{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Transformed Simplex Rounding Error Plots" ] }, { "cell_type": "markdown", "metadata": { "ExecuteTime": { "end_time": "2018-07-23T03:45:06.001725Z", "start_time": "2018-07-23T03:45:05.999724Z" } }, "source": [ "## Parameter Selection\n", "This script calculates errors from the simulations run by `rounding_simplex_sims.ipynb`. Here are the parameters:\n", "\n", "- `d`: dimension, one of `{10, 20, 40}`\n", "- `k`: power in transformation, one of `{1, 2, 3, 4}`. The standard simplex (corresponding to `k = 1`) has condition number $(d + 1)$. A linear transformation $L$ was chosen such that after transforming the standard simplex, the resultant covariance matrix has condition number $(d+1)^k$.\n", "- `n_steps`: An upper bound total of number of steps to consider. In the simulation script, `10**6` steps of the random walk was run, so this value must be less than or equal to `10**6`. \n", "- `n_min`: Minimum number of samples to consider when generating the mean and covariance error. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "ExecuteTime": { "end_time": "2018-08-16T07:05:06.316435Z", "start_time": "2018-08-16T07:03:30.147410Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 10 \t k = 1\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75 4. 4.25 4.5 4.75\n", " 5. ]\n", "[ 31 56 100 177 316 562 1000 1778 3162 5623\n", " 10000 17782 31622 56234 100000]\n", "[3.22580645e+04 1.78571429e+04 1.00000000e+04 5.64971751e+03\n", " 3.16455696e+03 1.77935943e+03 1.00000000e+03 5.62429696e+02\n", " 3.16255534e+02 1.77841010e+02 1.00000000e+02 5.62366438e+01\n", " 3.16235532e+01 1.77828360e+01 1.00000000e+01]\n", "mean errors:\n", " [[ 0.58711852 0.93197266 1.27350798]\n", " [ 0.56909864 0.9140366 1.25499646]\n", " [ 0.53763903 0.88565052 1.22427965]\n", " [ 0.4884468 0.84206692 1.17302945]\n", " [ 0.40901771 0.77194255 1.10100089]\n", " [ 0.30531507 0.67335634 0.99818302]\n", " [ 0.16064586 0.54323937 0.87041481]\n", " [-0.02134225 0.37446733 0.68984658]\n", " [-0.25684084 0.16811441 0.5273512 ]\n", " [-0.49131213 -0.04883976 0.22887318]\n", " [-0.6712158 -0.25751353 0.03741558]\n", " [-0.86078334 -0.54319486 -0.23685613]\n", " [-1.18869871 -0.76338929 -0.50152028]\n", " [-1.55264308 -0.97988079 -0.67902265]\n", " [-1.42104534 -1.19716231 -0.87217869]]\n", "cov errors:\n", " [[3.60419869 4.75198725 7.01589073]\n", " [3.21160332 4.3301789 6.59839322]\n", " [2.8594552 3.97722969 6.25947063]\n", " [2.53568145 3.60796379 5.90296733]\n", " [2.15218665 3.19589659 5.47084886]\n", " [1.75976105 2.68041144 4.93368627]\n", " [1.28767056 2.03561749 4.24125106]\n", " [0.9768844 1.38814762 3.30427659]\n", " [0.68814669 0.91900845 2.0383457 ]\n", " [0.47929667 0.63172792 0.89189511]\n", " [0.33054224 0.46361124 0.58891118]\n", " [0.27488914 0.33025255 0.41570546]\n", " [0.18039614 0.23775794 0.31283563]\n", " [0.12560741 0.17830751 0.21481897]\n", " [0.10440926 0.12773017 0.16181924]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 10 \t k = 2\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75 4. 