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English: Gaussian process posterior of function with gap visualized by mean function and confidence interval
Date
Source Own work
Author Physikinger
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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

def covMat(x1, x2, covFunc, noise=0):  # Covariance matrix
    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

numpy.random.seed(107)

covFunc1 = lambda d: 2*numpy.exp(-numpy.abs(numpy.sin(1.55*numpy.pi*d))**1.9/3 - d**2/7.)
covFunc2 = lambda d: 1*numpy.exp( - d**2/6.)
covFunc = lambda d: 1.5*numpy.exp(-numpy.abs(numpy.sin(1.55*numpy.pi*d))**1.9/3 - d**2/10.)

n=60
x = numpy.linspace(0, 10, 300)
y1 = numpy.random.multivariate_normal(x.ravel()*0, covMat(x, x, covFunc1, noise=0.00))
y2 = numpy.random.multivariate_normal(x.ravel()*0, covMat(x, x, covFunc2, noise=0.00))
x_known = numpy.concatenate([x[:n+1], x[-n:]])
y_known = numpy.concatenate([y1[:n+1], y2[-n:]])
x_unknown = x[n:-n+1]

Ckk = covMat(x_known, x_known, covFunc, noise=0.000001)
Cuu = covMat(x_unknown, x_unknown, covFunc, noise=0.00)
CkkInv = numpy.linalg.inv(Ckk)
Cuk = covMat(x_unknown, x_known, covFunc, noise=0.0)
m = 0 #numpy.mean(y)
covPost = Cuu - numpy.dot(numpy.dot(Cuk,CkkInv),Cuk.T)
y_unknown = numpy.dot(numpy.dot(Cuk,CkkInv),y_known)

fig = plt.figure(figsize=(4.0,2))
sigma = numpy.sqrt(numpy.diag(covPost))
plt.plot(x_unknown, y_unknown,  label=u'Prediction')
plt.fill_between(x_unknown.ravel(), y_unknown - sigma, y_unknown + sigma, color = '0.85')
plt.plot(x[:n+1], y1[:n+1],'k-')
plt.plot(x[-n:], y2[-n:],'k-')
plt.vlines([x[n], x[-n]],-3,3,colors='r', linestyles='--', alpha=0.5)
plt.axis([0,10,-3,3])
plt.savefig('Gaussianprocess_gapMean.svg')

fig = plt.figure(figsize=(4.0,2))
for c in 'C1 C4  C2'.split():
    y_random = numpy.random.multivariate_normal(x_unknown.ravel()*0, covPost)
    plt.plot(x_unknown, y_unknown + y_random, c, label=u'Prediction')
sigma = numpy.sqrt(numpy.diag(covPost))
plt.plot(x[:n+1], y1[:n+1],'k-')
plt.plot(x[-n:], y2[-n:],'k-')
plt.vlines([x[n], x[-n]],-3,3,colors='r', linestyles='--', alpha=0.5)
plt.axis([0,10,-3,3])
plt.savefig('Gaussianprocess_gap.svg')

# Uncertainty animation

numpy.random.seed(1)
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(covPost)))
cov = covPost+1E-5*numpy.identity(len(covPost))
rSmooth = numpy.linalg.cholesky(cov).dot(r.T)

images = []
fig = plt.figure(figsize=(4.0,2))
for ti in [0]+list(range(len(t))):
    plt.plot(x_unknown, y_unknown + rSmooth[:,ti], label=u'Prediction',alpha=1)
    #plt.fill_between(x_unknown.ravel(), y_unknown - sigma, y_unknown + sigma, color = '0.85')
    plt.plot(x[:n+1], y1[:n+1],'k-')
    plt.plot(x[-n:], y2[-n:],'k-')
    plt.vlines([x[n], x[-n]],-3,3,colors='r', linestyles='--', alpha=0.5)
    plt.axis([0,10,-3,3])
    plt.xlabel('t')
    #plt.tight_layout()
    fig.canvas.draw()
    s, (width, height) = fig.canvas.print_to_buffer()
    images.append(numpy.fromstring(s, numpy.uint8).reshape((height, width, 4)))
    fig.clf()

# Save GIF animation
imageio.mimsave('Gaussianprocess_gapUncertainty.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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26 February 2018

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Date/TimeThumbnailDimensionsUserComment
current12:06, 1 December 2019Thumbnail for version as of 12:06, 1 December 2019360 × 180 (26 KB)PhysikingerRandom Seed
22:53, 27 February 2018Thumbnail for version as of 22:53, 27 February 2018360 × 180 (32 KB)PhysikingerShifted line
21:49, 27 February 2018Thumbnail for version as of 21:49, 27 February 2018360 × 180 (26 KB)Physikingernew
21:11, 27 February 2018Thumbnail for version as of 21:11, 27 February 2018360 × 180 (26 KB)PhysikingerAsymmetric interpolation
23:13, 26 February 2018Thumbnail for version as of 23:13, 26 February 2018360 × 180 (27 KB)PhysikingerUser created page with UploadWizard

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