Last updated: 2018-05-21

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Simulate Gaussian Process

Here we simulate a GP with squared exponential kernel:

set.seed(1)
x = seq(0,1,length=100)
d = abs(outer(x,x,"-")) # compute distance matrix, d_{ij} = |x_i - x_j|
l = 1 # length scale
Sigma_SE = exp(-d^2/(2*l^2)) # squared exponential kernel
y = mvtnorm::rmvnorm(1,sigma=Sigma_SE)
plot(x,y)

Try making the covariance decay faster with distance:

l = 0.1
Sigma_SE = exp(-d^2/(2*l^2)) # squared exponential kernel
y = mvtnorm::rmvnorm(1,sigma=Sigma_SE)
plot(x,y)

Here is a plot of five different simulations:

plot(x,y,type="l",ylim=c(-3,3))
for(i in 1:4){
  y = mvtnorm::rmvnorm(1,sigma=Sigma_SE)
  lines(x,y,col=i+1)
}

The OU covariance function:

Here we use the covariance function for what is known as the “Ornstein–Uhlenbeck process”, which you can think of as a modified Brownian motion, where the modification tends to pull the process back towards 0. (Unmodified BM tends to wander progressively further from 0.)

Notice it produces much “rougher” functions (actually not differentiable)!

Sigma_OU = exp(-d/l) # OU kernel
y = mvtnorm::rmvnorm(1,sigma=Sigma_OU)
plot(x,y,type="l",ylim=c(-3,3))
for(i in 1:4){
  y = mvtnorm::rmvnorm(1,sigma=Sigma_OU)
  lines(x,y,col=i+1)
}

Matern covariance function

library("geoR")
--------------------------------------------------------------
 Analysis of Geostatistical Data
 For an Introduction to geoR go to http://www.leg.ufpr.br/geoR
 geoR version 1.7-5.2 (built on 2016-05-02) is now loaded
--------------------------------------------------------------
Sigma_M = matern(d,phi=l,kappa=1) 
y = mvtnorm::rmvnorm(1,sigma=Sigma_M)
plot(x,y,type="l",ylim=c(-3,3))
for(i in 1:4){
  y = mvtnorm::rmvnorm(1,sigma=Sigma_M)
  lines(x,y,col=i+1)
}

Session information

sessionInfo()
R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] geoR_1.7-5.2

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.16             knitr_1.20              
 [3] whisker_0.3-2            magrittr_1.5            
 [5] workflowr_1.0.1          MASS_7.3-49             
 [7] lattice_0.20-35          stringr_1.3.0           
 [9] tcltk_3.3.2              tools_3.3.2             
[11] RandomFields_3.1.50      grid_3.3.2              
[13] R.oo_1.22.0              git2r_0.21.0            
[15] RandomFieldsUtils_0.3.25 htmltools_0.3.6         
[17] yaml_2.1.18              rprojroot_1.3-2         
[19] digest_0.6.15            splancs_2.01-40         
[21] R.utils_2.6.0            evaluate_0.10.1         
[23] rmarkdown_1.9            sp_1.2-7                
[25] stringi_1.1.7            backports_1.1.2         
[27] R.methodsS3_1.7.1        mvtnorm_1.0-7

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