4.25 4.5 4.75\n", " 5. ]\n", "[ 31 56 100 177 316 562 1000 1778 3162 5623\n", " 10000 17782 31622 56234 100000]\n", "[3.22580645e+04 1.78571429e+04 1.00000000e+04 5.64971751e+03\n", " 3.16455696e+03 1.77935943e+03 1.00000000e+03 5.62429696e+02\n", " 3.16255534e+02 1.77841010e+02 1.00000000e+02 5.62366438e+01\n", " 3.16235532e+01 1.77828360e+01 1.00000000e+01]\n", "mean errors:\n", " [[ 0.58175368 0.92505367 1.26603932]\n", " [ 0.5679828 0.90785849 1.2444538 ]\n", " [ 0.54128772 0.88003286 1.21359866]\n", " [ 0.48941281 0.83394936 1.16270364]\n", " [ 0.41602735 0.76308538 1.08511866]\n", " [ 0.30397359 0.66555611 0.97635072]\n", " [ 0.15775132 0.53844477 0.84610731]\n", " [-0.03026052 0.36397307 0.68950971]\n", " [-0.25246357 0.156943 0.48867132]\n", " [-0.50938863 -0.07765496 0.30824897]\n", " [-0.65355625 -0.29934437 0.04082005]\n", " [-0.88321556 -0.47129584 -0.24919811]\n", " [-0.99755144 -0.74239404 -0.54513646]\n", " [-1.33725689 -1.00187501 -0.74718272]\n", " [-1.44673714 -1.19774744 -1.00131143]]\n", "cov errors:\n", " [[3.61662897 4.73383741 6.97622844]\n", " [3.20471421 4.31691734 6.55964899]\n", " [2.87179302 3.96754256 6.19131827]\n", " [2.51302653 3.59663053 5.83867113]\n", " [2.12417452 3.14942964 5.4287655 ]\n", " [1.72519976 2.66176462 4.99394375]\n", " [1.28919249 2.00752634 4.25386333]\n", " [0.93863826 1.38096964 3.05885832]\n", " [0.68744441 0.93209387 1.96614436]\n", " [0.47840057 0.63341815 0.91614294]\n", " [0.35180717 0.42059367 0.58335543]\n", " [0.26947468 0.32072713 0.46257762]\n", " [0.19392714 0.25531408 0.31126147]\n", " [0.1381515 0.16834214 0.20666566]\n", " [0.08679155 0.12768481 0.15475175]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 10 \t k = 3\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75 4. 4.25 4.5 4.75\n", " 5. ]\n", "[ 31 56 100 177 316 562 1000 1778 3162 5623\n", " 10000 17782 31622 56234 100000]\n", "[3.22580645e+04 1.78571429e+04 1.00000000e+04 5.64971751e+03\n", " 3.16455696e+03 1.77935943e+03 1.00000000e+03 5.62429696e+02\n", " 3.16255534e+02 1.77841010e+02 1.00000000e+02 5.62366438e+01\n", " 3.16235532e+01 1.77828360e+01 1.00000000e+01]\n", "mean errors:\n", " [[ 5.77586453e-01 9.24389876e-01 1.26859823e+00]\n", " [ 5.63145702e-01 9.06146812e-01 1.24946203e+00]\n", " [ 5.29542915e-01 8.76926170e-01 1.21890499e+00]\n", " [ 4.72591712e-01 8.29666886e-01 1.17056738e+00]\n", " [ 4.10238746e-01 7.61228724e-01 1.09265715e+00]\n", " [ 3.12578113e-01 6.67066746e-01 9.92937802e-01]\n", " [ 1.69609665e-01 5.31462314e-01 8.53046653e-01]\n", " [-1.92150838e-02 3.71126596e-01 6.83779227e-01]\n", " [-2.06687400e-01 1.65386877e-01 4.66334440e-01]\n", " [-3.72112316e-01 -2.71697027e-02 2.57316891e-01]\n", " [-7.03465235e-01 -2.66262674e-01 -5.81591579e-04]\n", " [-1.03363153e+00 -5.58184680e-01 -1.87565374e-01]\n", " [-1.11229595e+00 -7.71111970e-01 -3.94607976e-01]\n", " [-1.53243239e+00 -1.03485125e+00 -5.51778782e-01]\n", " [-2.06752412e+00 -1.43449507e+00 -7.28517170e-01]]\n", "cov errors:\n", " [[3.60752249 4.71794871 6.84585448]\n", " [3.19641232 4.30180147 6.43283589]\n", " [2.861998 3.94381201 6.07462532]\n", " [2.50577606 3.58269061 5.67583837]\n", " [2.12807006 3.16458292 5.2507579 ]\n", " [1.71000089 2.63951903 4.7161073 ]\n", " [1.32190355 1.98906894 4.11024786]\n", " [0.96984444 1.38112694 3.23880372]\n", " [0.67430588 0.92426057 1.85055225]\n", " [0.48583191 0.65306502 0.99401408]\n", " [0.34238769 0.45094769 0.62713876]\n", " [0.25287199 0.33537352 0.4358172 ]\n", " [0.18924873 0.22475257 0.30684019]\n", " [0.14957801 0.17655495 0.21821101]\n", " [0.10488388 0.12379323 0.1959764 ]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 10 \t k = 4\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75 4. 