Computing the marginal standard deviation explicitly and computationally efficiently

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truncate_to_two <- function(x) {
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Below we load the necessary libraries and define the auxiliary functions.

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

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

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

library(slackr)
source("keys.R")
slackr_setup(token = token) # token comes from keys.R

## [1] "Successfully connected to Slack"

capture.output(
  knitr::purl(here::here("functionality1.Rmd"), output = here::here("functionality1.R")),
  file = here::here("old/purl_log.txt")
)
source(here::here("functionality1.R"))

Again, recall the SPDE \[\begin{equation} \label{eq:spde} \tag{1} (\kappa^2 - \Delta_{\Gamma})^{\alpha/2}(\tau u) = \mathcal{W}, \quad \text{on $\Gamma$}, \end{equation}\] where \(\tau\) and \(\kappa\) are non constant.

1 Variance-stationary Whittle–Matérn fields

One feature of the Whittle–Matérn fields on metric graphs is that they are inherently non-isotropic (Bolin, Simas, and Wallin (2024a)). Their marginal variances and practical correlation ranges change over the graph even if \(\kappa\) and \(\tau\) are constants. That the practical correlation range is non-stationary is often realistic for applications (Bolin, Simas, and Wallin (2024a)). However, a non-constant variance might be less ideal in some applications. An example of the marginal standard deviations of a Whittle–Matérn field with \(\alpha=1\), \(\kappa=2\), and \(\tau=1\), on a simple graph are shown in Figure 1. One can note that the variance is higher close to vertices of degree \(1\) and lower close to vertices of degree larger than two.

An interesting option to mitigate this effect is a nonstationary model defined via \[\begin{equation} \label{eq:spde_variance_stat} \tag{2} (\kappa^2 - \Delta_{\Gamma})^{\alpha/2} (\sigma_{\kappa} u) = \sigma_0 \mathcal{W}, \quad \text{on } \Gamma, \end{equation}\] where \(\sigma_0 > 0\) and \(\alpha > 1/2\) are constants, \(\kappa\) satisfies Assumption 1, and \(\sigma_{\kappa}(s)\) represents the marginal standard deviations of a generalized Whittle–Matérn field with the same \(\kappa\) and \(\alpha\), but with \(\tau = 1\). This is a special case of \(\eqref{eq:spde}\) with \(\tau(s) = \sigma_0^{-1} \sigma_{\kappa}(s)\). Under this specification, the model parameters are \((\kappa, \sigma_0, \alpha)\), and it follows directly that the variance of \(u(s)\) is \(\sigma_0^2\) for all \(s \in \Gamma\). Indeed, letting \(v\) denote the generalized Whittle–Matérn field with marginal standard deviations \(\sigma_{\kappa}(s)\), by \(\eqref{eq:spde_variance_stat}\), we have \(v = \sigma^{-1}_0\sigma_\kappa u\). Since \(\mathbb{V}(v(s)) = \sigma_\kappa^2(s)\), it follows that \[ \mathbb{V}(u(s)) = \mathbb{V}(\sigma_0\sigma_\kappa^{-1}(s)v(s)) = \sigma_0^2\sigma_\kappa^{-2}(s)\mathbb{V}(v(s)) = \sigma_0^2\sigma_\kappa^{-2}(s)\sigma_\kappa^{2}(s) = \sigma_0^2. \] For this reason, we refer to this model as a variance-stationary Whittle–Matérn field. Figure 2 illustrates the marginal variance of the field for \(\sigma_0 = 1\), showing that the variance is indeed 1 across all locations.

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.

