t  ( bLabel
Fr
ac
tio
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of
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eq
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y
Si
gn
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va
lue
Do
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D
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UI
Figure 4.2: Statistics of domain annotation made with Segway augmented with GBR. (A) Heatmap
of association of IMR90 domain labels to IMR90 data sets used in training. (B) Fraction of
genome covered by each domain type. (C) Segment length distribution for each domain type. (D)
Label-specific histograms of values for five notable data sets in IMR90.
80
imr90_hic_compartment
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Figure 4.3: (Caption on following page.)
81
Figure 4.3: Characteristics of domain types. (A) Gene density in IMR90. (B) Gene expression
relative to average over 33 cell types, averaged over eight annotations (t-test 95% confidence interval
error bars). (C) Fraction of active genes also active in more than 15 other cell types, averaged over
eight annotations (binomial test 95% confidence interval error bars). All gene expression data is
from CAGE (Methods). (D-F) Histograms of (D) GC content, (E) conservation (PhyloP score)
[Siepel et al., 2005] and (F) Hi-C compartment eigenvalues in IMR90. Positive PhyloP scores
indicate evolutionary conservation, positive scores indicate accelerated evolution, and scores near
zero indicate neutral evolution. Hi-C compartment values are computed according to the method of
[Lieberman-Aiden et al., 2009] (Methods). Positive values indicate open compartment, negative
values indicate closed compartment. (G) Enrichment of each domain type with respect to IMR90
replication time (early vs. late) and replication time dynamics across cell types (constitutive vs.
switching) (V Dileep, F Ay, J Sima, WS Noble, DM Gilbert et al., unpublished data) (Methods). (H)
CTCF motif density for each domain type. Dashed line indicates genome-wide average. (I) Fraction
of CTCF motifs bound (overlapping a CTCF peak) in IMR90 (binomial test 95% confidence interval
error bars). Dashed line indicates average over all motifs.
QUI
CON
FAC
BRD
SPC
Lamin
0.8Mb 1Mb1Mb
0.22
-0.25
Enrichm
ent / depletion
Figure 4.4: Enrichment of each domain type with respect to lamina-associating domain boundaries.
X axis indicates position with respect to lamina-associating domains, with each domain stretched
or shortened to the median length of 0.8 Mb. Y axis indicates label enrichment or depletion
(log(obs/expected)).
82
Consistent!
boundaries
Cell type-specific!
boundaries
0.0
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2 4 6 8
Number of cell types
Fr
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ue
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y Type
Expected
Observed
A
B C D
Mb Mb
Chr11 (Mb)
30 3431 32 33
HUVEC
GM12878
IMR90
K562
H1-hESC
HeLa-S3
HEPG2
A549
v
v
v
v
Figure 4.5: (A) Example of consistent domains. (B) Distribution of number of overlapping
boundaries compared to a permutation control (binomial test 95% confidence interval error bars).
Vertical dashed line denotes consistent boundary threshold. (C,D) Cumulative density of distances
from consistent domain boundaries to the nearest (C) promoter and (D) distal CTCF motif. Distal
CTCF motifs are defined as all CTCF motifs more than 5 kb from a promoter.
83
Cell type D 
annotation
Cell type A 
annotation
Cell type A 
graph
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Ti r 3 ell 
line tracks
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Cell 
type A  
tracks
Segway
Construct 
graph
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Combine graphs from 
reference annotations
Segway
Construct 
EGBR graph
Segway + EGBR
K562
tracks
EGBR 
graph
Tier 3 cell 
line tracks
Segway +  
prior
GBR 
graph
Cell type D 
tracks
Cell type B 
graph
Cell type C 
graph
Bases (x1000)
0 100 200 300 400
0.
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
Bases
R
ec
al
l
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With EGBR
CTCF
0 20 40 60 80
0.
0
0.
1
0.
2
0.
3
0.
4
0.
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0.
6
Bases
R
ec
al
l
Without EGBR
With EGBR
P300
Bases (x1000)
A
B C
EP300
kb kb
Re
ca
ll
Re
ca
ll
Figure 4.6: (A) Strategy for using GBR to transfer information between cell types. (B,C) Perfor-
mance of predicting the locations of GM12878 (B) EP300 and (C) CTCF ChIP-seq peaks with and
without using GBR to integrate information from K562. Each plot shows the fraction of elements
detected as a function of the number of bases predicted (Methods). Results are shown on the test set
(10 Mbp). Model training and label ordering was performed on the training set (10 Mbp). The X
axis is plotted up to 1 kbp times the total number of elements. These plots can be interpreted, for
example, in the context of an enhancer validation experiment, in which case it shows how many
sequences would need to be tested in order to discover a certain number of enhancers.
84
4.3 Discussion
We introduced graph-based regularization (GBR), a method which allows probabilistic models to
integrate a pairwise prior while maintaining efficient inference. We used GBR to model chromatin
conformation data and thereby jointly model all available data types for the study of chromatin
domains. To our knowledge, this represents the first method for integrating chromatin conformation
information into SAGA methods without resorting to simplifying transformations. We showed that
modeling Hi-C data with GBR improved the annotation’s ability to predict replication time and
topological domains. In addition, because graph-based regularization is a general method, it will
likely prove useful for other applications involving dynamic Bayesian networks, such as methods
for locating genes or predicting copy number.
The ability to integrate Hi-C data into an annotation allowed us to study the relationship
between types of domains by integrating all available data into a single annotation (Figure 4.7A).
This analysis revealed a set of five domain types that encompass all previously-described domain
types: (1) quiescent domains, which lack any activity, (2) constitutive heterochromatin, which
represses permanently silent regions and is marked with the histone modification H3K9me3, (3)
facultative heterochromatin, which represses cell type-specific regions and is marked with the
histone modification H3K27me3, (4) broadly-expressed domains, which cover genes that are highly
expressed in all cell types, and (5) specifically-expressed domains, which exhibit high regulatory
activity and cover genes that are expressed in a small number of cell types.
To our knowledge, domains of specific expression (SPC) have not been identified previously
in human cells. These domains are likely the result of complex regulatory programs designed to
precisely control the condition and level of genes important for a certain cell state or function.
SPC domains are similar in some ways to dense clusters of regulatory elements important for cell
identity known as superenhancers [Whyte et al., 2013, Love´n et al., 2013]. However, there are
only small number of known superenhancers (∼300) and each is much smaller than a SPC domain
(∼10 kb compared to ∼200 kb). Therefore, SPC domains and superenhancers may result from
similar mechanisms, but on very different scales. However, the mechanisms underlying of both
85
SPC BRD
QUI FAC
H3K27me3H3K9me3 X
…
…
CON
Regulatory!
elements
A
CTCF
Nuclear lamina
Localizing !
element
Localizing !
element
CTCF
CTCF
Transcription!
factory
Co-regulated!
region
B
Figure 4.7: Models of (A) domain types and (B) co-regulated regions.
types of regions must be studied further in order to understand this relationship.
One likely mechanism of domain formation involves the spreading of heterochromatin [Weiler
and Wakimoto, 1995, Talbert and Henikoff, 2006]. Under this hypothesis, heterochromatin nu-
cleates at silencing elements such as telomeres, repeats or repressed promoters and sequentially
assembles along chromatin. Spreading heterochromatin has been demonstrated mechanistically
in Saccharomyces and Drosophila and for the SIR, HP1 and Polycomb complexes. While SIR is
unique to yeast, HP1 and Polycomb have orthologs in humans that drive constitutive and facultative
heterochromatin, respectively. Under the spreading hypothesis, heterchromatin can be halted by
the presence of a strong active element. This halting mechanism is consistent with the observation
that active domains (especially SPC) are strongly enriched directly outside of lamina-associating
domains.
The consistency of domain boundaries between cell types also suggests a model in which
core regions are regulated as a unit [Phillips and Corces, 2009, Dixon et al., 2012] (Figure 4.7B).
Under this hypothesis, these units self-interact as topological domains and are co-regulated through
availability of regulatory factors and elements such as enhancers. These co-regulated units are
thought to be delimited by localizing sequences, particularly CTCF sites. Under this model, each of
our annotated domains is actually composed of several such neighboring co-regulated regions with
the same state. Therefore, while profiling a small number of cell types has allowed us to define a
small number of consistent domain boundaries, profiling more cell types may lead to a complete
86
catalogue of potential boundary sites.
We have described five domain types because we this model allowed us to concisely summarize
domain regulation, but we do not claim that this represents the “true” number of domain types. It
is likely that new domain subtypes will be discovered in the future, thus increasing the number of
known domain types. In addition, methods that discover the optimal number of domain types or
that allow mixtures of domain types are an interesting direction for future work. To our knowledge,
all existing SAGA methods require either a fixed number of labels [Day et al., 2007, Hoffman
et al., 2012, Ernst and Kellis, 2010, Thurman et al., 2007, Lian et al., 2008, Filion et al., 2010] or a
hyperparameter that indirectly controls the number of labels [Ho et al., 2013]. A method that allows
for “mixed” domain labels at a given process could potentially circumvent the manual merging
process that we used to reduce an eight-label model to a five-label one.
Finally, we presented a method for transferring information from well-studied cell types using
GBR in order to improve the quality and interpretability of annotations of cell types with limited
available data. This method enables a new strategy for understanding cell types, in which a small
number of assays are performed on each cell type of interest to determine the unique characteristics
of this cell type, then Segway with GBR is used to combine this data with the large body of available
information from well-studied cell types. This method has the additional benefit of matching
the label semantics of the target cell types to the semantics of the reference annotations, which
allows the label interpretation process to be performed automatically. Because consortia such as
ENCODE and Roadmap Epigenomics are already analyzing a large number of cell types with a
small number of assays each, this strategy is immediately applicable. Determining which assays are
most informative as input to this strategy is an interesting question for future work.
4.4 Methods
4.4.1 Histone modification, open chromatin and replication timing signal data
We acquired histone ChIP-seq, DNase-seq, and FAIRE-seq data for A549, K562, H1-hESC,
GM12878, HeLa-S3, HepG2 and HUVEC from ENCODE and for IMR90 from Roadmap Epige-
87
nomics [ENCODE Project Consortium, 2012, Bernstein et al., 2010]. We used a uniform signal-
processing pipeline to generate a genome-wide vector for each data set, as described in [Hoffman
et al., 2013]. We also acquired Repli-seq data for IMR90 from ENCODE and smoothed this data
using wavelet smoothing as described in [Thurman et al., 2007]. We applied the inverse hyperbolic
sine transform asinh(x) = ln(x+
√
x2 + 1) to all signal data. This transform is similar to the log
transform in that it depresses the magnitude of extremely large values, but it is defined at zero and
amplifies the magnitude of small values less severely than the log transform does. This transform
has been shown to be important for reducing the effect of large values in analysis of genomics data
sets [Johnson, 1949, Hoffman et al., 2012].
We acquired transcription factor ChIP-seq data from ENCODE. Peaks were called for each
factor using MACS using an IDR threshold of 0.05 [Zhang et al., 2008, Landt et al., 2012].
We acquired CAGE expression data for 33 cell types from GENCODE [Harrow et al., 2012].
The full list of data sets used is available in Supplementary Table 4.1.
4.4.2 Hi-C data
We used publicly available Hi-C data sets for two human cell lines (IMR90 and H1-hESC) [Dixon
et al., 2012]. We processed raw paired-end libraries with a pipeline that combines reads from
two replicates per cell line, maps these reads, extracts the read pairs for which each end maps
uniquely, and removes potential PCR duplicates. We then partitioned the human genome into a
collection of non-overlapping 10 kb windows and assigned each end of a read pair to the nearest
10 kb window mid-point. This process yielded a 303,641×303,641 whole genome contact map.
These contact maps consisted of both intra and interchromosomal contacts and contained only
∼0.3% non-zero entries. We assigned statistical confidence estimates to these contact counts using
the method Fit-Hi-C, which jointly models the random polymer looping effect and technical biases
[Ay et al., 2014a]. First, we applied the bias correction method ICE to the contact map to estimate a
bias associated with each 10 kb locus, after eliminating all loci that have less than 50% uniquely
mappable bases [Imakaev et al., 2012]. Second, using these computed biases and raw contact maps
as input, we estimated a p-value of interaction for each pair of 10 kb loci with non-zero contact
88
counts (p-value was set to 1 for pairs with zero contacts). We used a slightly modified version of the
original Fit-Hi-C algorithm which handles inter- as well as intrachromosomal contacts and omits
the refinement step for fast computation. Because Fit-Hi-C normalizes for 1D genomic distance,
the majority of significant contacts were at long distances (Supplementary Figure 4.7). Note that
while the data sets we used have insufficient coverage to identify many high-confidence contacts at
10kb resolution, Segway with GBR aggregates information over roughly 400kb in order to make
each domain call, so individual high-confidence interactions are not necessary.
We computed the genome chromatin compartment using eigenvalue decomposition on the
normalized contact maps of IMR90 and H1-ESC cell lines at 1 Mb resolution as described in
[Lieberman-Aiden et al., 2009]. For each chromosome, we calculated the Pearson correlation
between each pair of rows of the intrachromosomal contact matrix and applied eigenvalue decom-
position to the correlation matrix. Similar to [Lieberman-Aiden et al., 2009], we used the second
eigenvector in cases where the first eigenvector values were either all positive or all negative to
define the compartments. We used average GC content to map signs of eigenvectors to either open
(higher GC content) or closed chromatin compartments.
4.4.3 Segway model
We used graph-based regularization to augment the Segway semi-automated genome annotation
method [Hoffman et al., 2012]. Segway uses a dynamic Bayesian network model to perform genome
annotation. The model is presented in detail in [Hoffman et al., 2012], but we describe it briefly
here.
• We define a latent label variable Yi ∈ {1..K} for each position i ∈ {1..N} in the genome,
where K is the user-specified number of labels and N is the number of positions.
• We define observed signal data variables Xi,j representing the value of signal data set j ∈
{1..M} at genomic position i, where M is the number of signal data sets. We downsample
the genome into bins of size R and average the signal data in each bin (after applying the
inverse hyperbolic sine transform), so N ≈ 3× 109/R. Because the sequencing depth of
89
existing Hi-C data sets is too low to achieve single base pair resolution, we used R = 10000
for experiments using GBR to integrate Hi-C data. We used R = 1 for experiments using
GBR to transfer information between cell types.
• The observed data variable Xi,j depends only on the label at position i, Yi. We model the
variable Xi,j as a Gaussian distribution with data set- and label-specific mean parameter µi,j
and data set-specific variance parameter σj . In the case that some data values are missing due
to mappability, we weight the observation of Xi,j by the proportion of mappable positions
X˚i,j in bin i in data set j.
• The label variable Yi depends only on the label at the previous position Yi−1. We model the
label transition from label a to label b using a transition parameter Qa,b.
• We model segment length, determined by self-transitions Qa,a, separately from label tran-
sitions, determined by Qa,b for a 6= b. The self-transition is weighted by a hyperparameter
λtransition, which weighs the importance of segment length relative to signal data.
• The parameters µ1:K,1:M , σ1:M and Q1:K,1:K are learned through EM.
The overall log-likelihood of the Segway model is defined as:
log Pr(X, Y | µ, σ,Q) =
N∑
i=1
M∑
j=1
X˚i,j logN(Xi,j | µYi,j, σj)
+ λtransition
N−1∑
i=1
1(Yi == Yi+1) logQYi,Yi
+
N−1∑
i=1
1(Yi 6= Yi+1)(λtransition log(1−QYi,Yi) + logQYi,Yi+1)
(4.1)
where µ`,j is the mean associated with signal data set j and label `; σj is the variance associated
with signal data set j (shared between all labels); X˚i,j is the proportion of mappable positions in bin
90
Segment 
label (XH)
Signal  
tracks (XO)
dependence pairwise prior (w)
…
Figure 4.8: GBR model. Squares and circles denote discrete and continuous random variables
respectively. Filled-in and unfilled shapes denote observed and unobserved variables, respectively.
i for data set j; Qa,b is the transition probability parameter from label a to label b; and λtransition is a
weight on the transitions relative to the emissions of the model.
4.4.4 Graph-based regularization
In a SAGA method, we are given a set of vertices V that index a set of n = |V | random variables
XV = {X1, . . . , Xn} and a probability distribution parameterized by θ, pθ(XH , XO). Different
SAGA methods employ different distributions pθ. Graph-based regularization could be applied
to any probabilistic model, but in this work we use the Segway model (Supplementary Methods)
because it can handle real-valued and missing data, and it can use non-geometric segment length
distributions. We denote random variables with capital letters (e.g., XH) and instantiations of
variables with lower-case (e.g., xH ∈ domain(XH)). We use capitals to denote sets and lowercase
to denote values (e.g., Xh for h ∈ H).
Training the model involves a set of observed data x¯O, where a subset of variables O ⊆ V is
observed, and the remainder H = V \O are hidden. The maximum likelihood training procedure
optimizes the objective
maximizeθ J(θ) , L(θ) +R(θ) (4.2)
where L(θ) , log pθ(x¯O) = log
∑
xH
pθ(xH , x¯O), (4.3)
91
whereR(θ) is a regularizer that expresses prior knowledge about the parameters. Many regularizers
are used in practice, such as the `2 or `1 norms, which encourage parameters to be small or sparse
respectively.
Dynamic programming algorithms such as the forward-backward algorithm can be used to
perform inference in SAGA models, because all such existing models have dependencies in the form
of a chain. That is, the variables associated with position i depend only on the variables associated
with positions i− 1 and i+ 1. Examples of such chain-structured models include hidden Markov
models and dynamic Bayesian networks. However, these dynamic programming algorithms do
not apply if a pairwise prior is added to the model, since the prior may have an arbitrary structure.
Several techniques have been proposed to handle models with arbitrary structure (Supplementary
Note 4.6.2). However, none of these techniques are optimal for expressing a pairwise prior.
Therefore, we instead employ a novel strategy based on posterior regularization [Ganchev et al.,
2010] to integrate this prior. This is done by introducing an auxiliary joint distribution q(XH),
placing a regularizer on q(XH), and encouraging q to be similar to pθ through a KL divergence
penalty. The regularizer is
RPR(θ) , max
q
R′PR(θ, q) (4.4)
R′PR(θ, q) , −D(q(XH)‖pθ(XH |x¯O)) + PR(q), (4.5)
where D(·‖·) is the KL divergence D(p(XH)‖q(XH)) =
∑
xH
p(xH) log(p(xH)/q(xH)) and
PR(q) is a posterior regularizer that expresses prior knowledge about the posterior distribution. KL
divergence measures the dissimilarity of probability distributions, such that D(p‖q) is zero if the
distributions are identical and can be arbitrarily large if they are not. Several posterior regulariz-
ers have been proposed in the past, such as those that require posteriors to satisfy constraints in
expectation [Ganchev et al., 2010].
We propose a new type of posterior regularizer that expresses a pairwise prior (Figure 4.8). We
are given a weighted, undirected regularization graph over the hidden variables GR = (H,ER),
where ER ⊆ H ×H is a set of edges with non-negative similarity weights w : ER → R+, such
92
that a large w(u, v) indicates that we have strong belief that Xu and Xv should be similar. (We
describe how we generate this graph in the next two sections.) For a distribution p(XH), let pMh (Xh)
indicate the marginal distribution over Xh, pMh (xh) =
∑
xH\h
p(xH). Let λG be a hyperparameter
controlling the strength of regularization. The posterior regularizer is
PRGBR(q) , −λG
∑
(u,v)∈ER
w(u, v)D(qMu (Xu)‖qMv (Xv)). (4.6)
Thus the full objective is
maximizeθ,q JGBR(θ) , L(θ)−D(q(XH)‖pθ(XH |x¯O))− λG
∑
(u,v)∈ER
w(u, v)D(qMu (Xu)‖qMv (Xv)).
(4.7)
We term this strategy of adding graph-based penalties graph-based regularization (GBR). See
Chapter 3 for more detail.
4.4.5 GBR optimization
We have developed a novel algorithm for efficiently optimizing JGBR in q. This algorithm alternates
between using a method for probabilistic inference such as the forward-backward algorithm and
applying a message passing algorithm over the regularization graph GR. In the inference step,
the model receives evidence from the message passing step in the form of a “virtual evidence”
distribution rMh (Xh) over each variable h. These virtual evidence distributions are used in conjunc-
tion with the original SAGA model to compute a posterior distribution over the labels using any
algorithm for probabilistic inference on dynamic Bayesian networks, such as belief propagation or
the forward-backward algorithm.
