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GaussianProcessDecomposition_Uncertainty.gif (420 × 300 pixels, file size: 381 KB, MIME type: image/gif, looped, 50 frames, 5.0 s)

Summary

Description
English: Uncertainties and crosscorrelations of decomposed signals shown as animated random fluctuations.
Deutsch: Unsicherheiten und Kreuzkorrelationen der zerlegten Signale, dargestellt als animierte Zufallsfluktuationen.
Date
Source Own work
Author Christian Schirm
GIF development
InfoField
 
This plot was created with Matplotlib.
Source code
InfoField

Python code

# This source code is public domain
# Author: Christian Schirm

import numpy, scipy.spatial
import matplotlib.pyplot as plt
import imageio

numpy.random.seed(50)

# Covariance matrix
def covMat(x1, x2, covFunc, noise=0):
    cov = covFunc(scipy.spatial.distance_matrix(numpy.atleast_2d(x1).T, numpy.atleast_2d(x2).T))
    if noise: cov += numpy.diag(numpy.ones(len(cov))*noise)
    return cov

# Decomposition of e.g. sum of signals into components
def decompose(xIn, yIn, xOut, covFuncIn, covFuncListOut):
    Ckk = covMat(xIn, xIn, covFuncIn, noise=0)
    n = len(covFuncListOut)
    N = len(xOut)
    Cuu = numpy.zeros((n*len(xOut), n*len(xOut)))
    Cuk = numpy.zeros((n*len(xOut), len(xOut)))
    for i,covOut in enumerate(covFuncListOut):
        Cuu[i*N:(i+1)*N, i*N:(i+1)*N] = covMat(xOut, xOut, covOut, noise=0)
        Cuk[i*N:(i+1)*N,:] = covMat(xOut, xIn, covOut, noise=0)
    CkkInv = numpy.linalg.inv(Ckk)
    y = Cuk.dot(CkkInv.dot(yIn))
    sigmaSplit = (Cuu - Cuk.dot(CkkInv.dot(Cuk.T)))
    return y, sigmaSplit

# Covariance function 1: smooth random signal underground
covFunc1 = lambda d: 2.7**2*numpy.exp(-((d/1.)**2))

# Covariance function 2: periodic signal
covFunc2 = lambda d: 2.7**2*numpy.exp(-0.4*numpy.abs((numpy.sin(numpy.pi*d/2.5))))

# Covariance function 3: white gaussian noise
covFunc3 = lambda d: d*0 + 0.8**2*(numpy.abs(d)<0.00001)

# Covariance function of sum
covFuncSum = lambda d: covFunc1(d) + covFunc2(d) + covFunc3(d)

x = numpy.linspace(0, 10, 300)

# Generate random signales
Y = []
for covFunc in covFunc1, covFunc2, covFunc3:
    y = numpy.random.multivariate_normal(x.ravel()*0, covMat(x, x, covFunc))
    Y += [y]

# perform decomposition
YSplit = []
YSigma = []
ySplit, sigmaSplit = decompose(x, Y[0]+Y[1]+Y[2], x, covFuncSum, [covFunc1, covFunc2, covFunc3])
YSplit = ySplit.reshape(3,len(x))

# set prior mean of signals 1 and 2
meanShift = 3
YSplit[0] += meanShift
Y[0] += meanShift
YSplit[1] -= meanShift
Y[1] -= meanShift

# Random gaussian process signals
fig = plt.figure(figsize=(4.2,3.0))
for i,c in (2,1), (0,0), (1,2):
    plt.plot(x, Y[i], color='C'+str(c), label=u'Prediction',alpha=1)
plt.axis([0,10,-10,10])
plt.xlabel('t')
plt.tight_layout()
plt.savefig('GaussianProcessDecomposition_3RandomSignals.svg')
plt.show()

# Sum of all 3 signals
fig = plt.figure(figsize=(4.2,3.0))
plt.plot(x, (Y[0]+Y[1]+Y[2]), 'r-', label=u'Prediction')
plt.axis([0,10,-10,10])
plt.xlabel('t')
plt.tight_layout()
plt.savefig('GaussianProcessDecomposition_SumOf3Signals.svg')
plt.show()

# plot figures
# Decomposion of sum into single signals
fig = plt.figure(figsize=(4.2,3.0))
for i,c in (2,1), (0,0), (1,2):
    plt.plot(x, Y[i], '--', color='C'+str(c), label=u'Prediction',alpha=0.4)
    plt.plot(x, YSplit[i], color='C'+str(c), label=u'Prediction',alpha=1)
plt.axis([0,10,-10,10])
plt.xlabel('t')
plt.tight_layout()
plt.savefig('GaussianProcessDecomposition_DecomposedSignals.svg')
plt.show()

# Uncertainty animation

t = numpy.arange(0, 1, 0.02)
covFunc = lambda d: numpy.exp(-(3*numpy.sin(d*numpy.pi))**2) # Covariance function
chol = numpy.linalg.cholesky(covMat(t, t, covFunc, noise=1E-5))
r = chol.dot(numpy.random.randn(len(t), len(sigmaSplit)))
cov = sigmaSplit+1E-5*numpy.identity(len(sigmaSplit))
rSmooth = numpy.linalg.cholesky(cov).dot(r.T).reshape(3,len(x),len(t))

images = []
fig = plt.figure(figsize=(4.2,3.0))
for ti in [0]+list(range(len(t))):
    for i,c in (2,1), (0,0), (1,2):
        plt.plot(x, YSplit[i] + rSmooth[i,:,ti], color='C'+str(c), label=u'Prediction',alpha=1)
    plt.axis([0,10,-10,10])
    plt.xlabel('t')
    plt.tight_layout()
    fig.canvas.draw()
    s, (width, height) = fig.canvas.print_to_buffer()
    images.append(numpy.array(list(s), numpy.uint8).reshape((height, width, 4)))
    fig.clf()

# Save GIF animation
fileOut = 'GaussianProcessDecomposition_Uncertainty.gif'
imageio.mimsave(fileOut, images[1:])

# Optimize GIF size
from pygifsicle import optimize
optimize(fileOut, colors=16)

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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29 July 2019

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Date/TimeThumbnailDimensionsUserComment
current20:35, 8 September 2021Thumbnail for version as of 20:35, 8 September 2021420 × 300 (381 KB)PhysikingerSmaller file size
05:58, 2 August 2019Thumbnail for version as of 05:58, 2 August 2019420 × 300 (698 KB)PhysikingerMissing frame
05:48, 2 August 2019Thumbnail for version as of 05:48, 2 August 2019420 × 300 (684 KB)PhysikingerCorrected periodicity
18:39, 1 August 2019Thumbnail for version as of 18:39, 1 August 2019420 × 300 (698 KB)PhysikingerSlower fluctuations
18:34, 1 August 2019Thumbnail for version as of 18:34, 1 August 2019420 × 300 (731 KB)PhysikingerSmaller file
18:21, 1 August 2019Thumbnail for version as of 18:21, 1 August 2019420 × 300 (1.64 MB)PhysikingerWith random seed
20:54, 29 July 2019Thumbnail for version as of 20:54, 29 July 2019420 × 300 (713 KB)PhysikingerUser created page with UploadWizard

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