Optimization Enabled Kalman Smoothing

Abstract

Kalman smoothing has tremendous importance in a wide range of time series analysis applications. Classic algorithms use Gaussian assumptions to simplify estimation but optimization tools can be used to unlock more modeling capabilities. We develop techniques that allow for efficient robust estimates in the presence of singular noise. Such models arise frequently in the presence of auto-correlated noise, bias, and constraint systems. We also consider simultaneous parameter estimation and provide second-order methods for these problems, even when covariance matrices are singular. This captures ARMA and many physics based models. We also develop second-order methods for a general class of convex-composite PLQ functions and use this to solve general robust nonlinear Kalman smoothing models. A number of applications are considered that apply the developed methods. In particular we consider problems in navigation, finance, ADCP current estimation, and flight testing using a combination of real and synthetic data sets. We also include open source implementation of our models and share these via GitHub.