In the message passing step, the algorithm updates rM to minimize the Kullback-Leibler
penalties in the objective function JGBR. This message passing step is itself performed using
an alternating optimization algorithm, which passes messages over the regularization graph GR.
This algorithm is similar to one originally developed for the field of semi-supervised learning
93
[Subramanya and Bilmes, 2011].
The inference and message passing steps are iterated until convergence. These two updates are
linear in the number of variables (for chain-structured models, which include all existing SAGA
methods) and linear in the degree of the regularization graph, respectively. The algorithm exhibits
monotonic convergence, similar to the EM algorithm. We derive the algorithm for optimizing JGBR
and prove its convergence in Chapter 3.
4.4.6 GBR graph for incorporating Hi-C data
When we are using GBR to incorporate Hi-C data, we are given a matrix of contact p-values
P ∈ Rn×n, generated from a matrix of contact counts as described above. To remove noise and
decrease the degree of the graph, we removed all contacts with uncorrected p-value p > 10−6
and multiplied the remaining p-values by 106, similar to a Bonferroni correction. Note that due to
the large number of hypotheses, performing a full Bonferroni correction would result in very few
contacts. Moreover, the graph weights allow the algorithm to take into account the strength of each
connection, so the choice of 106 was made for computational, not statistical, reasons. We computed
the weights as
w(i, j) , max(0,− loge(p(i, j)/106)). (4.8)
As with the graph for transferring information between cell types, the multiplicative scale of the
weights is arbitrary, since it is controlled by the graph weight hyperparameter λG. We used only
intrachromosomal contacts for forming the GBR graph. To produce a GBR graph representing cell
type-consistent chromatin conformation used in the domain annotation of eight cell types, we added
the edge weights from the IMR90 and H1-hESC Hi-C GBR graphs.
4.4.7 GBR graph for annotation of multiple cell types
When we are using GBR to transfer information about cell type A to improve annotation of cell
type B, we are given an annotation aA1:n ∈ {1 . . . k}n of cell type A, produced without GBR. We
construct a GBR graph from this annotation by connecting each pair of positions that received the
94
same label in aA with an undirected edge of weight 1. Note that the weight is arbitrary, since it
is scaled by the regularization parameter λG. To mitigate the problem of quadratic growth in the
degree of this graph, we randomly subsampled this graph such that each node had outgoing degree
17 ≈ loge(n). We chose this graph degree because a randomly-subsampled graph with n loge n
edges has the same connected components as the full graph with high likelihood [Erdo˝s and Re´nyi,
1960], and our experiments on synthetic data (not shown) showed that the sparse graph performed
similarly to a complete graph.
4.4.8 Circular permutation
As a null model for several experiments, we performed a circular permutation of the genome along
each chromosome arm as follows. We randomly choose a translation fraction θ ∈ [0, 1]. For each
coordinate i ∈ {1 . . . n} within a chromosome arm that spans the range [a, b), we translate i to t(i),
where
t(i) = mod(b−a)(i+ bθ(b− a)c) + a. (4.9)
To circularly permute a genome feature, such as an annotation or a Hi-C contact map, we translate
each element from position i to t(i). Thus, when a circularly-permuted feature is compared to an
un-permuted feature, all positional correspondence between permuted and un-permuted features
are removed, but each feature’s spatial patterns are preserved. In each case, we performed this
permutation 200 times and report the average over all permutations. If the feature includes any
centromere- or telomere-defined elements, we remove these as a preprocessing step.
4.4.9 Topological domain agreement
To evaluate the degree to which an annotation A matches a set of topological domains, we computed
the number cd,` of bases by which domain d is covered by label `. We then computed c∗d = max` cd,`
to be the number of bases covered by the highest-coverage label for domain d, and divided c∗d by the
length of d to produce f ∗d , the fraction of d covered by its plurality label. The agreement f
∗
d takes its
maximum value of 1 if the domain d is covered by exactly one annotation label. We computed the
95
raw genome-wide agreement fraw = (1/|d|)
∑
d f
∗
d .
This raw genome-wide agreement fraw can be improved simply by increasing the length of
segments and decreasing the number of labels. Therefore, we circularly permuted A to form Ap,
and used this permuted annotation to compute fpraw. Finally, we computed the topological domain
agreement a = fraw/fpraw as the ratio of unpermuted and permuted raw agreements. This normalized
agreement is large when the annotation has small segments that exactly match the topological
domains and is small when the annotation’s segments are not correlated with topological domains.
4.4.10 Signal variance explained
To evaluate the similarity between a genome-wide signal vector and a genome annotation, we use
the following measure, which we term the variance explained. We are given a genome annotation
with k labels a1:n ∈ {1 . . . k}n and a vector x1:n ∈ Rn. We compute the signal mean over the
positions assigned a given label ` as
µ` ,
∑n
i=1 1(ai = `)xi∑n
i=1 1(ai = `)
for ` ∈ {1 . . . k}. (4.10)
We define a predicted signal vector xpi = µai and compute the the prediction error as di = xi − xpi .
We compute the residual standard deviation of the signal vector as
s , stdev(d1:n) =
√√√√ 1
n
n∑
i=1
(di −mean(d1:n))2 =
√√√√ 1
n
n∑
i=1
d2i (4.11)
The last equality holds because mean(d1:n) = 0 by construction. We define the variance explained
(VE) for annotation a and signal vector x as VE , stdev(d1:n)− stdev(x1:n). VE is bounded by the
range [0, stdev(x1:n)]. VE is a measure of the extent to which a genome-wide signal data set and
annotation are similar, where higher values indicate better agreement.
96
4.4.11 Genomic element prediction
We form a classifier for a set of genomic elements based on an annotation using the following
strategy. We are given a genome annotation with k labels a1:n ∈ {1 . . . k}n and a set of positions
S ⊆ {1 . . . n} which represent some set of elements of interest, such as enhancers or CTCF binding
sites. Define A` = {i | ai = `} to be the positions annotated by label ` To avoid biases caused
by differing-size elements, we assume that each element occupies just 1 bp. In the case of larger
elements (such as MACS-called TF binding sites, which are ∼200 bp), we define each element as
the middle base pair of the range.
For each label, we compute the predictive precision of label ` as
precision(`) =
|S ∩ A`|
|A`| for ` ∈ {1 . . . k}. (4.12)
We rank the labels in decreasing order of their precision on a training set to get an order s1:k ∈
{1 . . . k}k.
Using this ordering, we form k predictors, Pj =
⋃j
i=1Asi for j ∈ {1 . . . k}. The true positives
and false positives of a predictor P are TP(P ) = P∩S and FP(P ) = P∩({1 . . . n}\S) respectively.
The predictors are in order of decreasing stringency—that is, Pj−1 ⊆ Pj .
We can trace out the full sensitivity-specificity tradeoff (such as for an ROC or PR curve), by
interpolating between each successive pair of predictors. To interpolate between a pair of predictors
Pj and Pj+1, we form an interpolated predictor Pj,j+1,θ by sampling each position i ∈ Pj \Pj−1 with
probability θ ∈ [0, 1]. The expected number of true positives and false positives of an interpolated
predictor Pj,j+1,θ can be shown to be
E[|TP(Pj,j+1,θ)|] = |TP(Pj)|+ θ|TP(Aj+1)| and (4.13)
E[|FP(Pj,j+1,θ)|] = |FP(Pj)|+ θ|FP(Aj+1)|, (4.14)
respectively. We report our performance using a test set disjoint from the training set used to order
the labels.
97
4.4.12 Developmental replication domains
In order to evaluate the replication timing dynamics of different types of domains, we used a
four-label (constitutive early/late, switching early/late) annotation of the human genome using
published replication timing data for 16 different human cell types, gathered by V Dileep, F Ay,
J Sima, WS Noble, DM Gilbert et al. (unpublished). This annotation first windowed replication
timing data into 40 kb bins and then determined for each window whether it replicates early (RT
value > 0.5) or late (RT value < -0.2) in all cell types. Such windows with consistent timing profiles
across all cell types were labeled as “constitutively early” and “constitutively late”, respectively.
The remaining windows were labeled as either switching or left unlabeled. Switching windows are
determined as those with an absolute value of replication timing larger than 0.5 in all cell types but
with an opposite sign than others in at least one cell type. Switching windows that are early and late
replicating in IMR90 were labeled as switching early and switching late, respectively.
4.4.13 Consistent domain boundaries
When we annotated domains in eight cell types, we found that domain boundaries were shared be-
tween annotations much more often than would be expected by chance. To identify developmentally-
consistent domain boundaries, we first formed a list of all segment boundaries that occurred in at
least one cell type. For each boundary, we computed the number of cell types with boundaries within
50 kb. We formed a set of representative by greedily selecting the boundary with the most nearby
boundaries as a representative, removing all boundaries near the representative from the list, and
repeating the process until no two boundaries in the list were within 50 kb of one another. While
this problem is an instance of the NP-hard set cover problem, the greedy approach is guaranteed
to result in a constant-factor approximation of optimal [Nemhauser et al., 1978a]. This yielded a
set of 13,906 boundary groups, each more than 50 kb from all other groups. We defined the 2,967
boundary groups composed of at least five boundaries as consistent boundaries.
98
4.5 Data access
Domain annotations and code for Segway with GBR is available as Supplemental Material and
online at http://noble.gs.washington.edu/proj/gbr.
4.6 Supplementary Notes
4.6.1 Review of existing SAGA methods for using data from multiple cell types
Existing methods for semi-automated genome annotation work well on data from a single cell
type, but annotating multiple cell types remains an active area of research. There are three simple
strategies for performing annotation of multiple cell types. First, the simplest strategy is to apply
the same model to both genomes (sometimes called “concatenated” annotation) [Sheffield et al.,
2013], but this requires that all cell types have the same set of available data, which is not generally
true. Moreover, in practice, experimental artifacts lead to poor performance for models which
model multiple data from multiple experiments with the same parameters, exhibiting effects such
as assigning separate sets of labels to each cell type in the model. Second, one could perform
annotation separately on each cell type and find a mapping between the labels (for example, by
using the Hungarian algorithm [Kuhn, 1955]). However, since different cell types generally have
different types of activity and different sets of signal data sets, such a mapping is generally very
poor. Third, one could use all data from all cell types in one model (sometimes called “stacked”
annotation), but this strategy must either give the same label to each position for every cell type or
use a separate label for each pattern of labels across cell types, which requires an exponentially-large
number of labels.
Two additional methods have been proposed to annotate multiple cell types. The first, called
hiHMM (“hierarchically-linked infinite HMM”) maintains a separate model for each cell type and
uses a regularization penalty to encourage the models to have similar parameters [Ho et al., 2013].
This addresses the problem of requiring the same set of data across cell types, but does not share any
position-specific information between cell types. The second method for performing multi-cell-line
annotation, called TreeHMM, is given a tree over cell types and models the transition between
99
labels between neighboring positions and also between neighboring developmental states [Biesinger
et al., 2013]. This model can integrate position-specific information between cell types, but requires
that each cell type has the same available data and is sensitive to any cross-experiment artifacts.
Moreover, the complexity of this model forced the authors to resort to approximate methods for
inference, which likely decreases the quality of the resulting annotations.
These two problems—requiring a common set of data and failure to integrate evidence—are
especially important because, although there are virtually limitless cell types and cell states that
one would like to understand, very limited numbers of experiments have been performed in most of
these cell types due to the cost of genomic experiments. For example, ENCODE has performed 335
experiments in its most-studied cell type, but has performed just 2-10 experiments in more than 100
cell types.
Transferring information with GBR removes the requirement for a common set of data across
cell types and does integrate position-specific evidence across cell types. Therefore, GBR provides
a method leveraging all available data in order to produce high-quality annotations of each cell type.
4.6.2 Related optimization methods
Clearly, the most straightforward way to express pairwise interactions in a graphical model is to
encode them in the underlying graph and to use approximate methods (reviewed in [Wainwright and
Jordan, 2008]) to enable inference. This form of interaction is quite general, in that when one adds a
factor φ(yi, yj) between two random variables Yi and Yj , these random variables may have any type
of interaction, expressed by φ(yi, yj). GBR, on the other hand, asks only for similarity between the
marginals, meaning that p(yi|·) and p(yj|·) should be similar. Alternatively, a factor could encode
such similarity, for example if φ(yi, yj) = λ1(yi = yj). Such factors added to an HMM or CRF
would result in a high treewidth model that can be dealt with using approximate inference. Doing
so, however, loses any guarantee of optimality (which we preserve with GBR).
The posterior regularization framework of Ganchev et al. [Ganchev et al., 2010] takes an ap-
proach similar to ours, augmenting a simple model in a way that maintains tractable inference. This
method adds a regularization term to an EM objective in order to require the posterior probabilities
100
to satisfy logical constraints in expectation. Ganchev et al. show how to optimize this combined
objective efficiently when the regularization term is linear in the posterior distribution of the model.
Unfortunately, pairwise similarity relationships cannot be expressed with such a linear regularization
term.
The most similar work to ours are the following three methods for graph regularization. First,
Altun el al. [Altun et al., 2005] describe a graph regularization applied to a max-margin model
applied to pitch-accent prediction and optical character recognition. However, this method involves a
matrix inversion step, and thus cannot scale to large models. Second, Subramanya et al. [Subramanya
et al., 2010] combine a temporal CRF with a regularizer that expresses pairwise squared-error
penalties derived from unlabeled data. They apply this method to the part-of-speech tagging task
[Subramanya et al., 2010] and later to related problems in natural language [Das and Petrov, 2011,
Das and Smith, 2011]. That work, however, resorts to a purely heuristic update step, and lacks
any optimality guarantees. Third, He et al. [He et al., 2013] present an approach based on an
exponentiated gradient descent algorithm. Like our approach, He’s approach exhibits monotone
convergence. Although He’s work has many similarities with our approach, our methods were
developed independently, and He’s work differs from ours in three important ways. First, He
et al. [He et al., 2013] use an exponentiated gradient descent strategy, while we use alternating
minimization. Second, He’s method uses a squared-error penalty, which is inappropriate for
probability distributions, unlike our use of the Kullback-Leibler divergence [Bishop, 1995, p. 226].
Third, the exponentiated gradient descent method is applied to handwriting recognition and part-of-
speech tagging, while we apply GBR to genome annotation.
4.7 Supplementary Figures/Tables
101
Data type URL
CAGE http://www.gencodegenes.org/releases/7.html
ENCODE (ChIP-seq, DNase, Replication timing) http://hgdownload.cse.ucsc.edu/goldenPath/hg19/encodeDCC/
ChIP-seq (Roadmap) http://www.roadmapepigenomics.org/data
Hi-C http://yuelab.org/hi-c/download.html
Topological domains http://www.cs.cmu.edu/∼ckingsf/software/armatus/
Supplementary Table 4.1: Data sources
IMR90 domain annotation Eight-cell type domain annotation GM12878 reduced annotation
Input data sets
DNase
H2aK5ac
H2aK9ac
H2a.Z
H2bK120ac
H2bK12ac
H2bK15ac
H2bK20ac
H2bK5ac
H3K14ac
H3K18ac
H3K23ac
H3K27ac
H3K27me3
H3K36me3
H3K4ac
H3K4me1
H3K4me2
H3K4me3
H3K56ac
H3K79me1
H3K79me2 H3K9ac
H3K9me1
H3K9me3
H4K20me1
H4K5ac
H4K8ac
H4K91ac
Repli-seq
DNase
H2a.Z
H3K27ac H3K27me3
H3K36me3
H3K4me1
H3K4me2
H3K4me3
H3K79me2
H3K9ac
H3K9me3
H4K20me1
H3K4me1
H3K4me2
H3K3me3
H3K9ac
H3K27ac
H3K27me3
H3K36me3
H4K20me1
Number of labels 8 8 25
Transition weight (λtransition) 48 12 1
Number of random EM initializations 10 10 10
GBR graph scale (λG) 1 1 1
GBR optimization hyperparameter (λR1) 1 1 10
Supplementary Table 4.2: Parameters of all genome annotations
102
GOID Bonferroni-corrected p-value Name
GO:0008150 5.44809636039661e-49 biological process
GO:0070647 5.36690477736568e-14 protein modification by small protein conjugation or removal
GO:0016567 2.53980288482968e-13 protein ubiquitination
GO:0032446 3.48958938873894e-13 protein modification by small protein conjugation
GO:0071840 1.2816846277191e-07 cellular component organization or biogenesis
GO:0048522 3.15663523300463e-07 positive regulation of cellular process
GO:0016043 1.55013794212494e-06 cellular component organization
GO:0050953 2.07559208909527e-06 sensory perception of light stimulus
GO:0007601 2.79978686922173e-06 visual perception
GO:0006996 2.18396515274592e-05 organelle organization
GO:0016071 7.40708736771575e-05 mRNA metabolic process
GO:0048519 0.000123587368672724 negative regulation of biological process
GO:0023056 0.000191430490530164 positive regulation of signaling
GO:0048518 0.000239550248137195 positive regulation of biological process
GO:0032270 0.000250487526217915 positive regulation of cellular protein metabolic process
GO:0018146 0.000336577942863192 keratan sulfate biosynthetic process
GO:0007005 0.000402299808664309 mitochondrion organization
GO:0010647 0.000414643467676553 positive regulation of cell communication
GO:0032268 0.000530771233104358 regulation of cellular protein metabolic process
GO:0043928 0.000623789428174407 exonucleolytic nuclear-transcribed mRNA catabolic process involved in deadenylation-dependent decay
GO:0000288 0.000670978259119087 nuclear-transcribed mRNA catabolic process, deadenylation-dependent decay
GO:1903320 0.00117566239446111 regulation of protein modification by small protein conjugation or removal
GO:0009967 0.00131477493050438 positive regulation of signal transduction
GO:1902533 0.00136644459633516 positive regulation of intracellular signal transduction
GO:0048523 0.00148158842395948 negative regulation of cellular process
GO:0051340 0.00150521495507962 regulation of ligase activity
GO:0000291 0.00193533449146964 nuclear-transcribed mRNA catabolic process, exonucleolytic
GO:0031401 0.00194625332376889 positive regulation of protein modification process
GO:1903047 0.00215993757988771 mitotic cell cycle process
GO:0042531 0.00237481842965934 positive regulation of tyrosine phosphorylation of STAT protein
GO:0007267 0.0039286364798486 cell-cell signaling
GO:0006401 0.00399526165008678 RNA catabolic process
GO:0031396 0.00443702013802013 regulation of protein ubiquitination
GO:0000278 0.00470584872871359 mitotic cell cycle
GO:0051351 0.00495154943903681 positive regulation of ligase activity
GO:0051438 0.00524041402868326 regulation of ubiquitin-protein transferase activity
GO:0042339 0.0053487952592411 keratan sulfate metabolic process
GO:0051247 0.0056999635780996 positive regulation of protein metabolic process
GO:0016265 0.00629144608895318 death
GO:0046427 0.00671456368410498 positive regulation of JAK-STAT cascade
GO:0008219 0.0067422659494943 cell death
GO:0000956 0.00698342075165014 nuclear-transcribed mRNA catabolic process
GO:0031399 0.00786100228743956 regulation of protein modification process
GO:0044770 0.00795426575038111 cell cycle phase transition
GO:0051246 0.00937868333257509 regulation of protein metabolic process
Supplementary Table 4.3: GO terms enriched for genes in BRD domains
103
GOID Bonferroni-corrected p-value Name
GO:0001944 2.9592753850599e-11 vasculature development
GO:0001568 1.34928597594565e-09 blood vessel development
GO:0072358 1.87785996344277e-09 cardiovascular system development
GO:0072359 1.94865728031881e-09 circulatory system development
GO:0048598 2.33761516882996e-09 embryonic morphogenesis
GO:0032501 4.29504418805107e-09 multicellular organismal process
GO:0044707 6.45108090271442e-09 single-multicellular organism process
GO:0007275 9.54873387980616e-09 multicellular organismal development
GO:0032502 9.87403900226897e-09 developmental process
GO:0009653 1.13900576119261e-08 anatomical structure morphogenesis
GO:0048646 1.27560082632202e-08 anatomical structure formation involved in morphogenesis
GO:0048856 3.44027824649974e-08 anatomical structure development
GO:0048514 3.64379056813118e-08 blood vessel morphogenesis
GO:0044767 4.96188838214423e-08 single-organism developmental process
GO:0008150 1.09920334986419e-07 biological process
GO:0009888 1.15062070196739e-07 tissue development
GO:0048731 1.34005756992352e-07 system development
GO:0030198 2.54744657740361e-07 extracellular matrix organization
GO:0043062 2.70008497648116e-07 extracellular structure organization
GO:0040011 4.09681657551741e-07 locomotion
GO:0048523 6.00924738320409e-07 negative regulation of cellular process
GO:0048513 1.27555684817328e-06 organ development
GO:0009887 1.67228271838845e-06 organ morphogenesis
GO:0060429 1.75248481849761e-06 epithelium development
GO:0009605 2.12388368291444e-06 response to external stimulus
GO:0048519 2.70812717394184e-06 negative regulation of biological process
GO:0048869 3.99639769208689e-06 cellular developmental process
GO:0009790 4.8142081410924e-06 embryo development
GO:0030154 8.17799427787657e-06 cell differentiation
GO:0048522 8.79037004023573e-06 positive regulation of cellular process
GO:0048870 9.70748540194382e-06 cell motility
GO:0051674 9.70748540194382e-06 localization of cell
GO:0048518 9.8762526690946e-06 positive regulation of biological process
GO:1902533 1.75047286259699e-05 positive regulation of intracellular signal transduction
GO:0071840 2.37299292646679e-05 cellular component organization or biogenesis
GO:0030334 2.38844411436045e-05 regulation of cell migration
GO:0007389 3.41057645381657e-05 pattern specification process
GO:0051270 4.15793772024666e-05 regulation of cellular component movement
GO:2000145 7.20700858370221e-05 regulation of cell motility
GO:0040012 7.2398136159996e-05 regulation of locomotion
GO:0016043 7.57778182419685e-05 cellular component organization
GO:0001501 9.16531463777006e-05 skeletal system development
GO:0008219 0.000103725100284498 cell death
GO:0001525 0.0001112117619674 angiogenesis
GO:0009966 0.000112414145595174 regulation of signal transduction
GO:0016265 0.000115956206339428 death
GO:0009967 0.000119265054523972 positive regulation of signal transduction
GO:0016477 0.000141195508435494 cell migration
GO:0048568 0.000148183903115669 embryonic organ development
GO:0001503 0.000148474000627697 ossification
GO:0006915 0.000174710834263385 apoptotic process
GO:0048729 0.000185804853094102 tissue morphogenesis
GO:0012501 0.000193299675551627 programmed cell death
GO:0010647 0.000264669391358318 positive regulation of cell communication
GO:0003007 0.000308615021011459 heart morphogenesis
GO:0010628 0.000312006014076936 positive regulation of gene expression
GO:0023056 0.000458831197316147 positive regulation of signaling
GO:0051239 0.000694243253157008 regulation of multicellular organismal process
GO:0031325 0.000823302329284296 positive regulation of cellular metabolic process
GO:0002009 0.000843967242091738 morphogenesis of an epithelium
GO:0048863 0.000980998073137653 stem cell differentiation
GO:0042325 0.00125849445147922 regulation of phosphorylation
GO:1902531 0.00144207328720399 regulation of intracellular signal transduction
GO:0044236 0.00151180971326667 multicellular organismal metabolic process
GO:0009893 0.00161958399151636 positive regulation of metabolic process
GO:0022603 0.00169126335423007 regulation of anatomical structure morphogenesis
GO:0035295 0.00223230684414793 tube development
GO:0006928 0.00230989986294592 cellular component movement
GO:0010604 0.00236857824547227 positive regulation of macromolecule metabolic process
GO:0010646 0.00252308440283661 regulation of cell communication
GO:0050793 0.0026847817189637 regulation of developmental process
GO:0048562 0.00285877724714378 embryonic organ morphogenesis
GO:0032879 0.00295955595643228 regulation of localization
GO:0080134 0.00335551717091664 regulation of response to stress
GO:0048705 0.00370439844279299 skeletal system morphogenesis
GO:0048864 0.00389984096973906 stem cell development
GO:0023051 0.0040158774598731 regulation of signaling
GO:0006334 0.00406223628626311 nucleosome assembly
GO:0045893 0.00414091776655817 positive regulation of transcription, DNA-templated
GO:0007167 0.00519580828357705 enzyme linked receptor protein signaling pathway
GO:0003002 0.00540223999981726 regionalization
GO:0051254 0.00549153180374605 positive regulation of RNA metabolic process
GO:1902680 0.00598393700399004 positive regulation of RNA biosynthetic process
GO:0023014 0.00676138544251521 signal transduction by phosphorylation
GO:0008283 0.0070310418956878 cell proliferation
GO:0048583 0.0072096008049822 regulation of response to stimulus
GO:0006333 0.00732272231577271 chromatin assembly or disassembly
GO:0007507 0.00736076227709712 heart development
GO:0042127 0.00771299785801664 regulation of cell proliferation
GO:2000026 0.008432344878453 regulation of multicellular organismal development
GO:0010941 0.00916678907171083 regulation of cell death
GO:0043408 0.00935020093242137 regulation of MAPK cascade
GO:0044259 0.00998773914798186 multicellular organismal macromolecule metabolic process
Supplementary Table 4.4: GO terms enriched for genes in SPC domains.
