Computing the marginal standard deviation explicitly and computationally efficiently

Below we set some global options for all code chunks in this document.

rm(list = ls())
# Set seed for reproducibility
set.seed(593) 
# 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)
}
# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
  floor(x * 100) / 100
}

Below we load the necessary libraries and define the auxiliary functions.

library(rSPDE)
library(MetricGraph)
library(sp)
library(osmdata)

library(dplyr)
library(plotly)
library(scales)
library(patchwork)
library(tidyr)

library(here)
library(rmarkdown)
# Cite all loaded packages
library(grateful)

rtilde_matern_interval <- function(kappa, tau, le, t1, t2) (1 / (2 * kappa * tau^2 * sinh(kappa * le))) * (cosh(kappa * (le - abs(t1 - t2))) + cosh(kappa * (t1 + t2 - le)))
sparse_A <- function(n) bandSparse(n-1, n, k = c(0, 1), diagonals = list(rep(1, n), rep(-1, n - 1)))

Let’s build a small graph to illustrate how to compute the marginal standard deviation.

{r}
edge1 <- rbind(c(0,0), c(1,0))[2:1, ]
theta <- seq(from = -pi, to = pi,length.out = 100)
edge2 <- cbind(1+1/pi+cos(theta)/pi, sin(theta)/pi)
edge3 <- rbind(c(-1,-1), c(1,0))[2:1, ]
edges <- list(edge1, edge2, edge3)
graph <- metric_graph$new(edges = edges)

{r}
edge <- rbind(c(0,0),c(1,0))
edges <- list(edge)
graph <- metric_graph$new(edges = edges)

{r}
edges_roots_of_unity <- function(n, length = 10) {
  x_coords <- cos(2 * pi * (0:(n-1)) / n)
  y_coords <- sin(2 * pi * (0:(n-1)) / n)
  return(lapply(1:n, function(i) {rbind(c(0, 0), c(length*x_coords[i], length*y_coords[i]))}))
}
n <- 10
my_length <- 2
graph <- metric_graph$new(edges = edges_roots_of_unity(n, length = my_length))

a <- 3
b <- 12
edge2_x <- runif(1, 8, 12)

edge1 <- rbind(c(0,0),c(-runif(1, a, b),0))
edge2 <- rbind(c(0,0),c(edge2_x,0))
edge3 <- rbind(c(edge2_x,0),c(edge2_x + runif(1, a, b),0))
edge4 <- rbind(c(edge2_x,0),c(edge2_x + runif(1, a, b),-runif(1, a, b)))
edge5 <- rbind(c(edge2_x,0),c(edge2_x + runif(1, a, b),runif(1, a, b)))
edge6 <- rbind(c(0,0),c(-runif(1, a, b),-runif(1, a, b)))
edge7 <- rbind(c(0,0),c(-runif(1, a, b), runif(1, a, b)))

edges <- list(edge1, edge2, edge3, edge4, edge5, edge6, edge7)
graph <- metric_graph$new(edges = edges)

{r}
edges <- logo_lines() 
graph <- metric_graph$new(edges = edges, perform_merges = TRUE) 

{r}
call <- opq(bbox = c(39.0884, 22.33, 39.115, 22.3056))
call <- add_osm_feature(call, key = "highway",value=c("motorway",
                                                      "primary","secondary",
                                                      "tertiary",
                                                      "residential"))
data <- osmdata_sf(call)
graph <- graph_components$new(data)
graph <- graph$get_largest()
graph$prune_vertices()

graph_copy <- graph$clone()
graph_pcr_1 <- graph$clone()
graph_pcr_2 <- graph$clone()
graph$build_mesh(h = 0.1)

sigma <- 1
range <- 1
nu <- 1/2

kappa <- sqrt(8*nu)/range
tau <-sqrt(gamma(nu) / (sigma^2 * kappa^(2*nu) * (4*pi)^(1/2) * gamma(nu + 1/2)))
alpha <- nu + 1/2 #alpha = 1

rtilde_int <- function(t1, t2, le) rtilde_matern_interval(kappa = kappa, tau = tau, le, t1, t2)

get_marginal_std <- function(kappa, tau, graph){
  
  V <- graph$V
  E <- graph$E
  degrees <- graph$get_degrees() 
  edge_lengths <- as.vector(graph$get_edge_lengths())
  VtE <- graph$mesh$VtE
  
