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Summary

Description
English: Three randomly generated signals that follow certain Gaussian processes.
Deutsch: Drei zufällig erzeugte Signale, die bestimmten Gaußprozessen folgen.
Date
Source Own work
Author Christian Schirm
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Source code
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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
imageio.mimsave('GaussianProcessDecomposition_Uncertainty.gif', images[1:])

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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3 September 2018

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Date/TimeThumbnailDimensionsUserComment
current18:18, 1 August 2019Thumbnail for version as of 18:18, 1 August 2019378 × 270 (30 KB)PhysikingerWith random seed
21:03, 29 July 2019Thumbnail for version as of 21:03, 29 July 2019378 × 270 (30 KB)Physikingernew
10:06, 4 September 2018Thumbnail for version as of 10:06, 4 September 2018378 × 270 (32 KB)Physikingerbigger label
09:44, 4 September 2018Thumbnail for version as of 09:44, 4 September 2018405 × 315 (32 KB)Physikingeraxis label
14:08, 3 September 2018Thumbnail for version as of 14:08, 3 September 2018405 × 315 (31 KB)PhysikingerImproved colors
09:11, 3 September 2018Thumbnail for version as of 09:11, 3 September 2018405 × 315 (31 KB)PhysikingerUser created page with UploadWizard

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