104
0.0
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SPC
-0.5 0 0.5
Value
Color Key
A B
Supplementary Figure 4.1: (A) Average squared difference between replication timing values
at left and right sides of significant contacts, as a function of genomic distance, relative to a
permutation control (t-test 95% confidence interval grey error regions). (B) Confusion matrix
of Segway annotation labels at left and right sides of significant contacts (without GBR). Color
depicts log2(obs/expected) relative to a permutation control (Methods). Pairs of annotation labels
at significantly interacting positions match more often than expected by chance (binomial test
p < 10−16).
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Supplementary Figure 4.2: Fraction of annotation changed by GBR. Y axis depicts the fraction of
positions that receive a different label between an annotation without GBR and an annotation with
GBR using a certain set of hyperparameters. X axis and color depict the λG and λR1 hyperparameters
respectively.
105
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Supplementary Figure 4.3: Correlation of Hi-C contact strength between IMR90 and H1-hESC. X
and Y axes are log p-values of association of a given pair of positions. Color indicates density of
points. Black lines indicate density contours in 0.1% bins. Spearman r = 0.57.
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BRD CON FAC
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Supplementary Figure 4.4: Distribution of domain labels across eight cell types. Y axis indicates
log2(bases covered by label ` in celltype A / (bases covered by label ` / 8)).
107
A
B
Supplementary Figure 4.5: Visualization of GO term enrichment for genes in IMR90 (A) BRD
domains and (B) SPC domains using REVIGO [Supek et al., 2011]. Each bubble represents a
cluster of related enriched GO terms. X and Y axes are projected semantic axes defined using
multidimensional scaling on the semantic similarity of each pair of terms.
108
A B
Supplementary Figure 4.6: Enrichment of consistent Segway boundaries for consistent replication
domain boundaries. (A) Fraction of consistent replication domain boundaries overlapping consistent
Segway domain boundaries as a function of the overlap distance. (B) Same as (A), but fraction of
Segway domain boundaries. We used replication domain boundaries called by Pope et al. [2014].
We defined replication boundaries occurring in more than 10 out of 18 cell types as consistent.
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Supplementary Figure 4.7: Genomic distance distribution of significant IMR90 Hi-C contacts
(q < 0.05).
109
Supplementary Figure 4.8: Schematic of the three formulations of the objective and the alternating
maximization strategy. Edges in this figure indicate KL terms, labeled according to their weight in
the objective. Boxed formulae are update steps. We perform two reformulations, first splitting q
into q and rM linked by a KL term of weight λR1, then splitting rM into rM and sM , linked by a KL
term of weight λR2.
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Supplementary Figure 4.9: Comparison between inference methods on synthetic data. The X
axis shows σ, a hyperparameter controlling the difficulty of inference. The Y axis shows the
average accuracy over 200 simulations of MAP inference on the model in question (95% Wilcoxon
test confidence intervals). Bars correspond to five different inference methods: 1) inference
on each position independently, with no chain model (Independent); 2) inference on the chain
alone, without using W (Chain); 3) loopy belief propagation on the chain plus extra factors of
Pr(Xi = Xj) = sigmoid(λwij), where λ controls the strength of these factors; (Loopy BP) 4) a
variant of GBR using the regularization graph W and a squared-error penalties as described in [He
et al., 2013] (SQ-GBR); and 5) GBR using the regularization graph W (our method, GBR).
110
Chapter 5
CHOOSING PANELS OF GENOMICS ASSAYS USING SUBMODULAR
OPTIMIZATION
Abstract
Due to the high cost of sequencing-based genomics assays such as ChIP-seq and DNase-seq, a
panel of at most 3–10 assays is usually performed on each cell type. We present submodular
selection of assays (SSA), a method for choosing a diverse panel of genomic assays that leverages
methods from the field of submodular optimization. SSA performs better than alternative strategies
in practice, is computationally efficient and extremely flexible, and is theoretically optimal under
certain assumptions. This application may also serve as a model for how submodular optimization
can be applied to other discrete problems in biology. SSA is available at http://github.com/
melodi-lab/Submodular-Selection-of-Assays.
5.1 Background
Genomics assays such as ChIP-seq, DNase-seq and RNA-seq can measure a wide variety of types of
DNA activity, but the cost of these assays limits their application. In principle, to fully characterize a
cell type, one would like to perform every possible type of assay, including ChIP-seq for a variety of
histone modifications and dozens of transcription factors, several chromatin accessibility assays, and
RNA-seq characterizing various types of RNA molecules. However, at current sequencing prices,
performing a single genomics assay with reasonable sequencing depth costs on the order of $400
[scienceexchange]. As a point of comparison, consider the ENCODE and Roadmap Epigenomics
consortia, which develop, perform and analyze genomics assays as their primary activity [Bernstein
et al., 2010, ENCODE Project Consortium, 2012]. As of January 2015, the two consortia had
performed a total of 216 types of assays on at least one cell type, and at least one assay on a total of
111
228 cell types (Methods). Applying all these assay types to all these cell types would require 49,248
assays; however, the two consortia have performed just 1,359 assays, 5% of the possible number
(encodedcc.org; 2014). These consortia are worldwide efforts with large budgets; a typical
lab might be able to perform at most several assays per cell type in order to analyze a particular
tissue or perturbation that they are interested in. Moreover, there are virtually limitless perturbations
and variations of a given cell type for which it would be interesting to examine the effect on DNA
activity, including drug treatments, age, differentiation, etc.
Consequently, selecting a small panel of assays to perform on each cell type of interest—a
problem we call assay panel selection—is a key step in any genomics project. To our knowledge,
there has been little discussion in the literature of how to choose such a panel. In consortia such as
ENCODE and Roadmap, the procedure for choosing which assay types to perform on each cell type
is typically ad hoc. These decisions are made by the investigators involved, based on their intuition
about the diversity of assay types, perhaps based on pairwise correlations between assays or similar
simple metrics. Ernst and Kellis [Ernst and Kellis, 2015] proposed that imputation methods can
evaluate the quality of a given of a given panel, but this approach cannot be used to select a panel
(see Discussion).
In this work, we propose a principled method to solve the assay panel selection problem. Qual-
itatively, the method aims to identify, on the basis of existing data sets, assay types that yield
complementary views of the genome. In practice, many pairs of assay types yield redundant infor-
mation. For example, the transcription factors REST and RCOR1 are cofactors and therefore bind
almost the same set of genomic positions [Andre´s et al., 1999]. Similarly, the histone modification
H3K36me3 primarily marks gene bodies, which are also transcribed and therefore measured by
RNA-seq. Therefore, a great deal of what can be learned from the full set of assays can likely be
learned by performing a small subset of the possible assays. This redundancy among assay types
suggests that a carefully chosen panel of assays is likely to produce most of the information that
would be obtained by performing all assays. Our solution to the assay panel selection problem is
composed of two parts: an objective function that defines the quality of a panel, and an optimization
algorithm that efficiently finds a panel that scores highly according to the objective function.
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We propose to use an objective function called facility location (defined mathematically below),
which measures what fraction of the information available in the full set of assay types is contained
within the panel. This function has been previously applied in many fields, including document
summarization [Lin and Bilmes, 2012], feature selection for machine learning [Liu et al., 2013],
and exemplar-based clustering [Mirzasoleiman et al., 2013]. This function also corresponds to
the objective function of the widely-used k-medoids clustering algorithm [Gomes et al., 2010].
Computing the facility location function requires a measure of similarity between assay types. For
this purpose, we use the Pearson correlation between the two assays, averaged over the cell types in
which the assays have been performed. We chose this objective function because it performs better
than or comparably to the other methods we tried (Supplementary Notes 5.9–5.12).
To optimize the facility location function, we borrow methods from the field of submodular
optimization. A simple approach to selecting a panel of assays would evaluate the facility location
function for every possible subset of assays and then choose the highest-scoring subset. Unfortu-
nately, 216 possible assay types yield 2216 ≈ 1065 possible panels of assays, so this approach is
not feasible in practice. Fortunately, an efficient alternative selection method exists because the
facility location function has the property of submodularity. The property of submodularity (defined
mathematically below) is analogous to the property of convexity but is defined on discrete set
functions rather than continuous functions. Submodular functions have a long history in economics
[Vives, 2001, Carter, 2001], game theory [Topkis, 1998, Shapley, 1971], combinatorial optimization
[Edmonds, 1970, Lova´sz, 1983, Schrijver, 2004], electrical networks [Narayanan, 1997], operations
research [Cornune´jols et al., 1990], and more recently, machine learning [Narasimhan and Bilmes,
2005, Krause et al., 2008, Liu et al., 2013, Wei et al., 2014], but they are not yet widely used
for problems in biology. Therefore, this application may serve as a model for how submodular
optimization can be applied to biological problems more generally.
We apply existing submodular optimization algorithms to the facility location function to effi-
ciently select a high-quality panel of assays, a method we call submodular selection of assays (SSA).
There exists a large literature of methods for optimizing submodular functions. The optimization
method we employ is very efficient and is theoretically guaranteed to find a solution whose quality
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comes within a constant factor of the quality of the optimal solution [Nemhauser et al., 1978b].
In addition to proposing solutions to the assay panel selection problem, an important contribution
of this work is development of three general methods for evaluating the quality of a selected panel
of assays. These three methods correspond to three distinct practical applications of the selected
panel: (1) the accuracy with which the panel can be used to impute the results of assays not included
in the panel; (2) the accuracy with which the panel can be used to detect functional elements such as
transcription factor binding sites, promoters, and enhancers; and (3) the quality of a whole-genome
annotation produced using the panel. These evaluation metrics share the property that an informative
and diverse set of assay types yields better performance, according to each metric, than does a
redundant set. Note that these evaluation metrics differ from the objective function because they
use information that is not available at the time a panel is chosen; therefore, the evaluation metrics
themselves cannot be used directly to choose a panel. These three metrics will be useful for any
future study of the quality of a panel of assays, independent of the particular procedure used to
choose such a panel.
We consider two variants of the assay panel selection problem. We are primarily interested in
the “future” variant, which arises when a researcher is interested in applying a panel of genomics
assays to a new tissue type or cellular condition. In this case, the researcher must use previously
performed assays in other cell types to choose a representative panel of assay types. However,
we also consider the “past” variant, which arises when a researcher is interested in applying a
computationally expensive analysis, such as a genome annotation method, that cannot efficiently be
run on all available data sets. In this setting, the researcher must therefore choose a representative
panel from the available data to use as input to the analysis. In this case, the researcher may use the
data from assays performed on the cell type in question to inform their choice. We propose a variant
of the method called SSA-past, which leverages this information to allow a researcher to choose
such a representative panel in this setting.
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Supplementary Figure 5.1: Schematic of the selection process performed by submodular selec-
tion of assays (SSA). The method takes as input all available existing genomics assays, where each
assay is represented as a real-valued track over the genome. In the SSA-past mode, SSA selects
a panel of already-performed assays to use as input to an expensive computational analysis. In
the SSA-future mode, SSA chooses a panel of assay types to be performed in a new cell type.
In both cases, the resulting data sets are provided as input to downstream analyses, which may
include imputing assays that weren’t performed, predicting the locations of functional elements, or
semi-automated genome annotation.
5.2 Results
5.2.1 Submodular selection of assays identifies diverse panels of genomics assays
Submodular selection of assays (SSA) takes as input a collection of genomics data sets (“assays”)
and identifies a high-quality subset of those assays (Methods, Figure 5.1). Each input data set is
represented as a real-valued signal vector over the genome. SSA begins by computing a pairwise
similarity matrix that contains, for each pair of assay types, the mean Pearson correlation over all
pairs of assays of those two types. The method employs a submodular function, called facility
location (Methods), to estimate the quality of any possible panel of assay types. The facility location
function takes a high value for a particular panel when all assay types have at least one similar
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representative in the panel. We chose this strategy because it performed the best of those we tried
according to our evaluation (Supplementary Notes 1-4). SSA then applies the greedy submodular
optimization algorithm (Methods) to efficiently choose a panel of assays that maximizes this facility
location function. The output of this method is an ordered list of assay types, where the top k assay
types in this list represent a high-quality panel of size k. A detailed description and theoretical
justification for the method is provided in Methods.
By analyzing data from the ENCODE and Roadmap Epigenomics Consortia, we found that
SSA results in assay panels with diverse genomic functions. Because researchers generally perform
panels of either histone modifications or transcription factor ChIP-seq assay types but rarely perform
mixed panels, we ran the method separately on transcription factor and histone modification types
(combined panels are shown in Supplementary Table 5.1). When choosing from transcription factors,
SSA chooses factors that engage in diverse regulatory pathways (Figure 5.2A). The vast majority of
transcription factors in our data set bind to promoters and enhancers and regulate the transcription
of RNA Pol II-transcribed genes. The top five transcription factors chosen by SSA include three
of these factors, each of which regulate very different regulatory pathways: SMARCB1, an ATP-
dependent chromatin remodeler; PML, a tumor suppression factor; and STAT5A, a factor involved
in developmental signal transduction [UniProt Consortium, 2014]. The top five also includes two
factors—CTCF and BRF2—that are not solely involved in RNA Pol II-mediated transcription.
CTCF, part of the cohesin complex, regulates chromatin conformation and enhancer-promoter
insulation, and only about half of its binding sites occur in promoters or enhancers. BRF2 is part of
the RNA Polymerase III complex, which transcribes rRNA, tRNA and other small RNAs. These
two assay types each have low objective scores when in a panel by themselves (“singleton scores”),
but are chosen by SSA because they measure different types of activity than the rest of the panel.
Therefore, they are important to include in a diverse panel.
When choosing a panel of histone modifications, SSA selects marks that cover diverse types
of genomic regions (Figure 5.2B,C). The top six histone modifications include a promoter mark
(H3K4me3), an enhancer mark (H3K4me1), a gene mark (H3K79me2) and marks associated with
both known types of repressive domains, facultative (H3K27me3) and constitutive (H3K9me3)
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A
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Singleton
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Objective
gain
1 SMARCB1 ATP-dependent chromatin remodeling 14.78 14.78
2 PML Tumor suppression 14.12 1.84
3 STAT5A Developmental signal transduction 14.17 0.91
4 CTCF Chromatin conformation and insulation 8.40 0.81
5 BRF2 RNA polymerase III initiation complex 8.88 0.55
B
C
Choice
order
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modification Association
Singleton
score
Objective
gain
Objective loss
if swapped in
1 H3K4me3 Promoters 3.18 3.18
2 H3K79me2 Transcription 2.40 0.97
3 H3K9me3 Constitutive heterochromatin 0.70 0.34
4 H3K27me3 Facultative heterochromatin 0.86 0.25
5 H3K36me3 Transcription 1.21 0.23
6 H3K4me1 Enhancers 1.86 0.18 0.05 (H3K36me3)
7 H3K4me2 Regulatory 3.12 0.08 0.07 (H3K36me3)
8 H3K9ac Regulatory 3.13 0.07 0.15 (H3K36me3)
9 H2A.Z Promoters 2.47 0.05 0.16 (H3K27me3)
10 H4K20me1 Transcription 1.41 0.005 0.22 (H3K36me3)
11 H3K27ac Regulatory 2.61 0 0.14 (H3K36me3)
Supplementary Figure 5.2: (Caption on following page)
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Supplementary Figure 5.2: Results of SSA-past on ENCODE+Roadmap data sets. (A) Panels
of transcription factors assays chosen by SSA-future. Each list is in the order assigned by SSA; for
any size k, the top k assay types in the list are the chosen panel of this size. The “singleton score”
is the objective value of a panel containing only the indicated assay type, and the “objective gain”
indicates the improvement in objective that results from SSA adding the indicated assay type to the
growing panel. Because there are 80 transcription factors, we display just the top five chosen by
SSA. Associations are summarized from UniProt [UniProt Consortium, 2014]. (B) Redundancy
in histone modification signal in the genome. The top five assay types chosen by SSA are boxed
in red. (C) Similar to (A), but for the histone modification assays. There are only eleven histone
modifications, so we can display the full list. Bold font indicates those histone modification assays
chosen by the Roadmap Epigenomics consortium.
heterochromatin. The two repressive marks, H3K27me3 and H3K9me3, have the lowest singleton
scores of all the histone modifications, but they give a high objective gain because they measure
distinct activity from the rest of the panel. In contrast, even though H3K27ac is sometimes
considered the best individual mark of enhancers and has a high singleton score, it is chosen last by
SSA because it is redundant with other assay types in the panel, such as H3K4me1 and H3K9ac.