4.25 4.5 4.75\n", " 5. ]\n", "[ 31 56 100 177 316 562 1000 1778 3162 5623\n", " 10000 17782 31622 56234 100000]\n", "[3.22580645e+04 1.78571429e+04 1.00000000e+04 5.64971751e+03\n", " 3.16455696e+03 1.77935943e+03 1.00000000e+03 5.62429696e+02\n", " 3.16255534e+02 1.77841010e+02 1.00000000e+02 5.62366438e+01\n", " 3.16235532e+01 1.77828360e+01 1.00000000e+01]\n", "mean errors:\n", " [[ 0.57851329 0.9285396 1.26974827]\n", " [ 0.56040041 0.91185913 1.25170836]\n", " [ 0.52984234 0.88100658 1.21921789]\n", " [ 0.48442708 0.83469148 1.17216543]\n", " [ 0.39671383 0.77022386 1.10538481]\n", " [ 0.30374996 0.66771372 1.00436604]\n", " [ 0.17380441 0.5369719 0.84298521]\n", " [ 0.01106339 0.37644928 0.69870458]\n", " [-0.21397475 0.18322184 0.4770149 ]\n", " [-0.40890842 -0.01763142 0.24274498]\n", " [-0.60701961 -0.29196782 -0.03631775]\n", " [-0.92017472 -0.51273593 -0.26341151]\n", " [-1.17390663 -0.78701528 -0.57570062]\n", " [-1.46155807 -1.02806289 -0.70454311]\n", " [-1.61206443 -1.3382997 -0.98304544]]\n", "cov errors:\n", " [[3.61544633 4.76527879 6.71492391]\n", " [3.20097501 4.34051376 6.29777036]\n", " [2.86562832 3.98893542 5.9391663 ]\n", " [2.52543448 3.6360819 5.59035695]\n", " [2.13087635 3.21264985 5.12978732]\n", " [1.75130231 2.69490222 4.6154808 ]\n", " [1.3089729 2.04151292 3.84105248]\n", " [0.94333327 1.36442793 3.04252631]\n", " [0.65956884 0.91103244 1.88959756]\n", " [0.48673662 0.64868437 1.11020072]\n", " [0.34874459 0.45984149 0.59484019]\n", " [0.2644898 0.31950034 0.41159749]\n", " [0.17119979 0.23187415 0.32619688]\n", " [0.12427948 0.17112367 0.21272082]\n", " [0.10483438 0.12283812 0.15158614]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 20 \t k = 1\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75]\n", "[ 89 189 400 845 1788 3782 8000 16917 35777 75659]\n", "[11235.95505618 5291.00529101 2500. 1183.43195266\n", " 559.28411633 264.41036489 125. 59.11213572\n", " 27.95091819 13.21719822]\n", "mean errors:\n", " [[ 0.97282931 1.23568563 1.50175147]\n", " [ 0.94156468 1.19902626 1.45378294]\n", " [ 0.87202432 1.13409107 1.39431871]\n", " [ 0.77891951 1.02075625 1.2740124 ]\n", " [ 0.61010123 0.86683167 1.10643374]\n", " [ 0.37386715 0.63456905 0.89757295]\n", " [ 0.0865023 0.40357128 0.59585932]\n", " [-0.16489247 0.06450749 0.33617145]\n", " [-0.4730199 -0.26206689 0.00503549]\n", " [-0.78777166 -0.55594528 -0.34108847]]\n", "cov errors:\n", " [[4.21636839 5.33748784 7.39511183]\n", " [3.74234242 4.86589332 6.91766918]\n", " [3.26428093 4.37436128 6.40467556]\n", " [2.71051801 3.79145289 5.89154193]\n", " [2.00890033 2.92911832 5.08372839]\n", " [1.34202328 1.89487279 3.55670686]\n", " [0.8989105 1.10210502 2.14027454]\n", " [0.60805595 0.70249723 0.90023819]\n", " [0.39178415 0.42959879 0.52325105]\n", " [0.27426918 0.29130013 0.36570373]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 20 \t k = 2\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75]\n", "[ 89 189 400 845 1788 3782 8000 16917 35777 75659]\n", "[11235.95505618 5291.00529101 2500. 