Using the KL expansion to approximate the covariance matrix for the interval and circle cases

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Let us set some global options for all code chunks in this document.

# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = FALSE,    
  # Disable warnings printed by R code chunks
  warning = FALSE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}

Import libraries

library(MetricGraph)
library(Matrix)
library(rSPDE)

library(dplyr)
library(tidyverse)
library(plotly)

library(grateful) # Cite all loaded packages

Interval graph

# function 1
gets_graph_interval <- function(n){
  edge <- rbind(c(0,0),c(1,0))
  edges = list(edge)
  graph <- metric_graph$new(edges = edges)
  graph$build_mesh(n = n)
  return(graph)
}


# true eigenfunctions
interval.eig <- function(k,graph){
  x <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[,2],1)
  if(k==0){
    f.0 <- rep(1,length(x)) 
    f1 = c(f.0[1],f.0[length(f.0)], f.0[2:(length(f.0)-1)])
    f = list(phi=f1) 
  }else{
    f.c <- sqrt(2)*cos(pi*k*x) 
    f2 = c(f.c[1],f.c[length(f.c)], f.c[2:(length(f.c)-1)])
    f <- list(psi=f2)
  }
  return(f)
}

# parameters
n = 333
rho = 1
m = 4
nu = 0.6
sigma = 1
n.overkill = 1000
graph = gets_graph_interval(n = n)

kappa = sqrt(8*nu)/rho
tau = sqrt(gamma(nu) / (sigma^2 * kappa^(2*nu) * (4*pi)^(1/2) * gamma(nu + 1/2)))  #sigma = 1, d = 1
alpha = nu + 1/2

# getting true covariance matrix
true_cov_mat <- matrix(0,nrow = dim(graph$mesh$V)[1],ncol = dim(graph$mesh$V)[1])
for(i in 0:n.overkill){
  if(i==0){
    phi <- interval.eig(i,graph)$phi
    true_cov_mat <- true_cov_mat + (1/(kappa^2 + (i*pi)^2)^(alpha))*phi%*%t(phi)
  }else{
    psi <- interval.eig(i,graph)$psi
    true_cov_mat <- true_cov_mat + (1/(kappa^2 + (i*pi)^2)^(alpha))*psi%*%t(psi) 
  }
  
}
true_cov_mat <- true_cov_mat/tau^2
true_cov_mat_reordered <- true_cov_mat[c(1,3:(n+2),2), c(1,3:(n+2),2)] # needs reorder


# getting approximate matrix 
op <- matern.operators(alpha = alpha, kappa = kappa, tau = tau,
                       m = m, graph = graph)
appr_cov_mat = op$covariance_mesh()
appr_cov_mat_reordered <- appr_cov_mat[c(1,3:(n+2),2), c(1,3:(n+2),2)] # needs reorder

# computing the errors
L_inf_error = max(abs(true_cov_mat - appr_cov_mat))
L_2_error = sqrt(as.double(t(graph$mesh$weights)%*%(true_cov_mat - appr_cov_mat)^2%*%graph$mesh$weights))
print(L_inf_error)
## [1] 0.0009502132
print(L_2_error)
## [1] 0.0001074629

# plot just for confirmation
plot(true_cov_mat_reordered[round(n/2)+1,], type="l", col = "blue", lty = 2) #true
lines(appr_cov_mat_reordered[round(n/2)+1,], type = "l", col = "red", lty = 3) #approx
legend("topright", legend = c("true", "approx"), col = c("blue", "red"), lty = 1)

Figure 1: True and approximate covariance matrix for the interval graph.

point  <- c(1, 0.5)
c_cov <- op$cov_function_mesh(matrix(point,1,2))
loc <- graph$coordinates(PtE = point)
m1 <- which.min((graph$mesh$V[,1]-loc[1])^2 + (graph$mesh$V[,2]-loc[2])^2)
p <- graph$plot_function(true_cov_mat[m1,], plotly = TRUE, line_width = 3)
graph$plot_function(c_cov,p=p, line_color = "red", plotly = TRUE,line_width = 3)

Figure 2: True and approximate covariance matrix for the interval graph.

Circle graph

r = 1/(pi)
theta <- seq(from=-pi,to=pi,length.out = 100)
edge <- cbind(1+r+r*cos(theta),r*sin(theta))
edges = list(edge)

n = 666

graph <- metric_graph$new(edges = edges)
graph$build_mesh(n = n)

#true eigenfunctions
circle.eig <- function(k,graph,L){
  x <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[,2],2)
  if(k==0){
    f.0 <- rep(1,length(x)) 
    f.0 = f.0[-length(f.0)]
    f = list(phi=f.0/sqrt(L)) 
  }else{
    f.c <- sqrt(2/L)*cos(pi*k*x/L) 
    f.c = f.c[-length(f.c)]
    f.s <- sqrt(2/L)*sin(pi*k*x/L) 
    f.s = f.s[-length(f.s)]
    f <- list(phi=f.c,psi=f.s)
  }
  return(f)
}

rho = 1
m = 4
nu = 0.6
sigma = 1
kappa = sqrt(8*nu)/rho
tau = sqrt(gamma(nu) / (sigma^2 * kappa^(2*nu) * (4*pi)^(1/2) * gamma(nu + 1/2)))  #sigma = 1, d = 1
alpha = nu + 1/2

L = 1
#check KL expansion
true_cov_mat <- matrix(0,nrow = dim(graph$mesh$V)[1],ncol = dim(graph$mesh$V)[1])
for(i in 0:1000){
  if(i==0){
    phi <- circle.eig(i,graph,L)$phi
    true_cov_mat <- true_cov_mat + (1/(kappa^2 + (i*pi/L)^2)^(alpha))*phi%*%t(phi)
  }else{
    eigen <- circle.eig(i,graph,L)
    psi <- eigen$psi
    phi <- eigen$phi
    true_cov_mat <- true_cov_mat + (1/(kappa^2 + (i*pi/L)^2)^(alpha))*psi%*%t(psi) + 
                           (1/(kappa^2 + (i*pi/L)^2)^(alpha))*phi%*%t(phi)
  }
  
}
true_cov_mat <- true_cov_mat/tau^2/2

# getting approximate matrix 
op <- matern.operators(alpha = alpha, kappa = kappa, tau = tau,
                       m = m, graph = graph)
appr_cov_mat = op$covariance_mesh()

# computing the errors
L_inf_error = max(abs(true_cov_mat - appr_cov_mat))
L_2_error = sqrt(as.double(t(graph$mesh$weights)%*%(true_cov_mat - appr_cov_mat)^2%*%graph$mesh$weights))
print(L_inf_error)
## [1] 0.0008314198
print(L_2_error)
## [1] 0.0002983357

#plot just for confirmation
plot(true_cov_mat[round(n/2),], type="l", col = "blue", lty = 2) #true
lines(appr_cov_mat[round(n/2),], type = "l", col = "red", lty = 3) #approx
legend("topright", legend = c("true", "approx"), col = c("blue", "red"), lty = 1)

Figure 3: True and approximate covariance matrix for the circle graph.

point  <- c(1,0)
c_cov <- op$cov_function_mesh(matrix(point,1,2))
loc <- graph$coordinates(PtE = point)
m1 <- which.min((graph$mesh$V[,1]-loc[1])^2 + (graph$mesh$V[,2]-loc[2])^2)
p <- graph$plot_function(true_cov_mat[m1,], plotly = TRUE, line_width = 3)
graph$plot_function(c_cov,p=p, line_color = "red", plotly = TRUE,line_width = 3)

Figure 4: True and approximate covariance matrix for the circle graph.

References

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