The top six includes two different marks of transcription, H3K79me2 and H3K36me3, but these
two modifications mark different parts of genes and are regulated differently relative to the gene’s
level of transcription [Li et al., 2007]. As expected, SSA ranks additional measures of regulation
(H3K4me2, H2A.Z, H3K9ac and H3K27ac) low on the list because these marks are redundant with
the regulatory marks H3K4me1 and H3K4me3.
Moreover, SSA almost exactly recapitulates the panel of histone modifications chosen by the
Roadmap Epigenomics consortium (boldface entries in Table 5.2C). This consortium chose a set
of five “core” histone modifications to assay across 111 human primary tissues. This choice was
made by the members of this consortium based on their collective, expert knowledge. These five
core histone modifications ranked among the top six modifications chosen by SSA. In fact, the
SSA-chosen and Roadmap-chosen panels of size 5 have very similar scores according to the facility
location function, ranking 1 and 16 respectively out of all
(
11
5
)
= 2772 possible panels of five
histone modifications. Therefore, SSA closely reproduces careful, manual selection by experts in an
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entirely automated and data-driven way.
In addition to its efficiency, SSA is flexible. First, in some circumstances, a researcher may have
already performed a few assays in a given cell type or be confident that they want to perform them,
and is therefore interested in choosing which assay types best complement these existing assays.
SSA can be used in this scenario simply by restricting the returned set to include these assay types.
For example, restricting the set of histone modifications to contain H3K27ac de-prioritizes other
enhancer marks, such as H3K9ac and H3K4me1 (Supplementary Table 5.2). Second, a researcher
might have a prior preference for certain classes of assays because they are easier to perform, less
expensive, or measure features the researcher is especially interested in. SSA can include this prior
preference by adding a weight to the objective for each assay type, reflecting the preference for that
assay type. For example, selecting from all assay types (transcription factor, histone modification
and DNA accessibility) jointly while placing a preference for (or against) histone modifications
increases (or decreases) the fraction of histone modifications chosen to for the panel (Supplementary
Table 5.3). Third, SSA can even alternatively be used to select cell types (instead of assay types) by
transposing the cell type-assay type matrix and running the same analysis. Doing so produces a
panel of cell types that includes a diverse mixture of mesodermal, endodermal and undifferentiated
cells, and both normal and cancer-derived cells (Supplementary Table 5.4).
5.2.2 Three metrics evaluate the quality of a set of genomics assays.
In order to quantitatively evaluate SSA, we developed an evaluation framework for assay panel
selection. We focused on three of the most common downstream applications of genomics data
sets: (1) imputing assays that have not been performed, (2) locating functional elements such as
promoters and enhancers, and (3) annotating the genome using a semi-automated method. We
describe each metric briefly here, with full details provided in Methods.
The first evaluation metric, assay imputation, measures how well a chosen panel of assays
can be used to predict assays that have not been performed (Figure 5.3A). We train a regression
model to predict each assay outside of the panel on the basis of the assays within the panel, using
random subsets of the genome for training and testing, respectively. High performance on the assay
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D
Supplementary Figure 5.3: Evaluation strategies. Schematics of the three evaluation metrics: (a)
assay imputation, (b) functional element prediction, and (c) annotation-based evaluation, as well as
(d) the overall cross-validation evaluation strategy.
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imputation metric indicates that the panel contains all of the information in the assays outside of the
panel. Moreover, recent work on imputation has showed that it is often effective to train a regression
model on data from reference cell types and apply it to a target cell type (Discussion).
The second evaluation metric, functional element prediction, is similar to the assay imputation
metric but focuses specifically on how well a chosen panel of assays can be used to locate functional
elements such as promoters and enhancers (Figure 5.3B). Because there are few validated examples
of each type of element, we use experimentally determined binding of transcription factors, as
measured by transcription factor ChIP-seq, as a proxy for functional elements. We train a classifier
model to predict the locations of these elements on the basis of the assays within the panel. High
performance on the functional element prediction metric indicates that a panel can be used to
accurately locate functional elements. Although both the assay imputation and functional element
prediction evaluation metrics aim to predict genomics data sets, functional element prediction
focuses on the small fraction of the genome corresponding to transcription factor binding sites.
The third evaluation metric, annotation-based evaluation, measures how effectively a given
panel can be used to annotate the genome through a semi-automated genome annotation (SAGA)
method (Figure 5.3C). SAGA methods, which include HMMSeg [Day et al., 2007], ChromHMM
[Ernst and Kellis, 2010], Segway [Hoffman et al., 2012] and others [Thurman et al., 2007, Lian
et al., 2008, Filion et al., 2010, Kharchenko et al., 2010, Jaschek and Tanay, 2009, Larson et al.,
2013, Biesinger et al., 2013], annotate the genome on the basis of a panel of genomics assays. They
simultaneously partition the genome and annotate each segment with an integer label such that
positions with the same label exhibit similar patterns of activity. These methods are semi-automated
because a person must interpret the biological meaning of each integer label. SAGA methods have
been shown to recapitulate known functional elements including genes, promoters and enhancers.
Given a particular panel of assays, we perform annotation-based evaluation by using this panel as
input to Segway and measuring how well the resulting genome annotation corresponds to patterns
observed in the assays outside of the panel. High performance on this metric indicates that the
chosen panel can be used to produce a comprehensive annotation of the genome.
Applying these metrics to evaluate a method for choosing panels is complicated by two factors.
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Supplementary Figure 5.4: Relationship between the facility location objective function and
evaluation metrics. Each dot corresponds to one of 40 randomly-chosen panels. Pink triangles
indicate results from maximizing the SSA-future facility location function; red diamonds indicate
the panel of most-frequently performed assay types (Supplementary Table 5.5). These results were
computed in GM12878, using panels of four assay types.
First, no cell type has had all assay types performed in it, so we perform evaluation separately on
each cell type in order to evaluate against all available assay types. Second, these evaluation metrics
must be used to compare to assays outside of the panel, so we use a cross-validation strategy in
which we hold out a target set for evaluation and choose panels from the remaining source set,
repeating this process for many choices of target set. This evaluation strategy enables principled
evaluation that compares all methods against the same held-out standard while using only the
available data sets (Methods, Figure 5.3D).
5.2.3 Panels chosen by SSA outperform alternative methods according to three evaluation metrics.
We applied this panel evaluation framework to evaluate SSA. First, to determine the most effective
objective function, we compared the facility location function and four other potential objective
functions based on the pairwise similarity matrix. We found that the facility location function had a
higher Spearman correlation with the three evaluation metrics than the other objective functions we
tried, and this trend was consistent across multiple cell types (Supplementary Note 5.9). In addition,
we found that the facility location function produced the highest correlation with the three evaluation
metrics when we defined the similarity between a pair of assay types as the mean Pearson correlation
122
between this pair, as opposed to the median, maximum or other aggregation function (Supplementary
Note 5.10). We also compared between two strategies for deriving the Pearson correlation. The first
computes the correlation based on the random samples of the genomics positions, and the second on
the DNase peaks positions only. We found that almost comparable performance is achieved on the
three evaluation metrics for the two strategies (Supplementary Note 5.11). We lastly compared the
effectiveness between Pearson correlation and Spearman correlation for computing the similarity
measure. We found that the consistently better performance is achieved when the similarity is
defined using the Pearson correlation (Supplementary Note 5.12). These observations led us to
choose our variant of facility location as the SSA’s objective function.
Next, we used our three evaluation metrics to compare SSA to alternative panel selection
approaches. As a baseline, we considered randomly selected panels of a given size. We also
considered the panel of most frequently performed assays (Supplementary Table 5.5) as a good
proxy for a likely data-driven choice. We found that the panels reported by SSA perform among
the top few percent out of the space of all possible panels, and this high performance is consistent
across panel sizes, evaluation cell types and performance metrics (Figure 5.5A). We found that
SSA also greatly outperforms the panel of most-frequently performed assay types. Indeed, this
commonly-performed panel actually performs worse than the average panel in many cases, which
may be a consequence of the fact that the most commonly-performed assay types measure broad
marks of regulation, such as histone modifications and DNA accessibility, which do not have the
specificity to identify pathway-specific elements. These results demonstrate quantitatively that
panels chosen by SSA are effective when applied to their most common downstream tasks.
5.2.4 SSA can select a subset of performed assays as input to an expensive analysis
So far we have considered panel selection in the “future” setting, where a researcher is planning
to experimentally perform a panel of assay types. Panel selection is sometimes also important in
the “past” setting, where a researcher wishes to apply a computationally expensive analysis that
cannot be efficiently applied to all assays together and therefore must be applied to a smaller panel.
For example, training a statistical model to perform semi-automated genome annotation jointly on
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Supplementary Figure 5.5: Performance of panel selection strategies. (a) Boxplots show the
distribution of evaluation metrics over 40 random panels on data from cell type GM12878. The
panels of most-frequently performed assays are composed of the top k most frequent assay types
available in our data set, where k is the size of the panel. Each evaluation metric is normalized to lie
within [0, 1] by subtracting the lowest value and dividing by the highest. (b) Scatter plot between
the performance of SSA-past and SSA-future across cell types K562, GM12878, and H1-hESC.
Each dot in the plot corresponds to the two variants of SSA for a panel size evaluated using a metric
in a cell type. The performance is measured as the fraction of panels that perform worse than the
SSA-chosen panel, estimated by comparing to 40 randomly-selected panels.
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dozens of assays across many cell types is computationally expensive. A strategy in which each
cell type is represented by a smaller panel of assays might yield very similar annotations using
a fraction of the computational resources. In this setting, the assays themselves are available to
the selection algorithm, so we compute the similarity matrix based on these values themselves
(SSA-past) rather than estimating the similarities by aggregating across cell types (SSA-future).
Importantly, in the selection of past assays case, a different panel can be selected for each cell type,
based on the available data. Alternatively, if the researcher would like to use the same panel of past
assays for analyses several cell types, SSA-future can be used to select a panel that maximizes the
average quality over all target cell types (Supplementary Table 5.6). To test SSA in the past setting,
we used the same evaluation strategy as in the future setting, but using the source assays themselves
to compute the similarity matrix. SSA performs consistently well according to these metrics, and it
performs slightly better on some cell types in the past than the future setting due to the availability
of this additional information (Figure 5.5B).
5.3 Discussion
The growing availability of a large number of types of genomics assays means that choosing a
panel of genomics assays is a key step in any genomics project. Previously, these panels were
chosen in an ad hoc fashion. We have developed submodular selection of assays (SSA), a method
for choosing high-quality panels using submodular optimization. This method is computationally
efficient, results in high-quality panels according to several quality measures, and is mathematically
optimal under some assumptions. By applying SSA, researchers can now easily choose a high-
quality panel of assay types to perform on any cell type of interest. These higher-quality panels will
allow researchers to achieve the same utility from performing fewer assays, saving thousands of
dollars in labor and reagent costs per cell type. This panel selection framework can also be used
partway through the investigation of a cell type, when several assays are already available. By
modifying the facility location function to include the availability of these assays, SSA can be used
to determine the most-informative next experiments to perform. In doing so, SSA will take into
account the information in these existing assays and choose additional assay types that measure
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distinct genomic features.
A key feature of the submodular optimization approach is the flexibility afforded by the broad
class of submodular objective functions, and the ability to encode appropriate prior knowledge into
the selection of the objective. In this manuscript, we focused on optimizing the facility location
function. However, the same submodular optimization framework can be used to optimize other
objective functions that may prove to be more relevant for certain applications. Several other
functions may be useful in practice. First, if some assays are more expensive or time consuming
than others, then the objective function can be modified to incorporate this cost. Second, if
some assays are inherently preferable to others, for example because they have better-established
processing pipelines, then the objective can incorporate this preference and trade off choosing
both diverse and established assay types. Third, entirely different types of panel attributes may
be valuable for a particular application, which can be formalized as a different objective function,
such as the alternatives we discuss in Supplementary Note 5.9. As long as the resulting objective
function remains submodular, it will be efficiently optimizable using either the greedy algorithm
for monotone non-decreasing functions or other efficient methods [Buchbinder et al., 2012, 2014]
for non-monotone functions. Moreover, such modifications are intuitive to design and easy to
implement.
The facility location function can also be used to guide manual assay panel selection. A
researcher may seek to optimize hard-to-quantify characteristics of a panel, such as familiarity with
the protocols involved or the panel’s concordance with panels performed on other cell types. In this
case, the researcher may choose to perform a panel that has slightly poorer quality as measured by
the facility location function in order to optimize these other criteria. Still, in such a setting, manual
investigation of the objective values associated with different panels can provide useful insights.
The framework presented here can also be extended to many related problems. We have
discussed two such variants that apply in the scenarios, “I would like to select a panel of assay
types to perform, taking cost into account”, and “I have carried out many assays in this cell type
and would like to choose a subset of this to use as input to an expensive computational analysis”.
This framework can also trivially be extended to the problem, “I have already performed several
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assay types in a particular cell type of interest and would like to select several more to perform,”
by simply restricting the output panel to contain the previously-performed assays (Supplementary
Table 5.2). The same framework also applies to the problem, “I have a set of assay types commonly
performed in several cell types and would like to choose several assay types from the set that are the
most informative and representative to study the cell types” by restricting the aggregation of the
assay type similarity to the cell types at hand (Supplementary Table 5.6). In addition, by deriving
a similarity measure between cell types, the methods presented here could be used to solve the
problem, “Based on orthogonal data such as gene expression profiles, which new cell types should I
perform assays in?” Finally, related methods may be able to apply to the problem “I have carried
out assays across a variety of cell types and assay types, and I would like to select a set of additional
assays to apply in any combination of cell and assay types.”
The evaluation strategy we introduce here can be used to evaluate any proposed strategy for
panel selection, whether or not this method is based on submodular optimization. This framework
is composed of two parts. First, a cross-validation strategy allows for principled comparison of
methods under the restriction that not all assays are available in all cell types, and that a panel must
not be evaluated on an assay type that it contains. Second, three distinct metrics capture the three
primary downstream applications of genomics data sets.
The problem of assay panel selection was posed previously by Ernst and Kellis [Ernst and Kellis,
2015]. These authors proposed the assay imputation evaluation metric for evaluating a panel of
genomics assays. However, this type of evaluation metric cannot be used to choose an assay panel
for two reasons. First, evaluating the accuracy of an imputation model requires that the target data
set already be performed. Second, even if such data sets were available, evaluating all possible
panels of size K from N assay types requires applying evaluation to
(
N
K
)
possible panels, which is
hopelessly computationally expensive. For example, the method of Ernst and Kellis was reported
to take roughly one hour to train per panel, so choosing a panel of five assay types from the 188
previously-performed assay types would take on the order of 105 processor-years. In contrast, the
facility location objective function we define does not require that the target data sets be available in
order to evaluate the function, and the submodular optimization approach requires only a polynomial
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number of evaluations. Moreover, we define two additional evaluation metrics (functional element
prediction and annotation-based evaluation) that complement evaluation by evaluation by assay
imputation.
In addition to be applicable to the assay panel selection problem, the same submodular optimiza-
tion framework may also be useful in the following scenario: “Rather than deciding which assay
types to choose, one wants to choose a set of informative and representative cell types to study”.
We call this problem cell type panel selection. Although this setting is out of scope of this work, it
can be viewed as a dual variant of the assay panel selection problem. To address this problem, a
similarity matrix among cell types can be derived in the same manner as the assay type similarity
matrix using the similar aggregation strategies. Instantiated on this cell type similarity matrix, a
submodular function (e.g., facility location function) may then serve as the optimization objective
for outputting the selection order of the cell types. The selection order of the cell types using the
SSA methodology is displayed in Supplementary Table 5.4. The flexibility in SSA (e.g., forcing
to choose certain assay types, and weighing certain assay types with higher preference) easily
carries over to this setting making the cell type selection framework also very general. Moreover,
the methodologies proposed here for evaluating the quality of a panel of assay types (e.g., assay
imputation, and annotation-based evaluation) can be easily extended to assess the goodness a cell
type panel selection approach.
One limitation of any data-driven analysis is that it is limited by any imperfections in the data
sets used. For example, if all available assays of a given type happen to be of particularly good
or poor quality, then the correlations associated with this assay type will appear to the algorithm
to be particularly strong or weak, respectively. Similarly, any mislabeled assays, batch effects, or
other artifacts may also influence whether certain assay types will be chosen in a panel. Future
assays of that type may not be expected to exhibit the same artifactual patterns, so the resulting
panels could be suboptimal. Therefore, it is always important to scrutinize the results of data-driven
approaches like this one to understand whether patterns in the available data are predictive of future
experiments. Modifications of this approach that, for example, find and remove faulty assays before
input into the algorithm might result in different panels. However, our evaluation metrics are also
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entirely data-driven, so we cannot use them to explore these issues.
As noted above, submodular optimization is widely used for discrete problems in other fields
but is not yet widely used in biology. We hope that the current work can serve as a model for how
submodular optimization can be applied to other problems in biology. As with convex optimization,
the same toolbox of submodular optimization methods can be applied to a wide variety of problems,
and any innovations to this toolbox improve all solutions. Therefore we expect that in the future,
submodular optimization will be used for other discrete problems in biology, such as for selecting
panels of DNA mutations to test in a functional screen or removing redundancy in protein sequence
data sets.
5.4 Methods
5.4.1 Genomics data
We acquired all public genomics data from the ENCODE (http://hgdownload.cse.ucsc.
edu/goldenPath/hg19/encodeDCC/) and Roadmap Epigenomics (https://sites.
google.com/site/anshulkundaje/projects/epigenomeroadmap) projects as of
January 2015. These data sets were processed by the two consortia into real-valued data tracks,
as described previously [Hoffman et al., 2013, Kundaje et al., 2015]. We omitted all assays with
more than 1% unspecified positions, which may indicate errors during processing or mapping. We
manually curated these assays to unify assay type and cell type terminology and, when multiple
assays were available, we arbitrarily chose a representative assay for each (cell type, assay type) pair.
This procedure resulted in a total of 1,359 assays comprised of a total of 216 assay types and 228 cell
types. The assay types include ChIP-seq with a variety of targets (both histone modification and tran-
scription factor), DNase-seq, FAIRE-seq, Repli-seq and RNA-seq. The full list of assays is given as
supplementary data. We applied the inverse hyperbolic sine transform asinh(x) = ln(x+
√
x2 + 1)
to all signal data. This function has the compressing effect of a function like log x for large values
of x but it is defined at zero and has much less of a compressing effect for small values. The asinh
transform has been shown to be important for reducing the effect of large values in analysis of
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genomics data sets [Johnson, 1949, Hoffman et al., 2012]. Transcription factor ChIP-seq peaks
were called by each consortium for each factor using MACS using an irreproducible discovery rate
(IDR) threshold of 0.05 [Zhang et al., 2008, Landt et al., 2012].