1183.43195266\n", " 559.28411633 264.41036489 125. 59.11213572\n", " 27.95091819 13.21719822]\n", "mean errors:\n", " [[ 0.97342673 1.2327624 1.49340484]\n", " [ 0.94256811 1.19578115 1.44958129]\n", " [ 0.883064 1.12767867 1.38717054]\n", " [ 0.77512663 1.02366693 1.26084632]\n", " [ 0.62450384 0.8668277 1.09508941]\n", " [ 0.41535841 0.64381212 0.89020476]\n", " [ 0.14913137 0.38379101 0.60144593]\n", " [-0.13648282 0.09616698 0.30021041]\n", " [-0.48112103 -0.21604354 0.04942508]\n", " [-0.87680972 -0.50599666 -0.32529491]]\n", "cov errors:\n", " [[4.20261107 5.36094387 7.40389278]\n", " [3.74153492 4.89175368 6.92115143]\n", " [3.27215262 4.41411444 6.48151809]\n", " [2.68643501 3.78091638 5.85792178]\n", " [2.01184029 2.96813421 4.98285125]\n", " [1.36425912 1.91216662 3.79421016]\n", " [0.91415359 1.13921371 1.83641704]\n", " [0.58910033 0.68381909 0.8853939 ]\n", " [0.38596889 0.44912118 0.63344431]\n", " [0.2686344 0.31418676 0.37294876]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 20 \t k = 3\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75]\n", "[ 89 189 400 845 1788 3782 8000 16917 35777 75659]\n", "[11235.95505618 5291.00529101 2500. 1183.43195266\n", " 559.28411633 264.41036489 125. 59.11213572\n", " 27.95091819 13.21719822]\n", "mean errors:\n", " [[ 0.97295146 1.22328705 1.49606072]\n", " [ 0.94135222 1.18543205 1.45335356]\n", " [ 0.87671804 1.12330988 1.38170376]\n", " [ 0.76097795 1.01648279 1.25684015]\n", " [ 0.61001901 0.85346831 1.08601594]\n", " [ 0.38363085 0.63019128 0.85654472]\n", " [ 0.13207396 0.35734654 0.56749695]\n", " [-0.18131737 0.04173814 0.28435993]\n", " [-0.44291982 -0.23663848 -0.10096328]\n", " [-0.80363823 -0.58890564 -0.35111187]]\n", "cov errors:\n", " [[4.15263177 5.28808311 7.42397503]\n", " [3.68543798 4.82712372 6.9519719 ]\n", " [3.18768249 4.34063431 6.51832198]\n", " [2.63984548 3.70229867 5.91231843]\n", " [1.9465972 2.89401622 5.04403638]\n", " [1.31488963 1.87085462 3.93948972]\n", " [0.89261218 1.11068108 1.66532249]\n", " [0.59013931 0.69088096 0.86604642]\n", " [0.38557702 0.48275891 0.52486519]\n", " [0.27530619 0.30614805 0.41673173]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 20 \t k = 4\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. 3.25 3.5 3.75]\n", "[ 89 189 400 845 1788 3782 8000 16917 35777 75659]\n", "[11235.95505618 5291.00529101 2500. 1183.43195266\n", " 559.28411633 264.41036489 125. 