  
  if(nrow(V) == 2 & nrow(E) == 1){
    aux_function <- function(VtErow) {
      edge <- VtErow[1]
      s <- VtErow[2]
      le <- edge_lengths[edge]
      return(rtilde_int(le*s, le*s, le))
  }
  } else {
    # Build matrix U
    U <- matrix(nrow = 0, ncol = 2)
    for (vertex in seq_len(nrow(V))) {
      if (degrees[vertex] > 1) {
        edges_leaving <- cbind(which(E[,1] %in% vertex), 0) 
        edges_entering <- cbind(which(E[,2] %in% vertex), 1) 
        if (sum(dim(edges_leaving)) == 2 ) {edges_leaving <- NULL}
        if (sum(dim(edges_entering)) == 2 ) {edges_entering <- NULL}
        constraints <- rbind(edges_leaving, edges_entering)
        ordered_constraints <- constraints[order(constraints[, 1], constraints[, 2]),]
        U <- rbind(U, ordered_constraints)
      }
    }
    # Build matrix A
    A <- bdiag(lapply(degrees[degrees >1], sparse_A))
    # Build matrix Sigma
    edge_lengths_in_U <- edge_lengths[U[, 1]] 
    s_diag <- edge_lengths_in_U*U[, 2] 
    diag <- as.vector(mapply(rtilde_int, t1 = s_diag, t2 = s_diag, le = edge_lengths_in_U))
    Sigma <- Diagonal(x = diag)
    for (edge in seq_len(nrow(E))) {
      pos <- which(U[, 1] == edge)
      if (length(pos) > 1) {
        Sigma[pos[1], pos[2]] <- Sigma[pos[2], pos[1]] <- rtilde_int(t1 = 0, t2 = edge_lengths[edge], le = edge_lengths[edge])
      }
    }
  
    ASigmaAt <- A %*% Sigma %*% t(A)
    
    aux_function <- function(VtErow) {
      edge <- VtErow[1]
      s <- VtErow[2]
      le <- edge_lengths[edge]
      pos <- which(U[, 1] == edge)
      cc <- mapply(rtilde_int, t1 = le*s, t2 = U[pos,2]*le, le = le)
      v <- if(length(pos) == 1) as.vector(A[, pos] * cc) else as.vector(A[, pos] %*% cc)
      return(rtilde_int(le*s, le*s, le) - t(v) %*% solve(ASigmaAt, v))
    }
  }
  return(sqrt(apply(X = VtE, MARGIN = 1, FUN = aux_function)))
}

compute_pcr_2v <- function(cov_matrix, graph, cor_threshold = 0.1) {
  
  # Compute the correlation matrix
  cor_matrix <-cov2cor(cov_matrix)
  # Compute the geodesic distance matrix
  graph$compute_geodist_mesh() 
  # Extract the geodesic distance matrix
  dist_matrix <- graph$mesh$geo_dist 
  # Initialize the vector to store the practical correlation range
  pcr <- numeric(dim(cor_matrix)[1]) 
  
  process_row <- function(row_index) {
    # For each row_index, create an auxiliary matrix aux_mat with the correlation and geodesic distance
    aux_mat <- cbind(cor_matrix[row_index, ], dist_matrix[row_index, ]) 
    # Order the auxiliary matrix by the geodesic distance
    ordered_aux_mat <- aux_mat[order(aux_mat[, 2]), ] 
    # Filter the auxiliary ordered matrix by the correlation threshold
    filtered_aux_mat <- ordered_aux_mat[ordered_aux_mat[,1] < cor_threshold, ] 
    if (nrow(filtered_aux_mat) > 0) {
      # Return the minimum geodesic distance such that the correlation is below the threshold
      return(filtered_aux_mat[1, 2]) 
    } else {
      # If the condition is not satisfied, record the minimum correlation value and return an error message
      min <- min(aux_mat[, 1])
      stop(paste0("The condition cor_matrix[row_index, ] < cor_threshold is not satisfied for row_index = ", 
                  row_index, ". Increase cor_threshold. The minimum value of cor_matrix[row_index, ] is ",
                  min))
    }
  }
  
  pcr <- sapply(1:dim(cor_matrix)[1], process_row)
  
  return(pcr)
}

std_cheap <- get_marginal_std(kappa = kappa, tau = tau, graph = graph)

mesh_loc <- graph$get_mesh_locations() %>% 
  as.data.frame() %>% 
  mutate(y = 1) %>% 
  rename(edge_number = V1, distance_on_edge = V2)

graph_copy$add_observations(mesh_loc,
                       edge_number = "edge_number",
                       distance_on_edge = "distance_on_edge",
                       data_coords = "PtE",
                       normalized = TRUE, 
                       clear_obs = TRUE)
graph_copy$observation_to_vertex()

Q <- spde_precision(kappa = kappa,
               tau = tau,
               alpha = alpha,
               graph = graph_copy,
               BC = 0) # BC = 1