5.4.2 Notation
We use the following notation below to facilitate the description of our method. We use the term
“assay type” to mean a particular genomics assay protocol that may be performed in any cell type (for
example “ChIP-seq targeting H3K27me3”) and “assay” to mean a particular assay type performed
in a particular cell type. The term “cell type” refers to any cellular state that may be queried with a
genomics assay, which may refer to any combination of cell line, tissue type, disease state (such as
cancer), individual, or drug perturbation. We refer to a cell type as c and the entire set of all cell
types as C (|C| = 228). We use a to refer to an assay type, A for a subset of assay types, and A for
the set of all assay types (|A| = 216). We use s to denote a single assay (that is, a given assay type
performed in a given cell type), S for a set of assays, and S as the set of all available performed
assays. Given any cell type c ∈ C we define the set of assay types performed in this cell type as
Ac and the corresponding assays as Sc. We define I = {1, . . . , n} as the set of all positions in a
genome. An assay s is represented as a vector of length n; i.e., s ∈ Rn. We denote its ith entry (i.e.,
the value of assay s at genomic position i) as s(i).
5.4.3 Submodular optimization
A submodular function [Fujishige, 2005] is defined as follows: given a finite size m set V =
{1, 2, . . . ,m}, a discrete set function f : 2V → R that offers a real value for any subset S ⊆ V is
submodular if and only if:
f(S) + f(T ) ≥ f(S ∪ T ) + f(S ∩ T ),∀S, T ⊆ V. (5.1)
Defining f(s|S) , f(s ∪ S) − f(S), submodularity can equivalently be defined as f(s|S) ≥
f(s|T ),∀S ⊆ T and s /∈ T . That is, the incremental gain of adding item s to the set decreases when
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the set to which s is added to grows from S to T . In this work, the whole set V represents a set of
genomics assays and the set function f(S) represents a measure of quality of a subset of assays
S ⊆ V .
Two other properties of set functions are relevant to this setting. First, a set function f is defined
as monotone non-decreasing if
f(s|S) ≥ 0,∀s ∈ V \ S, S ⊆ V. (5.2)
Second, we say that f is normalized if f(∅) = 0.
In this work we are interested in the problem of maximizing a submodular function subject to a
constraint on the size of the reported set. That is, we are interested in solving the problem
maximize f(S), subject to |S| ≤ k (5.3)
for some integer k ≤ |V |. In this work, we require that f is submodular, monotone-nondecreasing
and normalized.
While this problem is NP-hard, it can be approximately solved by a simple greedy algorithm
with a worst-case approximation factor (1− e−1) [Nemhauser et al., 1978a]. This is also the best
solution obtainable in polynomial time unless P = NP [Feige, 1998]. The algorithm starts with
the empty set S0 = ∅ and at each iteration i adds the element si that maximizes the conditional
gain f(si|Si−1) with ties broken arbitrarily (i.e., finding si ∈ argmaxe∈V \Si−1 f(e|Si−1)) and then
updates Si ← Si−1 ∪ {si}. The algorithm stops when the cardinality constraint is met with
equality. This algorithm has a time complexity of O(km) function evaluations. The running time of
this algorithm can be improved to a near-linear number without any performance loss by further
exploiting the submodularity property [Minoux, 1978].
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5.4.4 Facility location function
In this work we use the facility location function to measure the quality of panel of assay types. The
facility location function [Cornune´jols et al., 1990] ffac : 2V → R is defined as follows:
ffac(S) =
∑
s′∈V
max
s∈S
rs′,s, (5.4)
where rs′,s measures the pairwise similarity between assays s′ and s (defined below). Intuitively,
the facility-location function takes a high value when every assay in V has at least one similar
representative in S. To instantiate the facility location function, one needs to compute and store a
pair-wise similarity graph, which takes O(m2) memory complexity.
5.4.5 Assay type similarity
We use the following strategy to define the similarity between each pair of assay types in order
to use this similarity to define a facility location function. We define this similarity differently
depending on the application: In the selection of past assays setting, the particular assays performed
in the cell type of interest c are available, while in the selection of future assays setting we must
estimate this similarity from reference cell types.
In the selection of past assays setting, we directly use the signal vectors si and sj to derive the
similarity. We define this similarity as rsi,sj = |ρsi,sj | ∈ [0, 1], where ρsi,sj is the Pearson correlation
between the signal vector si and sj . Pearson correlation is frequently used to evaluate the similarity
between genomics assays [ENCODE Project Consortium, 2012]. For efficiency, we compute the
correlation measure ρsi,sj only across a subset of genomic positions I
′ ⊆ I , where I ′ is randomly
subsampled from I , and |I ′| ≈ 0.01|I|.
In the selection of future assays setting, the assays in the cell type c are not available, but the
assays performed in cell types other than c, S \ Sc, are available. Let ai, aj ∈ A be the assay types
associated with the assay si and sj , respectively. Let Sai be the set of assays in S with type ai. We
approximate the similarity between si and sj by aggregating the all similarity between the pairs in
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Sai \ si and Saj \ sj . We utilize the aggregation strategy by taking the average of these similarity
scores. Mathematically, the aggregated similarity is defined below:
rsi,sj ,
1
|Sai \ si|
1
|Saj \ sj|
∑
s∈Sai\si
.
∑
s′∈Saj \sj
|ρs,s′ |, (5.5)
We chose to use the average correlation r because the facility location function defined via the
similarities aggregated in this way correlated best with our evaluation metrics. We provide the
comparison of such aggregation strategy against other strategies in the Supplementary Note 5.10.
5.4.6 Evaluation cross-validation strategy
We would prefer to apply our method once to select a single panel of assay types. However, doing
so could result in a panel of assay types that have not been performed in any cell type (or very
few cell types), which would prohibit evaluating the quality of this panel. Therefore, we apply a
cross-validation strategy that repeatedly holds out a subset of assay types for evaluation and selects
a panel from the remaining assay types, and we perform this cross-validation separately for each
cell type in turn (Figure 5.3D). To evaluate the quality of our method with respect to a cell type c,
we restrict ourselves to selecting from the set of assays performed in c (Sc). We randomly partition
Sc into 10 equally-sized, disjoint folds. Of the 10 folds, a single fold is retained as the target set T c,
and the remaining 9 blocks are used as the source set Vc. We select a panel of assays S ⊆ Vc from
the source set Vc and evaluate the panel on the assays relative to the target set T c using the three
evaluation metrics described below. The process is then repeated ten times, with each of the ten
folds used once as the target set. We average the ten results are averaged to produce a single number
representing the performance.
5.4.7 Assay imputation
The assay imputation evaluation metric measures the ability of a panel of assay types to be used
to impute the results of other assay types outside the panel (Figure 5.3A). We formalize assay
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imputation metric as a regression problem in which the assays in the panel S are used as features to
predict the target set assays, s′ ∈ T c. In this regression problem we have one labeled example for
each position in the genome.
As our regression model, we use support vector regression with a Gaussian kernel. To construct
the training and test data, we randomly choose disjoint sets of genomic positions ITr, ITe ⊆ I ,
where ITr ∩ ITe = ∅. In our experiments, we set |ITr| = 5, 000 and |ITe| = 2, 000. Given the
panel S = {s1, . . . , s|S|}, a target assay s′ ∈ T c, and the training genomic positions ITr, we create
the training data as DTr = {xi, yi}i∈ITr , where xi = [s1(i), s2(i), . . . , s|S|(i)]T and yi = s′(i).
Similarly, the test data set is constructed as DTe = {xi, yi}i∈ITe . The hyperparameters of the
regression model are tuned using 5-fold cross validation. We measure the performance of the trained
model on the test data DTe as the squared correlation coefficient θs′ . We repeat this evaluation
process for every target assay in T c and report the performance of the panel S as the average squared
correlation coefficient θ = 1|T c|
∑
s′∈T c θs′ .
5.4.8 Functional element prediction
The functional element prediction evaluation metric evaluates how well a panel of assays can predict
the genomic locations of functional elements such as promoters, enhancers and insulators. Because
there are few validated examples of each type of element, we use experimentally-determined
binding of transcription factors, as determined by transcription factor ChIP-seq peaks, as a proxy
for functional elements. Most known types of functional elements can be characterized by the
binding of particular transcription factors [Visel et al., 2009, Burgess-Beusse et al., 2002]. Note that
functional element prediction is similar to assay imputation in the sense that both evaluation metrics
aim to predict the output of a genomics assay; however, functional element prediction focuses
on just transcription factor binding sites, whereas assay imputation focuses on the whole genome.
Similar to assay imputation, we consider this metric separately for each cell type. For an evaluation
cell type c, we denote the set of transcription factor ChIP-seq assays performed in c as Sˆc ⊆ Sc.
Given a bi-partition of Sc into the source set Vc and the target set T c, we choose from the source set
Vc a panel of assays, and we evaluate functional element prediction only on the target assays in the
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set Tˆ c = T c ∩ Sˆc, in contrast to the assay imputation metric where all assays in T c are used for
evaluation.
For a target transcription factor assay s ∈ Tˆ c, let p be a binary vector {0, 1}n indicating the
genomic positions where s has a peak as called by the peak-calling algorithm. That is, p(i) = 1
if there is a peak at position i, and p(i) = 0 otherwise. We use a support vector machine (SVM)
with Gaussian kernel to predict p given a panel of assays S ⊆ Vc. For a given testing factor p,
we refer to the positions where p = 1 as I+ and the set of positions where p = 0 as I− = I \ I+.
We randomly choose ITr+ ⊆ I+ and ITr− ⊆ I− as the positive and negative positions to generate
training samples. Similarly, the testing samples are randomly chosen from ITe+ ⊆ I+ \ ITr+ and
ITe− ⊆ I− \ ITr− . Given the panel S = {s1, . . . , s|S|} of assays and the set of positive training
genomic positions ITr+ , we construct the set of positive training samples as DTr+ = {xi,+1}i∈ITr+
where xi = [s1(i), . . . , s|S|(i)]T . Similarly, we construct the negative training samples, positive
test samples, and negative test samples as DTr− , DTe+ , and DTe− , respectively. The SVM is first
trained on the training data set DTr = {DTr+ ,DTr− }, and then evaluated on the testing data set
DTe = {DTe+ ,DTe− }.
Because there are far more genomic positions that are not a functional element than there are
positions that are, measures of predictive accuracy such as the total fraction of correct predic-
tions (“accuracy”) and the area under the receiver operating characteristic curve do not offer a
reasonable measure of performance. Instead, we compute the area under the curve of a precision-
recall plot (AUC-PR), which is particularly well suited for settings with imbalanced class distri-
butions [Craven and Bockhorst, 2005, Davis and Goadrich, 2006]. In our experiments we set
|ITr+ | = 200, |ITr− | = 20, 000, |ITe+ | = 100 and |ITe− | = 10, 000. We apply 5-fold cross validation
for tuning the hyperparameters of the SVM. Let γs′ be the normalized area under curve for the
precision-recall plot (i.e., γs′ ∈ [0, 1]) for each target assay s′ ∈ Tˆ c. We illustrate this procedure
schematically in Figure 5.3B. We report the performance as the average AUC-PR on all target
assays, i.e., γ = 1Tˆ c
∑
s′∈Tˆ c γs′ .
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5.4.9 Annotation-based evaluation
The annotation-based evaluation metric measures the quality of a panel of genomics assays ac-
cording to the quality of the genome annotation that is obtained by inputting the panel into a
semi-automated genome annotation (SAGA) algorithm. SAGA algorithms are widely used to jointly
model diverse genomics data sets. These algorithms take as input a panel of genomics assays and
simultaneously partition the genome and label each segment with an integer such that positions with
the same label have similar patterns of activity. These algorithms are considered “semi-automated”
because a human performs a functional interpretation of the labels after the annotation process.
Examples of SAGA methods include HMMSeg [Day et al., 2007], ChromHMM [Ernst and Kellis,
2010], Segway [Hoffman et al., 2012] and others [Thurman et al., 2007, Lian et al., 2008, Filion
et al., 2010]. These genome annotation algorithms have had great success in interpreting genomics
data and have been shown to recapitulate known functional elements including genes, promoters
and enhancers. We use the SAGA method Segway in this work.
In order to apply annotation-based evaluation to a panel of assays, we input this panel into a
SAGA algorithm and evaluate the resulting annotation (Figure 5.3C). Intuitively, a diverse panel of
assays input to a SAGA algorithm should more accurately capture important biological phenomena
than a redundant panel. To evaluate the quality of an annotation relative to a particular genomics
data set, we use the variance explained measure [Libbrecht et al., 2015]. Given an evaluation cell
type c we randomly partition Sc into a source set Vc and a target set T c. For a given panel of assays
S ⊆ Vc, we first train a Segway model based on the panel and then obtain an annotation y. Segway
outputs an annotation y ∈ Yn, where Y = {1, 2, . . . , k} is a set of k labels that an annotation can
take on at each genomic position. For each target assay s′ ∈ T c, we measure the quality of the
annotation y as how well it explains the variance of the assay s′. We first compute the signal mean
of s′ over the positions assigned a given label ` as
µ` ,
∑n
i=1 1(y(i) = `)s
′(i)∑n
i=1 1(y(i) = `)
for ` ∈ {1 . . . k}. (5.6)
We then define a predicted signal vector sˆ′ with sˆ′(i) = µy(i) and compute the prediction error as
136
di = sˆ
′(i)− s′(i). We compute the residual standard deviation of the signal vector as
σres , stdev(d1:n) =
√√√√ 1
n
n∑
i=1
(di −mean(d1:n))2 =
√√√√ 1
n
n∑
i=1
d2i . (5.7)
The last equality holds because mean(d1:n) = 0 by construction. σres measures the residual standard
deviation of the target assay s′ accounting for the annotation y. Let σov = stdev(s′(1 : n)) be the
overall standard deviation of the assay s′. The normalized variance explained by the annotation y is
then
αs′ =
σov − σres
σov
. (5.8)
Observe that σov always upper bounds σres. The measure αs′ ∈ [0, 1] represents the fraction of the
variance of the assay s′ explained by the annotation y, where larger values indicate better agreement.
In our experiments, we trained the Segway model with 10 EM random initializations (using
GMTK [Bilmes and Rogers, 2015]) and 15 labels at 100 base-pair resolution. We report the
performance as the averaged measure on all target assays as α = 1|T c|
∑
s′∈T c αs′ .
5.5 Source code
Source code for SSA and computed assay type similarity matrix are available as Supplemental Mate-
rial and online at http://github.com/melodi-lab/Submodular-Selection-of-Assays.
5.6 Author contributions
All authors devised the method and designed the analysis. KW and MWL analyzed the data,
implemented the method, developed the software and wrote the initial manuscript. All authors
edited and approved the final manuscript. WSN and JAB jointly supervised the project.
5.7 Competing interests
The authors declare no competing interests.
137
5.8 Supplementary Information
5.9 Supplementary Note 1: Comparison of potential objective functions
We compared four classes of submodular objectives for selecting panels of assays. All functions are
defined via a pairwise similarity graph, where we denote rv,v′ as the similarity measure between the
v and v′ ∈ V .
Facility location function:
ffac(A) =
∑
v∈V
max
a∈A
rv,a (5.9)
This function is monotone and submodular.
Saturated coverage function:
fsat(A) =
∑
v∈V
min{
∑
a∈A
rv,a, β
∑
v′∈V
rv′,v}, (5.10)
In this function, 0 ≤ β ≤ 1 is a hyperparameter that controls the saturation of the coverage for each
item v ∈ V . In the experiment we set β = k/n, where k is the target number of assays to select,
and n is the size of entire set V of all available assays. The saturated coverage function is also
monotone submodular.
Diversity function:
fdiv(A) = −
∑
a∈A
∑
a′∈A
ra,a′ (5.11)
The diversity function evaluates a subset A as the sum of the pairwise dissimilarity among all items
in the subset. Maximizing this function naturally encourages choosing a diverse set of items. The
diversity function is submodular, but not monotone.
138
Log determinant function:
flog-det(A) = log det(I + λSA) (5.12)
In this function, λ > 0 is a hyperparameter, and SA is the pairwise similarity matrix indexed
by the subset A ⊆ V . The log determinant function satisfies submodularity and monotonicity.
This function has been shown to naturally capture the notion of diversity in a data set; therefore,
maximizing this function always leads to a set of diverse items.
We measure the performance of these submodular objectives by examining how their objective
valuations correlate with the three proposed evaluation metrics (assay imputation, functional element
prediction, and annotation-based evaluation). The experiment is performed under the selection of
past assays setting, where we compute the similarity between a pair of assays using the Pearson
correlation. Given an evaluation metric, a cell type c, a selection budget K, and a submodular
objective f , we randomly draw n subsets {Ai}ni=1 of assays from the cell type c, where each subset
Ai is of size K. We then compute the Spearman correlation between the submodular valuation
{f(Ai)}ni=1 and the performance measure {yi}ni=1 under the evaluation metric. We report the
correlation measure averaged over budget constraints K = 3, 4, 5, 6. In the experiment we set
n = 20, and we test on the cell types K562, H1-hESC, and GM12878.
The results (Figure 5.6) suggest that the facility location function, in most cases, yields the
highest correlation with the three evaluation metrics.
5.10 Supplementary Note 2: Comparison among various similarity measures
In this section, we study how the performance of SSA for the selection of future assays scenario
varies with different choices of similarity aggregation strategies. We focus on strategies for defining
the pairwise similarity between assay types in an evaluating cell type c given the assays performed
on cell types other than c. We consider the six aggregation strategies r1, r2, r3, r4, r5, and r6, where
they take the average, 0th, 25th, 50th, 75th, and 100th percentile over the available similarity scores,
139
H1-hESC K562 GM12878
0
0.2
0.4
0.6
0.8 Assay imputation
H1-hESC K562 GM12878
Sp
ea
rm
an
 c
or
re
la
tio
n
0
0.2
0.4
0.6
0.8 Functional Element Prediction
H1-hESC K562 GM12878
0
0.2
0.4
0.6
0.8 Annotation-based Evaluation
Facility location
Saturated coverage
Diversity function
Log determinant
Supplementary Figure 5.6: Comparison among various submodular functions in terms of the
Spearman correlation between function valuation and the performance metrics.
140
respectively. For completeness, we also give their corresponding mathematical definition below:
rsi,sj ,
1
|Sai \ si|
1
|Saj \ sj|
∑
s∈Sai\si
.
∑
s′∈Saj \sj
|ρs,s′ |, (5.13)
r2si,sj , percentile({|ρs,s′| : s ∈ Sai , s′ ∈ Saj \ sj}, 0), (5.14)
r3si,sj , percentile({|ρs,s′ | : s ∈ Sai , s′ ∈ Saj \ sj}, 25), (5.15)
r4si,sj , percentile({|ρs,s′ | : s ∈ Sai , s′ ∈ Saj \ sj}, 50), (5.16)
r5si,sj , percentile({|ρs,s′ | : s ∈ Sai , s′ ∈ Saj \ sj}, 75), (5.17)
r6si,sj , percentile({|ρs,s′ | : s ∈ Sai , s′ ∈ Saj \ sj}, 100), (5.18)
where the function percentile(C, p) returns the pth percentile of the items in the list C sorted in
non-decreasing order.
We evaluate the performance of the aggregation strategies using the facility location function.
In particular we utilize the six different aggregation strategies to compute the pairwise similarity
measure for defining the facility location function. We test separately on three cell types: H1-hESC,
K562, and GM12878. Similar to the previous experiment for comparing among different submodular
objectives, we measure the performance of an aggregation strategy as how the valuation given by
facility location function defined via the similarity matrix computed using this strategy correlates
with the three proposed evaluation metrics. Following the same evaluation procedure we show the
Spearman correlation measure for each aggregation strategy on each cell type and evaluation metric
in Figure 5.7. We observe that the aggregation strategies of taking the mean or 75th percentile yield
the best performance for most cell types and most evaluation metrics. Between these two strategies
we observe that the aggregation by mean performs marginally better. Therefore we choose it as the
strategy for computing the similarity between assay types in the propose approach SSA under the
selection of future assays setting.