59.11213572\n", " 27.95091819 13.21719822]\n", "mean errors:\n", " [[ 0.97459553 1.2328299 1.49982892]\n", " [ 0.93720076 1.19796148 1.45448795]\n", " [ 0.8780571 1.13019727 1.3897512 ]\n", " [ 0.76136055 1.02761547 1.27358744]\n", " [ 0.60279386 0.8753283 1.10142502]\n", " [ 0.3968323 0.67276919 0.88448121]\n", " [ 0.18348761 0.39418229 0.61374317]\n", " [-0.21102225 0.07308043 0.29063308]\n", " [-0.42688042 -0.2409622 -0.06645151]\n", " [-0.86071358 -0.55131804 -0.43003192]]\n", "cov errors:\n", " [[4.18392105 5.3718791 7.38619428]\n", " [3.71592878 4.89906482 6.92697682]\n", " [3.23388734 4.41513377 6.47961274]\n", " [2.70739711 3.82700842 5.88241558]\n", " [2.00446134 2.96807589 5.16468807]\n", " [1.33048678 1.92260643 3.41334631]\n", " [0.90765724 1.1328501 1.93323838]\n", " [0.58238141 0.69636222 0.85189104]\n", " [0.39091478 0.44471395 0.51179907]\n", " [0.25262814 0.30328214 0.35815305]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 40 \t k = 1\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. ]\n", "[ 252 636 1600 4023 10119 25448 64000]\n", "[3968.25396825 1572.32704403 625. 248.57071837 98.82399447\n", " 39.29581892 15.625 ]\n", "mean errors:\n", " [[ 1.33687023 1.52634268 1.73482374]\n", " [ 1.27507354 1.46342778 1.65513247]\n", " [ 1.14449331 1.33634857 1.51010993]\n", " [ 0.94858571 1.1292226 1.30992465]\n", " [ 0.70653984 0.84084471 1.00954208]\n", " [ 0.32912352 0.49333295 0.60930885]\n", " [-0.05166376 0.16358722 0.3075007 ]]\n", "cov errors:\n", " [[4.84980065 5.94905196 8.18614681]\n", " [4.27734018 5.36791313 7.62055974]\n", " [3.51741702 4.63541704 6.81485838]\n", " [2.60216382 3.50197164 5.56022731]\n", " [1.5627951 2.08998433 4.01581051]\n", " [0.97008495 1.1410803 1.49429302]\n", " [0.62620675 0.67229904 0.79544231]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 40 \t k = 2\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. ]\n", "[ 252 636 1600 4023 10119 25448 64000]\n", "[3968.25396825 1572.32704403 625. 248.57071837 98.82399447\n", " 39.29581892 15.625 ]\n", "mean errors:\n", " [[ 1.34165584 1.53151223 1.72730459]\n", " [ 1.27963497 1.46504762 1.65025742]\n", " [ 1.16048094 1.33958114 1.51824381]\n", " [ 0.97435741 1.14236682 1.30232236]\n", " [ 0.66724758 0.85358308 1.02248723]\n", " [ 0.27522484 0.51519176 0.5831569 ]\n", " [-0.11884519 0.07457987 0.30704906]]\n", "cov errors:\n", " [[4.87061047 6.13735245 8.28813523]\n", " [4.28776348 5.58591865 7.70131622]\n", " [3.56066688 4.81912368 6.95271869]\n", " [2.66450716 3.71387909 6.19405146]\n", " [1.58811597 2.27713936 4.36087219]\n", " [0.97830014 1.0600491 1.58558471]\n", " [0.6104515 0.64762301 1.2033682 ]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 40 \t k = 3\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. ]\n", "[ 252 636 1600 4023 10119 25448 64000]\n", "[3968.25396825 1572.32704403 625. 248.57071837 98.82399447\n", " 39.29581892 15.625 ]\n", "mean errors:\n", " [[ 1.3277071 1.52252846 1.71960345]\n", " [ 1.26340558 1.45490527 1.64510235]\n", " [ 1.15055637 1.3290093 1.50947365]\n", " [ 0.95361056 1.12261114 1.30877008]\n", " [ 0.64323516 0.82710856 1.03565044]\n", " [ 0.26224419 0.47792752 0.59644687]\n", " [-0.05338366 0.06074467 0.26102341]]\n", "cov errors:\n", " [[4.84321494 6.02424067 7.95575092]\n", " [4.27353964 5.46760963 7.36840009]\n", " [3.54757738 4.73602871 6.59676276]\n", " [2.49767083 3.64933908 5.63373271]\n", " [1.5064635 2.04866999 3.60090243]\n", " [0.9799644 1.08632506 1.68535271]\n", " [0.58847181 0.65781738 0.76157381]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "------------------------------------------\n", "\t d = 40 \t k = 4\n", "------------------------------------------\n", "[1.5 1.75 2. 