Sigma <- solve(Q)
std_expensive <- sqrt(diag(Sigma))

nonstat_cov <-  Matrix::Diagonal(x = 1/std_cheap)%*% Sigma %*%Matrix::Diagonal(x = 1/std_cheap)
nonstat_est_sigma <- sqrt(Matrix::diag(nonstat_cov))

sum(abs(std_cheap - std_expensive))

## [1] 6.007417e-13

plot(std_cheap - std_expensive)

graph_pcr_1$build_mesh(h = 0.1)
graph_pcr_2$build_mesh(h = 0.1)
pcr_1 <- compute_pcr_2v(cov_matrix = Sigma, graph = graph_pcr_1, cor_threshold = 0.1)
pcr_2 <- compute_pcr_2v(cov_matrix = nonstat_cov, graph = graph_pcr_2, cor_threshold = 0.1)
sum(abs(pcr_1 - pcr_2))

## [1] 0

plot(pcr_1 - pcr_2)

The following are the plots for the marginal standard deviation.

a = 2.35
graph$plot_function(X = std_expensive, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = 3, edge_color = "black", edge_width = 3) %>% # true one
  graph$plot_function(X = std_cheap, p = ., vertex_size = 1, type = "plotly", line_color = "blue", line_width = 3, edge_color = "black", edge_width = 3) %>% # approx one
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\sigma(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

graph$plot_function(X = nonstat_est_sigma, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = 3, edge_color = "black", edge_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\sigma_u(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

graph$plot_function(X = pcr_1, vertex_size = 1, type = "plotly", line_color = "red", line_width = 3, edge_color = "black", edge_width = 3) %>% # true one
  graph$plot_function(X = pcr_2, p = ., vertex_size = 1, type = "plotly", line_color = "red", line_width = 3, edge_color = "black", edge_width = 3) %>% # approx one
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\rho(s), \\rho_u(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

The following are the corresponding plots for the practical correlation range.

a = 2.35
graph$plot_function(X = pcr_1, vertex_size = 1, type = "plotly", line_color = "blue", line_width = 3, edge_color = "black", edge_width = 3) %>% 
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\rho(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

graph$plot_function(X = pcr_2, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = 3, edge_color = "black", edge_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\rho_u(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

graph$plot_function(X = pcr_1 - pcr_2, vertex_size = 1, type = "plotly", line_color = "red", line_width = 3, edge_color = "black", edge_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\rho_u(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

Sigma <- cov2cor(Sigma)

graph$plot_function(X = Sigma[300,], vertex_size = 1, type = "plotly", line_color = "red", line_width = 3, edge_color = "black", edge_width = 3) %>%
  graph$plot_function(X = Sigma[5,], p = ., vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = 3, edge_color = "black", edge_width = 3) %>% 
graph$plot_function(X = Sigma[3,], p = ., vertex_size = 1, type = "plotly", line_color = "blue", line_width = 3, edge_color = "black", edge_width = 3) %>% 
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\rho_u(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

References

cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.4.1 (R Core Team 2024a) and the following R packages: cowplot v. 1.1.3 (Wilke 2024), ggmap v. 4.0.0.900 (Kahle and Wickham 2013), ggpubr v. 0.6.0 (Kassambara 2023), ggtext v. 0.1.2 (Wilke and Wiernik 2022), grid v. 4.4.1 (R Core Team 2024b), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 24.12.11 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018; Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.12.0.9002 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.48 (Xie 2014, 2015, 2024), latex2exp v. 0.9.6 (Meschiari 2022), Matrix v. 1.6.5 (Bates, Maechler, and Jagan 2024), MetricGraph v. 1.4.0.9000 (Bolin, Simas, and Wallin 2023b, 2023a, 2023c, 2024; Bolin et al. 2024), OpenStreetMap v. 0.4.0 (Fellows and JMapViewer library by Jan Peter Stotz 2023), osmdata v. 0.2.5 (Mark Padgham et al. 2017), patchwork v. 1.2.0 (Pedersen 2024), plotly v. 4.10.4 (Sievert 2020), plotrix v. 3.8.4 (J 2006), reshape2 v. 1.4.4 (Wickham 2007), rmarkdown v. 2.28 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2024), rSPDE v. 2.4.0.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.3.0 (Wickham, Pedersen, and Seidel 2023), sf v. 1.0.19 (E. Pebesma 2018; E. Pebesma and Bivand 2023), sp v. 2.1.4 (E. J. Pebesma and Bivand 2005; Bivand, Pebesma, and Gomez-Rubio 2013), tidyverse v. 2.0.0 (Wickham et al. 2019), viridis v. 0.6.4 (Garnier et al. 2023), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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