141
H1-hESC K562 GM12878
0
0.2
0.4
0.6
0.8
Assay imputation
H1-hESC K562 GM12878
Sp
ea
rm
an
 c
or
re
la
tio
n
0
0.2
0.4
0.6
0.8
Functional Element Prediction
H1-hESC K562 GM12878
0
0.2
0.4
0.6
0.8
Annotation-based Evaluation
Mean
25% Quantile
50% Quantile
75% Quantile
Max
Min
Supplementary Figure 5.7: Comparison among various similarity aggregation strategies in terms
of the Spearman correlation between the facility location function valuation instantiated by the
similarity measure and the performance metrics.
142
5.11 Supplementary Note 3: Comparison of different supports for similarity computation
In this section, we examine how the similarity measure may be affected if we change the support for
deriving the similarity between assay types. In the main paper, we derive the similarity between
assays as the Pearson correlation between the values of these two assays on a randomly chosen
subset of genomic positions. In this section, we explore deriving the similarity based only on the
set of genomic positions that are identified as DNase peaks, since the response in the transcription
factors is only captured on these DNase peaks positions.
In Figure 5.8, we report the performance of the DNase peaks based SSA on all evaluation metrics
as well as the three cell types. For completeness, we also show the corresponding performance of
the variant of SSA that we used in the main main paper in Figure 5.9. Note that this variant of SSA
has the similarity measure computed over a randomly sampled subset of genomic positions. To
better illustrate the comparison between these two variants, we plot the performance measure of
the DNase peaks approach (each bar in Figure 5.8) against the random genomic positions approach
(each bar in Figure 5.9) in the form of scatter plot (Figure 5.10). We observe that consistent and
significant improvements over the random selection baseline are achieved by the variant of SSA
using DNase peaks only. Between the two variants of SSA, it is hard to establish the superiority of
one variant over the other according to Figure 5.10.
5.12 Supplementary Note 4: Comparison of different correlation metrics as the similarity
measure
In this section, we examine how the similarity measure may be affected by different choice of
correlation metrics. For the results reported in the main paper, we derive the similarity between
assays via the absolute Pearson correlation. Here, we explore a variant of SSA whose similarity
measure is defined via the absolute Spearman correlation. In Figure 5.11, we report the performance
of such variant on all evaluation metrics as well as the three cell types. To contrast the performance
difference between the Pearson correlation-based SSA and Spearman correlation-based SSA, we
again show a scatter plot of the performance achieved between these two variants in Figure 5.12.
143
3 4 5 6
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K562
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GM12878
Assay imputation
Functional element prediction
Annotation-based evaluation
3 4 5 6
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GM12878
Supplementary Figure 5.8: Performance of the Dnase peaks-based SSA relative to an estimate
of the performance on all possible panels. The vertical axis shows the fraction (%) of panels
that perform worse than the SSA-chosen panel for a given setting, estimated by comparing to 40
randomly-selected panels.
We observe that the Pearson correlation based SSA significantly and consistently outperforms the
Spearman correlation counterpart suggesting that Pearson correlation is more effective for capturing
similarity between assays in our tasks.
5.13 Supplementary Tables
144
3 4 5 6
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40
60
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H1-hESC
3 4 5 6
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H1-hESC
3 4 5 6
0
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40
60
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100
K562
3 4 5 6
0
20
40
60
80
100
K562
3 4 5 6
0
20
40
60
80
100
GM12878
Assay imputation
Functional element prediction
Annotation-based evaluation
3 4 5 6
0
20
40
60
80
100
GM12878
Supplementary Figure 5.9: Performance of the random genomics positions-based SSA relative to an
estimate of the performance on all possible panels.
Choice order Assay type Singleton score Objective gain
1 NFATC1 36.38 36.38
2 ILF2 33.37 4.60
3 PHF8 27.01 3.06
4 POU5F1 22.36 2.26
5 SMARCC1 13.63 1.93
6 IRF4 28.22 1.42
7 HNF4G 15.78 1.33
8 HMGN3 24.93 1.27
9 ILF3 29.09 1.04
10 ZNF217 16.00 1.03
Supplementary Table 5.1: The choice order of SSA on all assay types.
145
DNase peaks only
0.4 0.5 0.6 0.7 0.8 0.9 1
Sa
m
pl
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fro
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ge
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 p
os
iti
on
s
0.4
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0.9
1
Supplementary Figure 5.10: Scatter plot between the two variants of SSA with different genomic
support for similarity computation. Each dot in the plot corresponds to the performance (again
measured as relative to an estimate of the performance on all possible panels) of the two variants
of SSA for a selection budget evaluated using a metric in a cell type (i.e., one bar in Figure 5.8
and its counterpart in Figure 5.5). Its x- and y-values are the performance measure for the DNase
peaks-based SSA and random genomic positions-based SSA, respectively.
146
3 4 5 6
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K562
3 4 5 6
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K562
3 4 5 6
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40
60
80
100
GM12878
Assay imputation
Functional element prediction
Annotation-based evaluation
3 4 5 6
0
20
40
60
80
100
GM12878
Supplementary Figure 5.11: Performance of the absolute Spearman correlation-based SSA relative
to an estimate of the performance on all possible panels.
147
Spearman correlation
0 0.2 0.4 0.6 0.8 1
Pe
ar
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Supplementary Figure 5.12: Scatter plot between the two variants of SSA with different correlation
metrics as the similarity measure. Each dot in the plot corresponds to the performance (again
measured as relative to an estimate of the performance on all possible panels) of the two variants of
SSA for a selection budget evaluated using a metric in a cell type (i.e., one bar in Figure 5.11 and
its counterpart in Figure 5.5). Its x- and y-values are the performance measure for the Spearman
correlation-based SSA and the Pearson correlation-based SSA, respectively.
148
Order None H3K4me1 H3K4me2 H3K9ac H2A.Z H4K20me1 H3K27ac
1 H3K4me3 H3K4me1 H3K4me2 H3K9ac H2A.Z H4K20me1 H3K27ac
2 H3K79me2 H3K4me3 H3K79me2 H3K79me2 H3K79me2 H3K4me3 H3K79me2
3 H3K9me3 H3K79me2 H3K4me3 H3K9me3 H3K4me3 H3K79me2 H3K4me3
4 H3K27me3 H3K9me3 H3K9me3 H3K27me3 H3K9me3 H3K9me3 H3K9me3
5 H3K36me3 H3K27me3 H3K27me3 H3K4me3 H3K36me3 H3K27me3 H3K27me3
6 H3K4me1 H3K36me3 H3K36me3 H3K36me3 H3K27me3 H3K36me3 H3K36me3
7 H3K4me2 H3K4me2 H3K4me1 H3K4me1 H3K4me1 H3K4me1 H3K4me1
8 H3K9ac H3K9ac H3K9ac H3K4me2 H3K4me2 H3K4me2 H3K4me2
9 H2A.Z H2A.Z H2A.Z H2A.Z H3K9ac H3K9ac H2A.Z
10 H4K20me1 H4K20me1 H4K20me1 H4K20me1 H4K20me1 H2A.Z H3K9ac
11 H3K27ac H3K27ac H3K27ac H3K27ac H3K27ac H3K27ac H4K20me1
Order None MAFF ZNF263 BDP1 E2F6 RBBP5
1 SMARCB1 MAFF ZNF263 BDP1 E2F6 RBBP5
2 PML SMARCB1 SMARCB1 SMARCB1 SMARCB1 PML
3 STAT5A PML PML PML PML SMARCB1
4 CTCF CTCF STAT5A STAT5A CTCF STAT5A
5 BRF2 STAT5A CTCF CTCF STAT5A CTCF
6 MAFF BRF2 BRF2 BRF2 BRF2 BRF2
7 ZNF263 ZNF263 MAFF MAFF MAFF MAFF
8 BDP1 BDP1 BDP1 ZNF263 ZNF263 ZNF263
9 E2F6 E2F6 E2F6 E2F6 BDP1 BDP1
10 RBBP5 RBBP5 RBBP5 RBBP5 RBBP5 E2F6
Supplementary Table 5.2: Performing SSA after one assay has already been performed. We
ran SSA while restricting the output panel to include each assay type in turn, by initializing the
submodular greedy algorithm with the assay type in question and then running the algorithm as
normal. As expected, restricting the panel to include a particular assay type de-prioritizes assay
types that measure similar types of activity. (Top) Choice order of histone modification assays by
SSA if one assay type is required to be included. (Bottom) Choice order of transcription factors
by SSA if one assay type is required to be included. In both tables, each assay type in the table is
encoded with a different color to allow easy comparison to different columns.
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Order α = 0 α = 0.5 α = 1 α = 5 α = 10
1 NFATC1 NFATC1 NFATC1 NFATC1 H2A.Z
2 ILF2 ILF2 ILF2 H3K4me3 H3K4me3
3 PHF8 PHF8 H3K4me3 H3K9me3 H3K79me2
4 POU5F1 POU5F1 POU5F1 H3K79me2 H3K9me3
5 SMARCC1 SMARCC1 SMARCC1 H3K27me3 H3K27me3
6 IRF4 IRF4 PML H3K36me3 H3K36me3
7 HNF4G HNF4G H3K79me2 H3K4me1 H3K4me2
8 HMGN3 HMGN3 MBD4 H2A.Z H3K4me1
9 ILF3 H3K4me3 H3K27me3 H3K4me2 H3K9ac
10 ZNF217 H3K79me2 HMGN3 H3K9ac H4K20me1
11 CCNT2 ILF3 H3K36me3 H4K20me1 H3K27ac
12 PML ZNF217 H3K9me3 H3K27ac NFATC1
13 CEBPZ CCNT2 H3K4me1 ILF2 ILF2
14 NR2F2 PML IRF4 POU5F1 POU5F1
15 MLL5 CEBPZ H3K4me2 PML PML
16 SREBF2 NR2F2 H3K9ac SMARCC1 SMARCC1
17 XRCC4 MLL5 H2A.Z MBD4 MBD4
18 IKZF1 SREBF2 ZNF217 HMGN3 HMGN3
19 MBD4 XRCC4 H4K20me1 IRF4 IRF4
20 PRDM1 IKZF1 CCNT2 ZNF217 ZNF217
Supplementary Table 5.3: Incorporating modular weighting to encourage certain assay types
to be chosen. We ran SSA with a variant that weights certain assay types more highly in the
selection process. In this experiment, we examine the choice order of assay types by changing the
weights on histone modification assays in a selection experiment involving both histone modification
and transcription factor assays. We implement this experiment by first defining a combined objective
hα(A) = f(A) + αm(A), where f is the facility location objective and m is the modular function,
with each modular score capturing the importance of the item. Note that a set function m is modular
if m(A) =
∑
a∈Am(a) holds for all A ⊆ V . We define a hyperparameter α > 0 that controls the
trade-off between f and m. We define the modular function m by assigning the score m(a) for
each histone mod assay as 1 and for other assay types as 0. Given a choice of α, we use the greedy
algorithm to optimize hα leading to an ordering of the assays. The table displays the resulting
order for various choices of α. All histone modification assay types are color encoded for ease of
comparison.
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Order Cell types Description Singleton value Objective gain
1 HBMEC brain microvascular endothelial cells (mesoderm blood vessel ) 48.31 48.31
2 LNCaP prostate adenocarcinoma (endoderm prostate cancer) 46.77 1.08
3 GM12864 B-lymphocyte (mesoderm blood) 43.74 0.39
4 IMR90 fetal lung fibroblasts (endoderm lung) 43.26 0.37
5 H7-hESC undifferentiated embryonic stem cells 40.13 0.33
Supplementary Table 5.4: The choice order of cell types using SSA methodology.
Assay Type
1 DnaseSeq
2 H3K4me3
3 CTCF
4 FaireSeq
5 POLR2A
6 H3K27me3
7 H3K36me3
8 H3K79me2
9 H3K4me2
10 H3K9me3
Supplementary Table 5.5: The 10 most frequently performed assay types.
Choice order Assay type Singleton score Objective gain
1 DnaseSeq 4.311 4.311
2 H3K4me3 4.224 1.338
3 CTCF 2.515 0.837
4 H3K79me2 2.997 0.825
5 FaireSeq 2.624 0.808
6 POLR2A 3.293 0.788
7 CEBPB 3.001 0.78
8 TAF1 3.047 0.773
9 REST 2.641 0.694
10 GABPB1 3.684 0.692
11 MAX 3.515 0.677
12 EP300 3.7 0.664
13 H3K9ac 2.84 0.62
14 RAD21 4.245 0.192
Supplementary Table 5.6: The choice order of SSA-past on common assay types shared among the
six ENCODE tier 1+2 cell types: K562, GM12878, H1-hESC, A549, HepG2, and HeLa-S3.
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Chapter 6
ELIMINATING REDUNDANCY AMONG PROTEIN SEQUENCES
USING SUBMODULAR OPTIMIZATION
Abstract
Submodular optimization, a discrete analogue to continuous convex optimization, has been used
with great success in many fields but is not yet widely used in biology. We apply submodular
optimization to the problem of removing redundancy in protein sequence data sets. This is a
common step in many bioinformatics and structural biology workflows, including creation of
non-redundant training sets for sequence and structural models as well as selection of “operational
taxonomic units” from metagenomics data. We demonstrate that the submodular optimization
approach results in representative protein sequence subsets with greater structural diversity than
sets chosen by existing methods. In particular, we compare to a widely used, heuristic algorithm
implemented in software tools such as CD-HIT, as well to as a variety of standard clustering
methods, using as a gold standard the SCOPe library of protein domain structures. In this setting,
submodular optimization consistently yields protein sequence subsets that include more SCOPe
domain families than sets of the same size selected by competing approaches. We also show how
the optimization framework allows us to design a mixture objective function that performs well for
both large and small representative sets. The framework we describe is theoretically optimal under
some assumptions, and it is flexible and intuitive because it applies generic methods to optimize
one of a variety of objective functions. This application serves as a model for how submodular
optimization can be applied to other discrete problems in biology. Source code is available at
https://github.com/mlibbrecht/submodular_sequence_repset.
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Toxins' membrane translocation domains
Bacterial photosystem II !
reaction centre, L and M subunits
Transmembrane beta-barrels;!
Porin
Voltage-gated !
potassium channels
Aquaporin-like
Family A G protein-coupled receptor-like
Leukocidin-like
Transmembrane beta-barrels;!
Ligand−gated protein channel
Clc chloride channel
Single transmembrane helix
Supplementary Figure 6.1: Protein sequence representative set selection. Points represent protein
sequences, projected into 2D based on their pairwise similarities according to BLAST using t-SNE
[Van der Maaten and Hinton, 2008]. All sequences are depicted from the SCOPe Class “Membrane
and cell surface proteins and peptides.” Color with text labels indicates SCOPe fold, and italicized
text indicates SCOPe family within a given fold. Black circles indicate the top ten representative
sequences chosen by the greedy algorithm on a mixture of facility-location (Rankprop sim; 0.5
weight) and sum-redundancy (percent ID sim; 0.5 weight). Numbers labeling these circles indicate
choice order. Choice #9 is colored red because it is the second chosen sequence within the same fold.
To produce the 2D embedding, we first computed a n× n pairwise distance matrix (for a set of n
sequences) with the formula: distance(i, j) = 1− percent id(i, j)/100. We used multidimensional
scaling (MDS) to project this distance matrix to a 30× n feature matrix . This MDS preprocessing
step is similar to the standard preprocessing step used by tSNE that applies principle component
analysis (PCA) to project a high-dimensional feature matrix to a 30 × n feature matrix [Van der
Maaten and Hinton, 2008]. We used tSNE to project this 30×n matrix into a 2×n matrix, which is
plotted. The diffuse points in the center are a common feature of tSNE plots, and are a consequence
of the algorithm imperfectly recapitulating the n-dimensional similarity matrix in two dimensions.
153
6.1 Introduction
Redundancy is ubiquitous in biological sequence data sets, and this redundancy often impedes
analysis. For example, because some proteins, such as those involved in disease or industrial
applications, are studied by many more researchers than proteins with less important or unknown
function, protein sequence databases contain hundreds or thousands of slight variants of well-studied
proteins and only one or a few copies of the sequences of less-studied proteins. Removing this
redundancy has two benefits. First, doing so removes computational and statistical issues introduced
by this redundancy. Second, members of a non-redundant set of sequences can be interpreted
as representatives of the whole set, such as using sequences to represent species or “operational
taxonomic units” in a metagenomics study [Human Microbiome Project Consortium, 2012].
Redundancy can be handled by selecting a representative subset of the original data set. This
representative subset is then used in downstream analysis. However, the task of choosing this subset
can be computationally challenging because finding the best representative subset out of N items
requires, in principle, considering all 2N possible subsets.
In many fields, representative subsets are selected using some form of submodular optimization.
The class of submodular functions is analogous to the class of convex functions. Submodular
functions, however, are defined over subsets of a given set of data items, whereas convex functions
are defined over real-valued vectors. Submodular functions are those that satisfy the property
of diminishing returns. If we think of a function f as measuring the quality of a given subset,
then the submodular property means that the incremental “value” of an item v decreases as the
context in which v is considered grows. Formally, for any two subsets A and B of the set V , where
A ⊆ B ⊂ V , and for any v ∈ V \B, f is submodular if and only if it satisfies f(A∪{v})−f(A) ≥
f(B∪{v})−f(B). In many applications, it is common to search for a subset of maximal quality, as
measured by a function f . The resulting optimization problem is hopelessly difficult for an arbitrary
set function, but when the set function is submodular, the quality can be maximized within a constant
factor of optimal in low-order polynomial time [Fisher et al., 1978, Nemhauser et al., 1978b, Minoux,
1978] (Methods). Moreover, these optimization algorithms are provably the best achievable in
154
polynomial time under some assumptions (Methods). For these reasons, submodular optimization
has a long history in economics [Vives, 2001, Carter, 2001], game theory [Topkis, 1998, Shapley,
1971], combinatorial optimization [Edmonds, 1970, Lova´sz, 1983, Schrijver, 2004], electrical
networks [Narayanan, 1997], and operations research [Cornune´jols et al., 1990]. Furthermore,
submodular optimization has recently been used with great success for selecting representative
subsets of text documents [Lin et al., 2007, Lin and Bilmes, 2011, 2012], recorded speech [Liu
et al., 2013, Wei et al., 2013, 2014], machine translation [Kirchhoff and Bilmes, 2014], and images
[Tschiatschek et al., 2014]. However, submodular optimization is not yet widely used in biology.
To illustrate how submodular optimization can be applied to biological data sets, we focus
on the problem of choosing representative subsets from a database of protein sequences. Many
methods have been developed for this problem [Hobohm et al., 1992, Holm and Sander, 1998, Li
et al., 2001, Edgar, 2010, Parsons et al., 1992, Altschul et al., 1997, Enright and Ouzounis, 2000,
Rice et al., 2000], including the popular CD-HIT [Weizhong and Adam, 2006] and PISCES [Wang
and Dunbrack, Jr., 2003] algorithms. These sequence selection methods are very widely used—for
example, the CD-HIT papers have been cited a total of >3,000 times (Google Scholar)—and are
a standard preprocessing step applied to data sets of protein sequences, cDNA sequences and
microbial DNA. All of these existing methods are based on the following simple algorithm: start
with an empty set; order the sequences by length; then for each sequence, add the sequence to the set
if no sequence currently in the set is more similar to it than some threshold (usually 90% sequence
identity). Despite the ubiquity of this algorithm, it is simply a heuristic, apparently without any
theoretical or empirical justification for its use over any other.
We propose a principled framework for representative protein sequence subset selection using
submodular optimization. This approach involves defining a submodular objective function that
quantifies the desirable properties of a given subset of sequences, and then applying a submodular
optimization algorithm to choose a representative subset that maximizes this function. For illustrative
purposes, we applied this optimization framework to protein domain sequences drawn from a single
class (“Membrane and cell surface proteins and peptides”) within the Structural Classification of
Proteins—extended (SCOPe) database [Murzin et al., 1995]. The availability of protein structures
155
provide an orthogonal gold standard for judging the quality of a given representative subset. The
results (Figure 6.1) suggest that the submodular approach does a good job of selecting protein
sequences with diverse structural properties. In particular, among the nine folds that comprise
our selected class, the submodular method successively chooses one exemplar sequence from fold
before selecting two sequences from the same fold. The only exception is that the method chooses
two “transmembrane beta-barrel” protein domains before selecting the first “single transmembrane
helix” domain; however, as illustrated in Figure 6.1, this choice is not particularly surprising, given
that the “transmembrane beta-barrel” fold is comprised of two superfamilies with quite different
structural properties.
