File:GaussianProcessDecomposition Uncertainty.gif
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GaussianProcessDecomposition_Uncertainty.gif (420 × 300 pixels, file size: 381 KB, MIME type: image/gif, looped, 50 frames, 5.0 s)
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Summary[edit]
DescriptionGaussianProcessDecomposition Uncertainty.gif |
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[edit]
I, the copyright holder of this work, hereby publish it under the following license:
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.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 20:35, 8 September 2021 | 420 × 300 (381 KB) | Physikinger (talk | contribs) | Smaller file size | |
05:58, 2 August 2019 | 420 × 300 (698 KB) | Physikinger (talk | contribs) | Missing frame | ||
05:48, 2 August 2019 | 420 × 300 (684 KB) | Physikinger (talk | contribs) | Corrected periodicity | ||
18:39, 1 August 2019 | 420 × 300 (698 KB) | Physikinger (talk | contribs) | Slower fluctuations | ||
18:34, 1 August 2019 | 420 × 300 (731 KB) | Physikinger (talk | contribs) | Smaller file | ||
18:21, 1 August 2019 | 420 × 300 (1.64 MB) | Physikinger (talk | contribs) | With random seed | ||
20:54, 29 July 2019 | 420 × 300 (713 KB) | Physikinger (talk | contribs) | User created page with UploadWizard |
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