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$new(edges = edges)
graph_copy <- graph$clone()
graph_copy2 <- graph$clone()

h <- 0.1
graph$build_mesh(h = h)
graph_copy2$build_mesh(h = h)

range <- 1
nu <- 1/2

kappa <- sqrt(8*nu)/range
tau <- 1
sigma <- sqrt(gamma(nu) / (tau^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)

2 Checking constant variance for variance-stationary Whittle–Matérn fields (without FEM approximation)

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)))
}

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))
plot(std_cheap - std_expensive)

a = 2.8
msdwmf <- graph$plot_function(X = std_cheap, vertex_size = 1, type = "plotly", line_color = "#0000C8", line_width = gsw, edge_color = "black", edge_width = gsw) %>% # approx one
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\sigma_{\\kappa}(\\cdot)"), y = 0.8),
         font = list(family = "Palatino"),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))


msdvswmf <- graph$plot_function(X = nonstat_est_sigma, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = gsw, edge_color = "black", edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\sigma_0"), y = 0.8),
         font = list(family = "Palatino"),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

save(msdwmf, file = here::here("data_files/msdwmf.Rdata"))
save(msdvswmf, file = here::here("data_files/msdvswmf.Rdata"))

The following are the plots for the marginal standard deviation.

load(here::here("data_files/msdwmf.Rdata"))
msdwmf

Figure 1: Marginal standard deviation \(\sigma_\kappa(\cdot)\) of a Whittle–Matérn field with \(\tau = 1\).

load(here::here("data_files/msdvswmf.Rdata"))
msdvswmf

Figure 2: Marginal standard deviation \(\sigma_0\) of a Generalized Whittle–Matérn field with \(\tau(\cdot) = \sigma_\kappa(\cdot)\).

3 Checking constant variance for variance-stationary Whittle–Matérn fields (with FEM approximation)

B.tau = cbind(0, log(tau), 0, 0, 0)
B.kappa = cbind(0, 0, log(kappa), 0, 0)
op <- rSPDE::spde.matern.operators(graph = graph_copy2,
                                      B.tau = B.tau,
                                      B.kappa =  B.kappa,
                                      parameterization = "spde",
                                      theta = c(1,1,0,0),
                                      alpha = alpha)
# Compute the covariance matrix
est_cov_matrix <- covariance_mesh(op)
# Compute the standard deviation
est_sigma_s <- sqrt(Matrix::diag(est_cov_matrix))

B.tau = cbind(0, log(est_sigma_s), 0, 0, 0)
B.kappa = cbind(0, 0, rep(log(kappa), length(log(est_sigma_s))), 0, 0)
op <- rSPDE::spde.matern.operators(graph = graph_copy2,
                                      B.tau = B.tau,
                                      B.kappa =  B.kappa,
                                      parameterization = "spde",
                                      theta = c(1,1,0,0),
                                      alpha = alpha)
# Compute the covariance matrix
est_cov_matrix <- covariance_mesh(op)
# Compute the standard deviation
est_sigma_0 <- sqrt(Matrix::diag(est_cov_matrix))

msdwmf_fem <- graph_copy2$plot_function(X = est_sigma_s, vertex_size = 1, type = "plotly", line_color = "#0000C8", line_width = gsw, edge_color = "black", edge_width = gsw) %>% 
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\sigma_{\\kappa}(\\cdot)"), y = 0.8),
         font = list(family = "Palatino"),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))


msdvswmf_fem <- graph_copy2$plot_function(X = est_sigma_0, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = gsw, edge_color = "black", edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\sigma_0"), y = 0.8),
         font = list(family = "Palatino"),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
           aspectratio = list(x = 1, y = 1, z = 1),
camera = list(
      eye = list(x = -0.7*a, y = 0.7*a, z = 0.7*a))))

save(msdwmf_fem, file = here::here("data_files/msdwmf_fem.Rdata"))
save(msdvswmf_fem, file = here::here("data_files/msdvswmf_fem.Rdata"))

load(here::here("data_files/msdwmf_fem.Rdata"))
msdwmf_fem

Figure 3: Marginal standard deviation \(\sigma_\kappa(\cdot)\) of a Whittle–Matérn field with \(\tau = 1\).

load(here::here("data_files/msdvswmf_fem.Rdata"))
msdvswmf_fem

Figure 4: Marginal standard deviation \(\sigma_0\) of a Generalized Whittle–Matérn field with \(\tau(\cdot) = \sigma_\kappa(\cdot)\).

4 References

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