In the remainder of this paper, we expand upon this example, demonstrating that submodular
optimization method does a better job of selecting representative protein sequence sets than compet-
ing methods, when evaluated relative to protein structure gold standards. We also demonstrate how
optimization-based method development offers practical advantages over an approach in which the
algorithm and the property it tries to optimize (its objective function) are inextricably intertwined.
In particular, we demonstrate how a hybrid optimization function allows us to design a method that
excels at selecting either large or small representative sets.
Note that the task of representative subset selection is distinct from that of clustering. Clustering
focuses on sets of similar sequences, whereas representative subset selection focuses on the indi-
vidual sequences chosen to represent a larger set. A large number of methods have been proposed
for protein sequence clustering, including those based on Markov clustering [Enright et al., 2002],
connected component detection [Enright and Ouzounis, 2000], generative modeling [Yang and
Wang, 2003], minimal spanning trees [Guan and Du, 1998], and spectral clustering [Paccanaro et al.,
2006]. Applying these methods to representative subset selection requires specifying, in addition, a
procedure for selecting an exemplar sequence from each cluster. In practice, clustering methods
are rarely used for reducing redundancy in protein sequence sets. Nonetheless, we empirically
demonstrate below that our submodular selection approach outperforms several commonly used
clustering algorithms.
156
6.2 Methods
6.2.1 Submodularity
The property of submodularity is important for the optimization of the set functions defined below.
A submodular function [Fujishige, 2005] is defined as follows: given a finite set V = {1, 2, . . . , n},
a discrete set function f : 2V → R is submodular if and only if:
f(S ∪ {v})− f(S) ≥ f(T ∪ {v})− f(T ), ∀S ⊆ T ⊆ V, v /∈ T. (6.1)
In other words, the incremental gain of adding item s to the set decreases when the set to which s is
added to grows from S to T . Intuitively, most reasonable measures of the quality or “informativeness”
of a set of items are submodular because the gain from adding a given item is reduced when the set
already contains other items similar to it.
Two other properties of set functions are relevant to their optimization. First, a set function f is
defined as monotone non-decreasing if
f(S ∪ {v})− f(S) ≥ 0, ∀v ∈ V \ S, S ⊆ V. (6.2)
Second, we say that f is normalized if f(∅) = 0.
6.2.2 Similarity functions
The measures of subset quality described below are based on pairwise similarities of sequences.
Such similarity measures are real-valued functions s(u, v) defined on pairs of sequences. We
considered two measures to quantify the similarity between a pair sequences: (1) the fraction of
residues that match between the two sequences (fraction identical), and (2) the similarity measure
used by the Rankprop method, defined as exp(−E-value/100) [Weston et al., 2004]. Both measures
have been used successfully in previous work on protein sequences, and can be calculated efficiently
using an index-based method like BLAST [Altschul et al., 1990]. In both cases, we define the
157
similarity between all pairs of sequences not reported by BLAST as 0. Note that, due to differing
sequence lengths, both measures are not symmetric.
6.2.3 Subset quality measures
Let V be the full set of sequences and let s(u, v) be a measure of similarity between a pair of
sequences u, v ∈ V . A subset quality function is defined over subsets f(A), where A ⊆ V . We
consider several such subset quality functions, each of which measures some property of a set of
sequences. In each case, a higher value of the function is desired.
First, we define the facility-location function as
ffacility-location(A) ,
1
|V |
∑
v∈V
max
a∈A
s(a, v). (6.3)
Intuitively, this function takes a high value when every sequence in V has at least one similar
representative in A. This function is also the objective function of the k-medoids clustering method.
The facility-location function is submodular, normalized and monotone non-decreasing.
Second, we define the sum-redundancy function as
fsum-redundancy(A) , κsum −
∑
a1,a2∈A
s(a1, a2), (6.4)
where κsum =
∑
a,b∈V s(a, b). Intuitively, for sets A of a given size |A|, this function takes a very
small value when A includes many pairs of similar sequences, and it takes a large value when all the
sequences in A are very dissimilar from one another. In other words, the sum-redundancy function
penalizes the redundancy in A. Optimizing the sum-redundancy function with the monotone greedy
algorithm (defined below) is identical to the commonly-used representative subset heuristic of
iteratively choosing the example that is most dissimilar from all previously-chosen examples. The
sum-redundancy is submodular and normalized, but not monotone non-decreasing (in fact, it is
monotone non-increasing).
158
Third, we define the max-redundancy function as
fmax-redundancy(A) , κmax −
∑
a1∈A
max
a2∈A,a2 6=a1
s(a1, a2). (6.5)
where κmax =
∑
a1∈V maxa2∈V,a2 6=a1 s(a1, a2). The max-redundancy function is very similar to
sum-redundancy, except that it penalizes only the most similar neighbor for each sequence in A
instead of all neighbors. The max-redundancy is submodular and normalized, but not monotone
non-decreasing and (like above) is monotone non-increasing.
6.2.4 Optimization algorithms
We are interested in choosing a subset of sequences A that maximizes a given measure of quality
of the subset. Because all of the set functions defined above are either monotone increasing or
monotone decreasing in the size of the set, optimizing the function over all subsets would result in
either the empty set or the full set, depending on the function. Therefore, we additionally control
the size of the returned set using one of two strategies. In the first setting, we constrain the size of
the returned set to a specific size k and solve the optimization problem
maximize f(A) (6.6)
subject to |A| = k.
In the second setting, we add a term to the objective that is proportional to the size of the subset,
scaled by a hyperparameter λ, and we optimize the unconstrained function
maximize f(A) + λ|A|. (6.7)
We use the constrained strategy when using the monotone greedy algorithm (defined below) because
that algorithm supports such constraints. The bidirectional greedy algorithm (defined below) works
only for unconstrained problems, so we use the second strategy for that algorithm, varying λ to
159
indirectly control the resultant size. In practice, if a particular subset size (or some other feature of
the set such as minimum objective value) is desired, this can be achieved using binary search.
Maximizing an arbitrary set function in general requires considering all 2|V | possible subsets and
is therefore impractical for large sets of sequences. In fact, even when this set function is submodular,
maximizing it exactly is NP-hard. However, submodular functions can be efficiently approximated
within a constant factor of optimal. In this work we apply the following two optimization algorithms.
The first algorithm is the monotone greedy algorithm (Supplementary Algorithm 3). At each
iteration, the monotone greedy algorithm computes the difference in the objective function obtained
by adding each sequence and adds the sequence with the largest difference. It turns out that when the
objective function is submodular, monotone nondecreasing and normalized, this simple algorithm is
guaranteed to produce a solution within a factor of 1− 1/e ≈ 0.63 of optimal [Nemhauser et al.,
1978b]. This is the best approximation ratio achievable in polynomial time unless P=NP [Feige,
1998]. Furthermore, this algorithm can be sped up using the fact that the submodular property
guarantees that, if adding a sequence v to A would result in a difference of d in the objective, then
adding v to a larger set A ∪B will result in an objective difference d′ ≤ d. This property allows the
algorithm to skip considering many sequences, and in practice often results in near-linear running
time [Minoux, 1978].
The second algorithm is the bidirectional greedy algorithm (Supplementary Algorithm 4). This
algorithm, which is randomized, maintains a growing set, which is initialized to the empty set,
and a shrinking set, which is initialized to the full set. It considers each sequence v in turn, and
either adds v to the growing set or removes it from the shrinking set randomly with probability
proportional to the gain in the objective resulting from either change, respectively. At the end of
the algorithm, the growing set and shrinking set are identical, and this set is returned. When the
objective function is submodular and normalized (but not necessarily monotone nondecreasing), the
mean of the objective values found over many randomized runs of this algorithm is guaranteed to be
at least 1
2
of optimal [Buchbinder et al., 2012]. This is the best approximation ratio achievable in
polynomial time (independent of whether P=NP) [Feige et al., 2011].
In the experiments below, we applied the monotone greedy algorithm to optimize the (mono-
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tone) facility-location function (1− 1/e approximation guarantee). We applied the bidirectional
greedy algorithm optimize the (nonmonotone) sum-redundancy and max-redundancy functions
(1/2 approximation guarantee). The bidirectional greedy algorithm can be used on mixtures of
these functions to maintain a 1/2 approximation guarantee. However, we achieved better results
on the development data set using the greedy algorithm to optimize these mixtures (no theoretical
guarantee; Supplementary Figure 6.13), so we used this strategy in our subsequent experiments.
Note that the 1/2 approximation guarantee can be maintained by running both algorithms and
choosing the result with the better objective value. Moreover, although the greedy algorithm does
not have a mathematical guarantee for non-monotone problems, we have found that for many
problems it nonetheless performs better than the mathematically-guaranteed bidirectional greedy
algorithm (results not shown). As future work, we are interested in investigating mathematical
properties of these functions that may explain why the greedy algorithm is performing better than
the current theory predicts.
6.2.5 Comparison to threshold algorithm (CD-HIT)
We call the algorithm used by most previous sequence subset selection methods, including CD-HIT
and PISCES, the threshold algorithm (Supplementary Algorithm 2). This algorithm starts with an
empty return set. It considers each sequence in decreasing order of sequence length, adding a given
sequence to the return set if there is no sequence currently in the return set with similarity greater
than some threshold τ . The value of τ controls the size of subset returned: a small τ results in a
small set, while a large τ results in a large set.
In order to facilitate comparison, we implemented our own version of CD-HIT. The primary
difference between our implementation and the CD-HIT source is that our implementation uses pre-
computed BLAST alignments, while the CD-HIT source uses an on-the-fly index-based alignment
method similar to BLAST. The main advantage of this on-the-fly method is that it does not require
an expensive database construction preprocessing step. However, when comparing multiple methods
on a single data set, it is much more efficient to perform a single alignment step. Using a single
set of alignments also removes the possibility for performance differences due to slightly different
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alignment results. Note that the on-the-fly method could in principle be used in the context of either
the CD-HIT or submodular optimization. Our implementation of CD-HIT and the CD-HIT source
result in nearly-identical subsets (Supplementary Note 1).
6.2.6 Protein sequence data
To test these methods, we downloaded the full Structural Classification Of Proteins–extended
(SCOPe) database version 15 from http://scop.berkeley.edu [Murzin et al., 1995].
SCOPe is a hierarchical structural classification database, consisting of four levels with increasing
granularity: class, fold, superfamily, and family. In this work, we focus primarily on the family and
superfamily levels because these represent a reasonable level of granularity for subset selection.
We also downloaded the ASTRAL database from http://scop.berkeley.edu [Brenner
et al., 2000]. ASTRAL is a database of protein domain sequences, some of which are associated
with SCOPe categories. We used only ATOM-derived sequence records, and we chose the subset of
ASTRAL sequences that are associated with a SCOPe family, removing eight sequences that were
labeled with multiple SCOPe families. This resulted in a set of 78,048 sequences.
We also downloaded the results from all-by-all BLAST on the ASTRAL sequences from http:
//scop.berkeley.edu. BLAST was run using the command line blastpgp -d DBNAME
-i SEQ.fa -j 1 -h 1e-2 -v 5000 -b 5000 -a2 -z 100000000 -J T -O OUTPUT.
This results in all pairwise relationships with an E-value less than 0.01, where the similarity of each
pair is measured by an E-value and a fraction sequence identity.
In order to overfitting to SCOPe, we performed all development on just the 16,880 ASTRAL
sequences with the SCOPe Class “All beta proteins.” Unless otherwise noted, the figures show per-
formance on all of ASTRAL; we computed this performance only when preparing this manuscript.
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Supplementary Figure 6.2: (A) Average redundancy in representative sets. The vertical axis
is calculated for a representative set A as (1/|A|)∑a1,a2∈A s(a1, a2), where s(a1, a2) is percent
identity. Vertical lines correspond to the sizes of subsets selected by CD-HIT using 40% and 90%
sequence identity thresholds. (B) Histogram of pairwise similarities within representative sets.
We produced subsets of size 12,614 sequences (the size reported by CD-HIT with a 40% identity
threshold) for all methods by finding the smallest subset larger than the target size and randomly
removing sequences to reach this size. Vertical axis indicates the number of pairs of sequences
with a given percent identity within each size-12,614 representative set, in 5% bins. Vertical line
indicates 40% identity threshold used by CD-HIT.
6.3 Results
6.3.1 Submodular optimization effectively eliminates redundancy in sequence data sets
To evaluate how well submodular optimization can remove redundancy in protein sequence data sets,
we applied it to ASTRAL sequences. We focus on the sum-redundancy function with percent identity
objective function because this function is most comparable to CD-HIT and it performed best in our
later evaluation (see below). As expected, submodular optimization produced representative subsets
with less total redundancy (Figure 6.2A). This because submodular optimization is a theoretically-
based algorithm that explicitly aims to minimize redundancy, while the threshold algorithm is a
heuristic that lacks any guarantee. The improvement primarily comes from the fact that the threshold
method chooses many pairs of sequences with identity just under the threshold, whereas optimizing
sum-redundancy chooses very few sequences with any detectable identity at the cost of choosing a
small number of closely-related pairs (Figure 6.2B). The small number of closely-related pairs that
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Greedy on facility location
CD−HIT
Greedy on sum−redundancy
Bidirectional greedy
on sum−redundancy
Greedy on mixture of
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Greedy on mixture of
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Maximum possible
4000 4200 4400 4600 4800
Number of SCOPe Families covered:
area under the curve
Al
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Supplementary Figure 6.3: Coverage of SCOPe families. (A) The vertical axis indicates is the
number of families with at least one representative, normalized to the range [0, 1] using the formula
f ′(x) = f(x)−E(x)
M(x)−E(x) , where x is a subset size, f(x) is the value for the method in question on that
subset size, E(x) is expected (random mean) andM(x) is maximum possible (defined asM(x) = x
if x < F , and M(x) = F otherwise, where F = 4,994 is the total number of SCOPe families).
Vertical lines correspond to the sizes of subsets selected by CD-HIT using 40% and 90% sequence
identity thresholds. (B) Area under the curve comparison. Bars indicate selection algorithms,
ordered by their area under the curve. Horizontal axis indicates area under the “family” coverage
curve, calculated as follows. Let V be the full set of sequences, i ∈ 1 . . . |V | be a subset size, m be
a particular method, A(m)i be the subset of size i chosen by m, and f(A) be the number of SCOPe
Families covered by A. Area under the curve = 1|V |
∑|V |
i=1 f(A
(m)
i ). For subset sizes for which we
did not compute a subset, f() is imputed with linear interpolation.
are accepted is likely due to the randomized nature of the algorithm.
6.3.2 Representative subsets chosen by submodular optimization exhibit high structural diversity
In order to evaluate the quality of a subset of sequences independently of a specific objective
function, we compared these subsets to the Structural Classification Of Proteins–extended (SCOPe)
database. SCOPe contains 78,048 protein domain sequences, each assigned to one of 4,994 protein
families. These assignments are based on experimentally determined protein structures, so they are
independent from the sequence-based information used by the optimization methods. Intuitively, a
164
good representative subset should include one sequence from each family, whereas a poor subset may
include many sequences from the same family while leaving other families with no representatives.
Therefore, we defined the quality of a subset as the number of SCOPe protein families with at least
one sequence from that family in the subset.
Among the three submodular objective functions that we considered, we found that optimizing
the sum-redundancy function produced the best results on this evaluation metric. Our evaluation
considered 12 possible approaches: two submodular optimization algorithms on three different
objective functions, each computed using two different measures of sequence similarity. To avoid
overfitting to SCOPe, we performed all development and comparison of objective functions on a
subset of the data (∼21%) and applied only the three best methods to the full data set. Optimizing
sum-redundancy with the bidirectional greedy algorithm results in much better coverage of SCOPe
families than the threshold heuristic method that is employed by methods such CD-HIT (Figure 6.3).
For representative sets of size 12,614 (the size chosen by CD-HIT with threshold 40%), the set
chosen by submodular optimization leaves 27% fewer SCOPe families uncovered (276 compared
with 379) than the threshold method does. We reached a similar conclusion when performing the
evaluation on SCOPe superfamilies and folds rather than families (Supplementary Figures 6.8-6.13).
6.3.3 A mixture of submodular objective functions outperforms any single component objective
Although optimizing sum-redundancy performs best on the SCOPe-based evaluation metric for
large subsets, we found that optimizing a different function resulted in better performance for very
small subsets (<5,000 sequences; Figure 6.4A). Specifically, this good performance resulted from
(1) optimizing a function called facility-location instead of sum-redundancy, and (2) using the
similarity function used by the Rankprop method [Weston et al., 2004] that takes into account the
statistical confidence of an alignment, instead of percent identity. The facility-location function
is defined as ffacility-location(A) , 1|V |
∑
v∈V maxa∈A s(a, v) where V is the set of all sequences.
Intuitively, this function takes a high value when every sequence in V has at least one similar
representative in A. This function makes up for a weakness of redundancy-minimizing objectives
such as sum-redundancy: such functions cannot differentiate representative sets with no detectable
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Greedy on facility
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Bidirectional greedy
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Facility−location sim: percent ID; Sum−redundancy sim: Rankprop
Facility−location sim: Rankprop; Sum−redundancy sim: percent ID
Facility−location sim: Rankprop; Sum−redundancy sim: Rankprop
Supplementary Figure 6.4: (A) Same as 6.3A, but including mixture objectives. (B) Comparison of
mixture weights. Each mixture objective is a combination of facility-location and sum-redundancy
objectives, each with either a percent ID or Rankprop similarity function. Horizontal axis indicates
weight on facility-location (weight on sum-redundancy equals one minus the weight on facility-
location). Line color indicates mixture objective. Vertical axis indicates area under the curve for
SCOPe family coverage (see Supplementary Figure 6.10). To avoid overfitting, the statistics in (A)
were computed on the subset of sequences with the SCOPe Class “All beta proteins”.
similarity between any representatives. In contrast, facility-location can differentiate sets of unrelated
representatives by how well they cover unpicked sequences.
These observations suggest that sum-redundancy with the percent identical similarity metric and
facility-location with the Rankprop similarity metric may provide different types of information.
To test this hypothesis, we optimized a mixed objective function using the greedy algorithm and
evaluated the resulting representative subsets against the SCOPe evaluation metric. We found that
this mixture performed much better on SCOPe coverage than either function individually, both for
small and large subsets (Figures 6.3B and 6.4A). Moreover, this performance is quite insensitive to
mixture weight (Figure 6.4B). These results show that a mixture of objective functions can better
represent the quantity of interest than a single function. Optimizing mixtures of objective functions
this way is very natural using the submodular optimization framework, but would be very difficult
in a non-optimization-based framework.
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Greedy on facility location (K−medoids++)
Greedy on facility location (K−medoids++)
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Affinity propagation;
random representative
Agglomerative clustering;
random representative
Affinity propagation with rankprop sim;
central representative
CD−HIT
Supplementary Figure 6.5: Performance of clustering methods repurposed for representative set
selection. Plot is same as Figure 6.3, but includes clustering-based methods instead of submodular
optimization-based. Results are computed on our development set, composed of the subset of the
data (∼21%) with the SCOPe Class “All beta proteins”.
6.3.4 Representative subset selection methods perform better than repurposed clustering methods
An alternative to applying a representative subset selection method is to apply a clustering method
and then choose a representative from each identified cluster. Using the SCOPe evaluation metric,
we compared representative subset selection using submodular optimization to eight variants of this
clustering-based approach. We compared two similarity functions: percent identity and Rankprop;
three clustering methods: k-medoids++, affinity propagation and agglomerative hierarchical clus-
tering; and two methods for choosing a representative from each cluster: “random” and “central”.
Central representative selection chooses the cluster member with the highest total similarity to the
other cluster members (i.e. it optimizes a facility-location objective for a single item). Random
representative selection chooses a random member. Note that k-medoids++ is identical to optimizing
facility-location with the submodular greedy algorithm.
We found that none of these clustering methods performed as well as either the submodular
optimization-based methods or the threshold heuristic (Figure 6.5 and Supplementary Figure 6.13).
The exception, as noted above, is that optimizing facility-location with the Rankprop similarity
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metric performs well for small representative sets. This is likely because clustering methods aim to
optimize the quality of the entire cluster, whereas representative set selection methods focus solely
on identifying good representatives.
6.4 Discussion
The submodular optimization framework we present here results in theoretically guaranteed per-
formance in some circumstances, performs well in practice, and has the flexibility to express a
wide variety of desired properties of subsets of sequences. This approach is therefore preferable to
existing, widely used, heuristic methods for selecting representative subsets of biological sequences.
In this work we have explored only a small number of submodular functions; the submodular
optimization framework supports many other objective functions that may provide even better
results, including those based on cluster coverage, those based on the sequences directly rather than
alignments, and others. Because the SCOPe evaluation function is itself submodular, a submodular
objective function could also be efficiently learned from training data to correlate well with it.
Moreover, this application may serve as a model for how submodular optimization can be applied
to other discrete problems in biology, such as selecting representative sets of protein structures,
genomic loci, or mass spectra.
Acknowledgements This work was supported by the National Institutes of Health award R01 CA180777.
Supplementary Note 1: Implementation of threshold algorithm
In order to verify that our implementation of the threshold algorithm matches existing packages,
we ran our implementation of the threshold algorithm and the CD-HIT packages on the same data
set. The methods use slightly different sequence alignments, so we cannot expect them to choose
exactly the same subsets. Instead, we compared the values of relevant summary statistics: our three
submodular objective functions and SCOPe coverage. As expected, we found that the methods
compared very similarly (SI Appendix, Fig. 6.6). Both implementations performed nearly identically
on the three submodular objective functions, with a slight advantage to our implementation due
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Supplementary Figure 6.6: Comparison of the CD-HIT source and our implementation of the same
algorithm. Horizontal axis indicates number of sequences in a given subset, focused on the region
between 40% and 90% percent ID thresholds (horizontal black lines indicates these thresholds).
Line color indicates quality measure; vertical axis indicates quality value. All quality measures are
linearly transformed to fall between zero and one. Solid lines with triangles indicate performance
with our implementation; dotted lines with circles indicate the CD-HIT source.
to the fact that these objectives and our implementation use the same sequence alignments. Our
implementation performed somewhat better on SCOPe coverage, perhaps due to more accurate
sequence alignments. Importantly, our implementation performed at least as well or better on all
metrics.
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Supplementary Figures/Algorithms
Algorithm 2 Threshold algorithm. V is the set of sequences and s : V × V → R is a similarity
function over V . N : V → 2V is the set of neighbors of v with nonzero similarity. τ is a
hyperparameter threshold determining the size of the returned subset (low τ results in a small
subset).
1: function THRESHOLD(V, s)
2: Order V into a list (v1, v2, . . . , vn) in decreasing order of sequence length.
3: A← {v1}
4: for i = 2 . . . N do
5: m← max{s(vi, v′) : v′ ∈ A ∩N(vi)}
6: if m < τ then
7: A← A ∪ {vi}
8: end if
9: end for
10: return A
11: end function
Algorithm 3 Monotone greedy submodular maximization algorithm for optimizing a function f(A).
V is the set of sequences, and k is the size of subset to return.
1: function MONOTONE GREEDY(V, f )
2: A← {}
3: for 1 . . . k do
4: for v ∈ V \ A do . Can be accelerated by considering only certain v ∈ V (Methods).
5: δ[v]← f(A ∪ {v})− f(A)
6: end for
7: v∗ ← argmaxv∈V \Aδ[v]
8: A← A ∪ {v∗}
9: end for
10: return A
11: end function
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Algorithm 4 Bidirectional greedy maximization algorithm for optimizing a function f(A). V is the
set of sequences.
1: function BIDIRECTIONAL GREEDY(V, f )
2: A← {}, B ← V
3: for v ∈ V do
4: δA ← max(0, f(A ∪ {v})− f(A))
5: δB ← max(0, f(B \ {v})− f(B))
6: if δA = δB = 0 then
7: p = 1/2
8: else
9: p = δA/(δA + δB)
10: end if
11: if random(0, 1) ≤ p then
12: A← A ∪ {v}
13: else
14: B ← B \ {v}
15: end if
16: end for
17: return A (or B, equivalently)
18: end function
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Supplementary Figure 6.7: Optimization performance on each submodular objective: (A) facility-
location with percent ID sim; (B) facility-location with Rankprop sim; (C) sum-redundancy with
percent ID sim. Horizontal axis indicates size of representative set. Vertical axis indicates facility-
location objective value (higher is better). Color and point type indicates choice method. The highly
similar lines in panel (B) result from the fact that the Rankprop similarity function has a small
dynamic range (∼0.9-1.0).
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Algorithm
l Random mean
Greedy on facility location
Greedy on facility
location with Rankprop sim
Bidirectional greedy
on sum redudancy
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facility−location (Rankprop sim; 0.5) and
sum−redundancy (percent ID sim; 0.5)
CD−HIT with Rankprop sim
CD−HIT
(b)
l
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um
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Greedy on mixture of
facility−location (Rankprop sim; 0.5) and
sum−redundancy (percent ID sim; 0.5)
CD−HIT with Rankprop sim
CD−HIT
(c)
Supplementary Figure 6.8: Coverage of SCOPe categories: (A) Families; (B) Superfamilies, and
(C) Folds. Horizontal axis indicates size of representative set. Vertical axis indicates number of
SCOPe categories with at least one representative. Color and point type indicates choice method.
Black curve indicates maximum possible performance. Vertical black lines indicate subset sizes
reported by CD-HIT for thresholds of 40% and 90%.
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0.00
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location with Rankprop sim
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on sum redudancy
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sum−redundancy (percent ID sim; 0.5)
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on sum redudancy
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sum−redundancy (percent ID sim; 0.5)
CD−HIT with Rankprop sim
CD−HIT
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d) AlgorithmGreedy on facility
location with Rankprop sim
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on sum redudancy
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facility−location (Rankprop sim; 0.5) and
sum−redundancy (percent ID sim; 0.5)
CD−HIT with Rankprop sim
CD−HIT
(c)
Supplementary Figure 6.9: Same as Supplementary Figure 6.8, but vertical axis is normalized as in
Figure 2.
174
Random mean
CD−HIT with Rankprop sim
Greedy on facility
location with Rankprop sim
Greedy on facility location
CD−HIT
Greedy on sum redudancy
Bidirectional greedy
on sum redudancy
Greedy on mixture of
facility−location (0.1)
and sum−redundancy (0.9)
Greedy on mixture of
facility−location (Rankprop sim; 0.5) and
sum−redundancy (percent ID sim; 0.5)
Maximum possible
4000 4200 4400 4600 4800
Number of SCOPe Families covered: area under the curve
Al
go
rit
hm
Supplementary Figure 6.10: Area under the curve for SCOPe Family coverage. Bars indicate
selection algorithms, ordered by their area under the curve. Horizontal axis indicates area under the
curve, calculated as follows. Let V be the full set of sequences, i ∈ 1 . . . |V | be a subset size, m be
a particular method, A(m)i be the subset of size i chosen by m, and f(A) be the number of SCOPe
Families covered by A. Area under the curve = 1|V |
∑|V |
i=1 f(A
(m)
i ). For subset sizes for which we
did not compute a subset, f() is imputed with linear interpolation.
175
Random mean
Greedy on facility location
CD−HIT with Rankprop sim
CD−HIT
Greedy on sum redudancy
Greedy on mixture of
facility−location (0.1)
and sum−redundancy (0.9)
Bidirectional greedy
on sum redudancy
Greedy on facility
location with Rankprop sim
Greedy on mixture of
facility−location (Rankprop sim; 0.5) and
sum−redundancy (percent ID sim; 0.5)
Maximum possible
1800 1900 2000 2100 2200
Number of SCOPe Superfamilies covered: area under the curve
Al
go
rit
hm
Supplementary Figure 6.11: Area under the curve for SCOPe Superfamily coverage. Plot is same as
Supplementary Figure 6.10, except that horizontal axis indicates Superfamily coverage.
176
Random mean
Greedy on facility location
CD−HIT with Rankprop sim
CD−HIT
Greedy on sum redudancy
Bidirectional greedy
on sum redudancy
Greedy on mixture of
facility−location (0.1)
and sum−redundancy (0.9)
Greedy on facility
location with Rankprop sim
Greedy on mixture of
facility−location (Rankprop sim; 0.5) and
sum−redundancy (percent ID sim; 0.5)
1150 1200 1250 1300 1350
Number of SCOPe Folds covered: area under the curve
Al
go
rit
hm
Supplementary Figure 6.12: Area under the curve for SCOPe Fold coverage. Plot is same as
Supplementary Figure 6.10, except that horizontal axis indicates Fold coverage.
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Nonmonotone on facility−location with Rankprop sim
Agglomerative clustering with rankprop sim; central representative
Agglomerative clustering; central representative
Agglomerative clustering with rankprop sim; random representative
Agglomerative clustering; random representative
Random mean
Affinity propagation with rankprop sim; random representative
Affinity propagation with rankprop sim; central representative
Affinity propagation; central representative
Affinity propagation; random representative
Greedy on facility−location (K−medoids++) with Rankprop sim
CD−HIT using Rankprop sim
Greedy on max−redundancy with Rankprop sim
Greedy on max−redundancy with percent ID sim
Nonmonotone on facility−location with percent ID sim
Greedy on facility−location (K−medoids++) with percent ID sim
Greedy on sum−redundancy with Rankprop sim
CD−HIT using percent ID sim
Nonmonotone on sum−redundancy with Rankprop sim
Greedy on sum−redundancy with percent ID sim
Nonmonotone on sum−redundancy with percent ID sim
Maximum possible
600 700 800 900
Number of SCOPe Families covered:
area under the curve
Al
go
rit
hm
Supplementary Figure 6.13: Area under the curve for SCOPe Family coverage on development set.
Plot is same as Supplementary Figure 6.10, except that results are computed on our development
set, composed of the subset of the data (∼21%) with the SCOPe Class “All beta proteins”.
178
Chapter 7
CONCLUSION
In this conclusion section I propose some promising directions for future work in this field.
7.1 Genome annotation
Chapters 3 and 4 covered the topic of genome annotation using dynamic Bayesian networks. There
are a number of directions in which these types of methods could be extended and improved.
7.1.1 Annotation with multiple length scales
The genome exhibits different patterns of activity at different length scales. Most research so far
has focused on elements on the scale of 100-1000 bp, such as promoters and enhancers. At a finer
scale, individual regulatory elements can be divided into individual TF binding sites (∼ 10 bp) and
these TF binding sites can be further subdivided into their characteristic DNase accessibility profile
[Hesselberth et al., 2009], which vary at a single-base pair scale. At a coarser scale, I discussed
in Chapter 4 how the genome is composed of five types of domains at the scale of 100-1000 kb.
However, the SAGA approach in Chapter 4 models only these large-scale patterns. Ideally, a single
model would be able to analyze multiple scales at the same time.
The only existing work that simultaneously models multiple scales is [Larson et al., 2013], in
which researchers defined two random variables at each position: a fine-scale label (promoters,
enhancers) and a coarse-scale label (active and inactive domains). The fine-scale label was allowed to
vary every 200 bp; the large-scale label was allowed to vary only every 10 kb. Each large-scale label
was associated with a certain probability distribution over fine-scale labels: for example, the coarse-
scale “active domain” label would have a higher probability for the fine-scale “enhancer” label than
would the coarse-scale “heterochromatin” label. However, this work had two disadvantages. First,
179
the model ended up learning near-uniform distributions for each coarse-scale label; this is likely
due to the fact that the model included no explicit penalty for this behavior. Second, the model was
an extension of ChromHMM [Ernst and Kellis, 2010], which has a minimum resolution of 200 bp,
which prevented the authors from additionally adding a third scale at the resolution of ∼1 bp. It
would be valuable to extend this approach to explicitly encourage low-entropy label distributions
and adding an additional 1 bp resolution scale.
An alternate design for this model might instead associate each coarse-scale label with an
entirely independent set of fine-scale labels. For example, a 100 bp-scale “enhancer” label might
be composed of 1 bp-scale elements of “H3K27ac region upstream of enhancer TF binding site”,
“enhancer nucleosome free TF binding site” and “downstream eRNA transcribed region”. This
approach has the disadvantage that it results in more sub-labels to manually interpret, but it may
give the model more power to uncover diverse types of elements.
7.1.2 Hierarchical label complexity
Our understanding of genome elements has multiple levels of complexity. At the simplest level, one
might divide the genome into three categories: regulatory, transcribed, and inactive. Regulatory
elements could be further divided into promoters and enhancers; enhancers could be divided into
strong, weak and poised, and so on. Existing SAGA elements do not respect this hierarchy; they
learn the parameters associated with each label independently.
A better SAGA method might learn a hierarchical tree of labels rather than a flat collection.
Each node (or leaf, alternatively) in this tree would correspond to a set of emission parameters.
Transition probabilities would be defined over the tree, such that nodes nearby in the tree have high
transition probabilities to one another. This tree could be learned dynamically as part of training,
by splitting nodes that are assigned to many diverse data patterns in the training data and merging
nodes that learn redundant patterns.
Such a tree-based model would greatly aid interpretation of the learned parameters. For example,
if the researcher is attempting to interpret a given node and has already verified that the nearby
nodes correspond to known sub-compartments of an enhancer, it is very likely that the node in
180
question represents another (potentially novel) enhancer sub-component. In general, when trying to
understand the biological function of a given pattern (i.e. node in the tree), a researcher can use the
nearby nodes to guide their interpretation. Moreover, even without a tree structure, this splitting and
vanishing strategy for initializing label parameters is likely to be less prone to local optima because
labels would be initialized to data-dense parts of the parameter space (i.e. similar parameters to a
label that was a assigned to many of positions), rather than random parameters.
7.1.3 Multiple cell types
There is a great deal of room for further work in simultaneously annotating multiple cell types.
Two primary strategies have been “stacked” and “concatenated” annotation. In stacked annotation,
instead of inputting data from just one cell type to the method, the user inputs all available data
from all cell types. However, this approach is generally not effective because, in order to represent
all the possible patterns of five activity types across ten cell types, it takes 510 ≈ 107 labels, each of
which must be interpreted independently. The second approach, “concatenated” annotation, applies
the same model to a number of cell types. (The name concatenated comes from the fact that this
strategy can be implemented by concatenating the data from each cell type as through it came from
additional chromosomes of the first cell type.) Concatenated annotation works well when all the
data has been produced consistently, but it has two disadvantages. First, it requires that all cell types
have the same set of data available, which does not hold in practice. Second, artifactual experimental
differences can result in learning cell type-biased labels. Several extensions have been proposed to
these framework, such as the ones I developed in Chapter 4 using EGPR, but there are several other
potential directions for improvement.
Latent activity mixture model. One way to better model activity in multiple cell types would
be to learn “potential activity types” at each genomic position. For example, a given position
might exhibit enhancer activity in 75% of cell types and be quiescent in 25%. In general, the
model would learn a vector of probabilities over the different annotation labels at each position, and
suppose that the annotation label in a given cell type is drawn from this distribution. Such a model
181
would discourage one cell type from calling a position “enhancer” (say) if no other cell types show
evidence for having an enhancer in that position.
A model similar to this was proposed in [Biesinger et al., 2013]. These researchers defined a tree
over cell types and modeled transition probabilities between neighboring cell types in the tree (in
addition to the usual transition probabilities between neighboring genomic positions). However, this
work had two disadvantages. First, like in [Larson et al., 2013], the model learned a near-uniform
transition distribution, so the multiple cell type aspect of the model had little effect. Second, when
this work was performed, there was only data available in nine human cell types. It would be
valuable to extend this work to explicitly encourage low-entropy label distributions and to apply an
approach like this to the hundreds of cell types that now have available data.
State space model. An alternative approach would be to take a “stacked” approach, but switch
to a continuous (rather than discrete) model. Such a model would assign a vector of real values
x1 . . . xk, called factors, to each position instead of a discrete label. The expected emission for a
given track t under this model would be
∑k
i=1 xiµit, where each µit is a learned parameter associated
factor i and track t. This type of model is called a state space model, and can be optimized with
dynamic programming approaches comparable to the forward-backward algorithm used for discrete
HMMs. A factor in this model might represent, for example, “the extent to which this position is an
enhancer active in mesodermal cells.” The advantage of this model is that, because each position
is modeled as a mixture of multiple types of activity, it would likely extend to hundreds of cell
types more naturally than a discrete stacked approach. This type of model is similar to the class of
methods known as “topic modeling” that have been successful in natural language tasks.
7.2 Statistical evaluation of differences between genomics assays
In order to understand genome activity, it is important to be able to ask whether or not a given
drug, tissue, or disease produces a statistically significant change in activity relative to a particular
cellular condition. This question is difficult to answer because the output of a genomics assay—the
number of reads mapping to a given genomic position—has no natural units. In particular, a
182
difference between 0 and 30 reads may have a very different statistical importance from a difference
between 1,000 and 1,030 reads. This property is due to a relationship between the mean of the
data-generating process (i.e. the genomics assay) and its variance. Currently-used methods that
do not model this mean-variance relationship, such as a Poisson model, result in poor evaluation
of statistical differences. It would be valuable to develop methods to learn the mean-variance
relationship of functional genomics data sets using the variation among biological replicates. Such a
method would make it possible to produce a statistically principled measure of differences between
functional genomics data sets. Such measurements would enable interrogation of the effects of
cellular perturbations, yielding insights into the mechanisms of disease, aging and drug activity.
Learning the mean-variance relationship of these data sets has the additional benefit that it would
make it possible to place genomics data in interpretable units. The lack of natural units for genomics
data sets makes it difficult to analyze these data sets by eye and confounds any downstream analyses
that does not internally model the mean-variance relationship (including SAGA methods). Currently,
to attempt to handle this phenomenon, many analyses of genomics data sets transform the data
using a log or inverse hyperbolic sine transform to attempt to normalize for the mean-variance
relationship of the sequencing read counts. However, there is no theoretical basis for using these
particular transformations rather than any other, so they almost certainly do not properly account for
the mean-variance relationship as intended. Given a known mean-variance relationship, a data set
can be put into interpretable units using the variance-stabilizing transformation f(x) =
∫
1
σ(u)
du,
where σ(x) is the standard deviation of a variable with a mean of x. The resulting signal is in
units of standard deviation, so it has the useful property that all data points have a 95% confidence
intervals of∼2 units. Therefore, variance-stabilized data sets are a natural by-product of a principled
method for evaluating statistical differences. These data sets are much more easily analyzed by eye,
and are necessary for any downstream analysis that does not internally model the mean-variance
relationship.
183
7.3 Improved submodular objective for choosing panels of genomics assays
In Chapter 5, I proposed a method based on submodular optimization for choosing panels of assays
to perform. This method was based around the facility location submodular objective function.
This function determines the putative quality of a panel of assays that is optimized, and therefore is
central to the behavior of the method. There are several potential improvements to the objective
function I proposed.
First, the similarity function used by the facility location was defined as the pairwise correlation
over all pairs of assays of each type. This calculation involves computing the correlation of assays
performed in different cell types. This is not optimal, because even identical assay types might have
low correlation when performed in very different cell types. However, it is impossible to directly
compute the correlation between each pair of assay types without cross-cell type comparisons,
because there is no one cell type in which all assay types have been performed. Instead, it may be
better to compute correlations between the subset of pairs of assay types that have been performed
in the same cell type, and then use a matrix completion method to impute rest of the correlations.
Second, the proposed method did not take into account our prior information about assay types,
cell type or genomic positions. Taking into account this prior information might suggest different
choices of assay types. For example, if assay type A is exactly predictable based on assay type B
and the genome sequence, it does not make sense to include A and B in a panel together. Our prior
information about the genome sequence can be incorporated into the submodular framework by
including the genome sequence as a “free” assay performed in each cell type. Likewise, it might
be possible to induce a similarity between ChIP-seq assays for transcription factors A and B by
comparing the structure of the two proteins, even if ChIP-seq has never been performed targeting
those factors. These types of prior knowledge were not taken into account by our previous method,
which considered each assay as an abstract vector over the genome. Incorporating them would
further improve the panels identified by this approach.
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7.4 Future problems in computational genomics
Upcoming technologies for single-cell measurement, better measurement of genome conformation
and genome editing promise to present new opportunities for computational methods. As with
existing types of genomics data sets, analyzing such data requires methods that are scalable and
that accurately model the underlying phenomena while remaining interpretable enough to afford
biological understanding. In addition to their original purposes, the strategies presented in this
dissertation may provide launching points to address these new problems.
185
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