2.25 2.5 2.75 3. ]\n", "[ 252 636 1600 4023 10119 25448 64000]\n", "[3968.25396825 1572.32704403 625. 248.57071837 98.82399447\n", " 39.29581892 15.625 ]\n", "mean errors:\n", " [[ 1.34202115 1.52363255 1.71782175]\n", " [ 1.27722274 1.45613406 1.64137123]\n", " [ 1.15768812 1.33385708 1.50562788]\n", " [ 0.94729218 1.12643031 1.31140119]\n", " [ 0.68397542 0.82142001 1.00805551]\n", " [ 0.30650331 0.49553501 0.67030004]\n", " [-0.01770109 0.14536727 0.22185113]]\n", "cov errors:\n", " [[4.78966507 5.90529117 7.29565186]\n", " [4.20849221 5.34078071 6.65350092]\n", " [3.52667992 4.57513632 5.9260332 ]\n", " [2.54971895 3.60785503 4.71833635]\n", " [1.56776643 2.09953015 3.22406441]\n", " [0.96836319 1.09887984 1.81133661]\n", " [0.59943999 0.66939482 0.78180227]]\n" ] }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "from scipy.linalg import fractional_matrix_power\n", "from scipy.stats.mstats import mquantiles\n", "import pickle\n", "import os\n", "import time\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "sns.set()\n", "\n", "###########################################################\n", "# dimension, k value, conifidence levels, data load\n", "###########################################################\n", "d_vals = [10, 20, 40]\n", "k_vals = [1, 2, 3, 4]\n", "conf_level = 0.90\n", "alpha = 1 - conf_level\n", "quantiles = np.array([alpha/2, .5, 1 - alpha/2])\n", "\n", "\n", "# loop through each d, k pair and produce plots\n", "for d in d_vals:\n", " for k in k_vals:\n", " print('------------------------------------------')\n", " print('\\t d = {}'.format(d), '\\t k = {}'.format(k))\n", " print('------------------------------------------')\n", " # load result (dictionary with keys d, k, X, and rejections)\n", " filename = os.path.join(os.path.normpath('E:/Data/Rounding'), \n", " 'simplex_d{}_k{}.pkl'.format(d, k))\n", " with open(filename, 'rb') as file:\n", " result = pickle.load(file)\n", "\n", " X = result['X']\n", "\n", " ############################################################\n", " # form population parameters\n", " ############################################################\n", " # population mean\n", " a = np.ones(d + 1)\n", " a0 = a.sum()\n", " mu_full = a/a0\n", " mu = mu_full[:d]\n", " # population covariance\n", " const = (1/(a0**2 * (a0 + 1)))\n", " sigma_full = np.outer(-a * const, a)\n", " sigma_full[np.diag_indices(d + 1)] = a*(a0 - a)*const\n", " sigma = sigma_full[0:d, 0:d]\n", " # transformation matrix\n", " if k == 1:\n", " sigma_new = sigma\n", " mu_new = mu\n", " else:\n", " kappa = (d + 1)**k\n", " vals, vecs = np.linalg.eigh(sigma)\n", " U = vecs.copy()\n", " D = np.diag([np.sqrt(a0/kappa)] + [1]*(d -1))\n", " L = np.matmul(U, np.matmul(D, U.T))\n", " sigma_new = np.matmul(L, np.matmul(sigma, L.T))\n", " mu_new = np.matmul(L, mu)\n", " # matrix for scaling (used in covariance error later)\n", " scale_mtx = fractional_matrix_power(sigma_new, -.5)\n", "\n", " n_steps = 10**6\n", " powers = np.arange(1.5, 5.5, step=.25)\n", " n_vals = np.floor(d**powers).astype('int')\n", " n_sims = n_steps/n_vals\n", " powers = powers[n_sims >= 10]\n", " n_vals = n_vals[n_sims >= 10]\n", " n_sims = n_sims[0:len(n_vals)]\n", " print(powers)\n", " print(n_vals)\n", " print(n_sims)\n", "\n", " ############################################################\n", " # form error matrix\n", " ############################################################\n", " n_steps = 10**6\n", " err_mtx = np.zeros((n_steps, 3)) \n", " i = 0\n", "\n", " for n in n_vals:\n", " for start_idx in range(0, n_steps, n):\n", " if (start_idx + n <= n_steps):\n", " idx = range(start_idx, start_idx + n)\n", " X_sub = X[idx]\n", " mu_hat = X_sub.mean(axis=0)\n", " sigma_hat = np.cov(X_sub, rowvar=False, bias=True)\n", " mu_err = np.linalg.norm(scale_mtx @ (mu_new - mu_hat))**2\n", " sigma_err_mtx = scale_mtx @ sigma_hat @ scale_mtx\n", " smallest = np.linalg.norm(sigma_err_mtx, -2)\n", " largest = np.linalg.norm(sigma_err_mtx, 2)\n", " sigma_err = max(1/smallest, largest)\n", " err_mtx[i, :] = (n, mu_err, sigma_err)\n", " i = i + 1\n", " err_mtx = err_mtx[err_mtx[:, 0] != 0, :]\n", "\n", " # mean and cov error matrices with columns: lower, median, upper \n", " mean_errs = np.zeros((len(n_vals), 3)) \n", " cov_errs = np.zeros((len(n_vals), 3))\n", " for (i, n) in enumerate(n_vals):\n", " err_mtx_sub = err_mtx[err_mtx[:, 0] == n]\n", " mean_errs[i, :] = np.log10(mquantiles(a=err_mtx_sub[:, 1], prob=quantiles))\n", " cov_errs[i, :] = np.log10(mquantiles(a=err_mtx_sub[:, 2], prob=quantiles))\n", " print('mean errors:\\n', mean_errs)\n", " print('cov errors:\\n', cov_errs)\n", "\n", " save_dir = os.path.normpath('C:/Users/Adam/OneDrive/Thesis')\n", " covfig_filename = 'simplex_cov_n{}_k{}.pdf'\n", " covfig_filename = covfig_filename.format(d, k)\n", " covfig_filename = os.path.join(save_dir, covfig_filename)\n", "\n", " fig, ax = plt.subplots()\n", " fig.set_size_inches(8, 8)\n", " txt = 'Covariance Error (n = {}, k = {})'\n", " ax.set_title(txt.format(d, k), pad=20, fontsize=16)\n", " ax.plot(powers, cov_errs[:, 1], linewidth=2, marker='o', color='darkred')\n", " ax.fill_between(powers, cov_errs[:, 0], cov_errs[:, 2], alpha=.3, color='red')\n", " plt.yticks(np.arange(0, 11))\n", " plt.xticks(powers)\n", " ax.set_xlabel(r'$\\gamma$', fontsize=14)\n", " ax.set_ylabel(r'$\\log_{10}\\, C$', fontsize=14)\n", " plt.savefig(covfig_filename)\n", " plt.show()\n", "\n", "\n", " meanfig_filename = 'simplex_mean_n{}_k{}.pdf'\n", " meanfig_filename = meanfig_filename.format(d, k)\n", " meanfig_filename = os.path.join(save_dir, meanfig_filename)\n", "\n", " fig, ax = plt.subplots()\n", " fig.set_size_inches(8, 8)\n", " txt = 'Mean Error (n = {}, k = {})'\n", " ax.set_title(txt.format(d, k), pad=20, fontsize=16)\n", " ax.plot(powers, mean_errs[:, 1], linewidth=2, marker='o', color='darkred')\n", " ax.fill_between(powers, mean_errs[:, 0], mean_errs[:, 2], alpha=.3, color='red')\n", " #plt.yticks(np.arange(0, 11))\n", " plt.xticks(powers)\n", " ax.set_xlabel(r'$\\gamma$', fontsize=14)\n", " ax.set_ylabel(r'$\\log_{10}\\, \\epsilon$', fontsize=14)\n", " plt.savefig(meanfig_filename)\n", " plt.show()\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.5" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }