The rol of \(\tau\)

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

library(rSPDE)
library(MetricGraph)

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"))

gets_f_in_dist_mesh <- function(graph, f_values){
  VtE <- graph$mesh$VtE
  E <- graph$E
  nE <- graph$nE
  nV <- graph$nV
  E_ext <- data.frame(edge_number = 1:nE, vertex_start = E[,1], vertex_end = E[,2])
  
  original_vertex <- VtE[1:nV,]
  f <- f_values[1:nV]
  
  no_vertices <- cbind(VtE[(nV+1):length(f_values),], f_values[(nV+1):length(f_values)])
  
  same_vertex_list <- list()
  for (i in 1:nE) {
    same_vertex_list[[i]] <- E_ext %>% 
      filter(vertex_start == i | vertex_end == i) %>%
      mutate(lor = case_when(vertex_start == i ~ 0,vertex_end == i ~ 1)) %>% 
      dplyr::select(edge_number, lor) %>% 
      as.matrix()
  }
  
  for (i in seq_len(nrow(original_vertex))) {
    current_row <- original_vertex[i, ]
    # Find which matrix in same_vertex_list contains this row
    for (j in seq_along(same_vertex_list)) {
      if (any(apply(same_vertex_list[[j]], 1, function(x) all(x == current_row)))) {
        # Add a new column with the corresponding value from f
        same_vertex_list[[j]] <- cbind(same_vertex_list[[j]], f[i])
        break
      }
    }
  }
  
  result_matrix <- do.call(rbind, same_vertex_list)
  return(rbind(no_vertices, result_matrix))
}

Let’s build a small graph to illustrate the effect of \(\tau\) on the graph.

edge1 <- rbind(c(0,0),c(1,0))
edge2 <- rbind(c(0,0),c(0,1))
edge3 <- rbind(c(0,1),c(-1,1))
theta <- seq(from = pi,to = 3*pi/2,length.out = 50)
edge4 <- cbind(sin(theta),1+ cos(theta))
edges <- list(edge1, edge2, edge3, edge4)

h = 0.01
graph <- metric_graph$new(edges = edges)
aux_graph <- graph$clone()
graph$build_mesh(h=h, continuous = TRUE)
edge_number <- graph$mesh$VtE
XY_graph <- graph$mesh$V

aux_graph$build_mesh(h=h, continuous = FALSE)
aux_edge_number <- aux_graph$mesh$VtE
aux_XY_graph <- aux_graph$mesh$V

Let \(f(s) = \mathrm{edge.number}(s)/4\) and \(g(s) = 0.5 \cdot (x^2(s) - y^2(s)) + 0.5\), where \((x(s), y(s))\) denote the Euclidean coordinates on the plane (see Figures 1 and 2). Clearly, \(f\) is continuous, whereas \(g\) is discontinuous. These functions serve as covariates in the analysis.

f <- edge_number[,1]/4 # discontinuous covariate
g <- 0.5*(XY_graph[, 1]^2 - XY_graph[, 2]^2) + 0.5 # continuous covariate

aux_f <- aux_edge_number[,1]/4# discontinuous covariate
aux_g <- 0.5*(aux_XY_graph[, 1]^2 - aux_XY_graph[, 2]^2) + 0.5 # continuous covariate

a = 2

1 Plot the covariates.

cov_f <- aux_graph$plot_function(aux_f, vertex_size = 1, type = "plotly", line_color = "#0000C8", line_width = gsw, continuous = FALSE, interpolate_plot = FALSE, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("f(s) = \\mathrm{edge.number}(s)/4"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

cov_g <- graph$plot_function(g, vertex_size = 1, type = "plotly", line_color = "red", line_width = gsw, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("g(s) = 0.5 \\cdot (x^2(s) - y^2(s)) + 0.5"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

save(cov_f, file = here::here("data_files/cov_f.Rdata"))
save(cov_g, file = here::here("data_files/cov_g.Rdata"))

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

Figure 1: Discontinuous covariate \(f(s) = \mathrm{edge.number}(s)/4\).

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

Figure 2: Continuous covariate \(g(s) = 0.5 \cdot (x^2(s) - y^2(s)) + 0.5\).

normal_sample <- rnorm(length(g))

2 Both covariates are continuous

We now consider the case where both covariates are continuous, that is \(\tau(s) = \exp(g(s))\) and \(\kappa(s) = \exp(g(s))\). Using these, we simulate \(u(s)\). See Figures 3,4, and 5.

# Define the matrices B.tau and B.kappa
B.tau =   cbind(0, 1, 0, g, 0)
B.kappa = cbind(0, 0, 1, 0, g)
# Log-regression coefficients
theta <- c(0, 0, 1, 1) 
# Choose alpha
nu = 2.5
alpha = nu + 1/2
# Compute the operator
op <- rSPDE::spde.matern.operators(graph = graph,
                                      B.tau = B.tau,
                                      B.kappa =  B.kappa,
                                      parameterization = "spde",
                                      theta = theta,
                                      alpha = alpha)
# Simulate the non-stationary field
Sigma <- precision(op)
R <- chol(Sigma)
u = solve(R, normal_sample)

model_for_tau <- exp(B.tau[,-1]%*%theta)
model_for_kappa <- exp(B.kappa[,-1]%*%theta)

kkk1 <- graph$plot_function(model_for_tau, vertex_size = 1, type = "plotly", line_color = "#0000C8", line_width = gsw, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\tau(s) = \\exp(g(s))"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk2 <- graph$plot_function(model_for_kappa, vertex_size = 1, type = "plotly", line_color = "red", line_width = gsw, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\kappa(s) = \\exp(g(s))"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk3 <- graph$plot_function(X = u, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = gsw, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("u(s)"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))


save(kkk1, file = here::here("data_files/kkk1.Rdata"))
save(kkk2, file = here::here("data_files/kkk2.Rdata"))
save(kkk3, file = here::here("data_files/kkk3.Rdata"))

load(here::here("data_files/kkk1.Rdata"))
kkk1

Figure 3: Model for \(\tau(s)\).

load(here::here("data_files/kkk2.Rdata"))
kkk2

Figure 4: Model for \(\kappa(s)\).

load(here::here("data_files/kkk3.Rdata"))
kkk3

Figure 5: Simulated field \(u(s)\).

3 Both covariates are discontinuos

We now consider the case where both covariates are discontinuous, that is \(\tau(s) = \exp(f(s))\) and \(\kappa(s) = \exp(f(s))\). Using these, we simulate \(u(s)\). See Figures 6,7, and 8.

tau <- 1
# Define the matrices B.tau and B.kappa
B.tau <- cbind(0, 1, 0, log(tau)*rep(1, length(f)), 0)
realB.tau <- cbind(0, 1, 0, f, 0)
B.kappa = cbind(0, 0, 1, 0, f)
# Log-regression coefficients
theta <- c(0, 0, 1, 1) 
# Compute the operator
op1 <- rSPDE::spde.matern.operators(graph = graph,
                                      B.tau = B.tau,
                                      B.kappa =  B.kappa,
                                      parameterization = "spde",
                                      theta = theta,
                                      alpha = alpha)

realmodel_for_tau <- exp(realB.tau[,-1]%*%theta)
normal_sample1 <- normal_sample #rnorm(length(g))

Sigma1 <- precision(op1)
R1 <- chol(Sigma1)
u1cont <- solve(R1, normal_sample1)
u1 <-u1cont*tau/realmodel_for_tau

op2 <- rSPDE::spde.matern.operators(graph = graph,
                                      B.tau = realB.tau,
                                      B.kappa =  B.kappa,
                                      parameterization = "spde",
                                      theta = theta,
                                      alpha = alpha)
Sigma2 <- precision(op2)
R2 <- chol(Sigma2)
u2 = solve(R2, normal_sample1)

# To plot the models for tau and kappa
aux_B.tau =   cbind(0, 1, 0, aux_f, 0)
aux_B.kappa = cbind(0, 0, 1, 0, aux_f)
model_for_tau <- exp(aux_B.tau[,-1]%*%theta)
model_for_kappa <- exp(aux_B.kappa[,-1]%*%theta)

gg <- gets_f_in_dist_mesh(graph, u1cont)
aux_df <- data.frame(.edge_number = gg[,1], .distance_on_edge = gg[,2], f = gg[,3])
another_df <- cbind(tau/model_for_tau, aux_graph$mesh$VtE)
colnames(another_df) <- c("tau_rel", ".edge_number", ".distance_on_edge")
another_df <- as.data.frame(another_df)
# Step 1: Sort aux_df by the four columns
aux_df_sorted <- aux_df %>%
  arrange(.edge_number, .distance_on_edge)

another_df_sorted <- another_df %>%
  arrange(.edge_number, .distance_on_edge)
# Merge
merged_df <- cbind(aux_df_sorted, another_df_sorted)[c(1:4)] %>% mutate(final_f = f*tau_rel) %>% dplyr::select(-tau_rel, -f) %>% rename(edge_number = .edge_number, distance_on_edge = .distance_on_edge, f = final_f)

prod <- aux_graph$process_data(data = merged_df, normalized = TRUE)

kkk4 <- aux_graph$plot_function(model_for_tau, vertex_size = 1, type = "plotly", line_color = "#0000C8", line_width = gsw, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\tau(s) = \\exp(f(s))"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk5 <- aux_graph$plot_function(model_for_kappa, vertex_size = 1, type = "plotly", line_color = "red", line_width = gsw, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("\\kappa(s) = \\exp(f(s))"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk6 <- aux_graph$plot_function(data = "f", newdata= prod, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = gsw, continuous = FALSE, edge_width = gsw) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = list(text = TeX("u(s)"), y = 0.85),
font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))


save(kkk4, file = here::here("data_files/kkk4.Rdata"))
save(kkk5, file = here::here("data_files/kkk5.Rdata"))
save(kkk6, file = here::here("data_files/kkk6.Rdata"))

load(here::here("data_files/kkk4.Rdata"))
kkk4

Figure 6: Model for \(\tau(s)\).

load(here::here("data_files/kkk5.Rdata"))
kkk5

Figure 7: Model for \(\kappa(s)\).

load(here::here("data_files/kkk6.Rdata"))
kkk6

Figure 8: Simulated field \(u(s)\).

4 The covariate for \(\tau\) is discontinuous and the covariate for \(\kappa\) is continuous

We now consider the case where the covariate for \(\tau\) is discontinuous and the covariate for \(\kappa\) is continuous, that is \(\tau(s) = \exp(f(s))\) and \(\kappa(s) = \exp(g(s))\). Using these, we simulate \(u(s)\). See Figures 12, 13, and 14.

# Define the matrices B.tau and B.kappa
B.tau <- cbind(0, 1, 0, log(tau)*rep(1, length(f)), 0)
realB.tau <- cbind(0, 1, 0, f, 0)
B.kappa = cbind(0, 0, 1, 0, g)
# Log-regression coefficients
theta <- c(0, 0, 1, 1)
# Compute the operator
op1 <- rSPDE::spde.matern.operators(graph = graph,
                                      B.tau = B.tau,
                                      B.kappa =  B.kappa,
                                      parameterization = "spde",
                                      theta = theta,
                                      alpha = alpha)

realmodel_for_tau <- exp(realB.tau[,-1]%*%theta)
normal_sample2 <- normal_sample#rnorm(length(g))

Sigma1 <- precision(op1)
R1 <- chol(Sigma1)
u1cont <- solve(R1, normal_sample2)
u1 <-u1cont*tau/realmodel_for_tau

op2 <- rSPDE::spde.matern.operators(graph = graph,
                                    B.tau = realB.tau,
                                    B.kappa =  B.kappa,
                                    parameterization = "spde",
                                    theta = theta,
                                    alpha = alpha)
Sigma2 <- precision(op2)
R2 <- chol(Sigma2)
u2 = solve(R2, normal_sample2)

# To plot the models for tau and kappa
aux_B.tau =   cbind(0, 1, 0, aux_f, 0)
model_for_tau <- exp(aux_B.tau[,-1]%*%theta)
model_for_kappa <- exp(B.kappa[,-1]%*%theta)

gg <- gets_f_in_dist_mesh(graph, u1cont)
aux_df <- data.frame(.edge_number = gg[,1], .distance_on_edge = gg[,2], f = gg[,3])
another_df <- cbind(tau/model_for_tau, aux_graph$mesh$VtE)
colnames(another_df) <- c("tau_rel", ".edge_number", ".distance_on_edge")
another_df <- as.data.frame(another_df)
# Step 1: Sort aux_df by the four columns
aux_df_sorted <- aux_df %>%
  arrange(.edge_number, .distance_on_edge)

another_df_sorted <- another_df %>%
  arrange(.edge_number, .distance_on_edge)
# Merge
merged_df <- cbind(aux_df_sorted, another_df_sorted)[c(1:4)] %>% mutate(final_f = f*tau_rel) %>% dplyr::select(-tau_rel, -f) %>% rename(edge_number = .edge_number, distance_on_edge = .distance_on_edge, f = final_f)

prod <- aux_graph$process_data(data = merged_df, normalized = TRUE)

kkk7 <- aux_graph$plot_function(model_for_tau, vertex_size = 1, type = "plotly", line_color = "#0000C8", line_width = gsw, edge_color = "black", edge_width = gsw) %>%
  config(mathjax = 'cdn') %>%
  layout(font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk8 <- graph$plot_function(model_for_kappa, vertex_size = 1, type = "plotly", line_color = "red", line_width = gsw, edge_color = "black", edge_width = gsw) %>%
  config(mathjax = 'cdn') %>%
  layout(font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk9 <- aux_graph$plot_function(data = "f", newdata= prod, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = gsw, continuous = FALSE, interpolate_plot = FALSE, edge_color = "black", edge_width = gsw) %>%
  config(mathjax = 'cdn') %>%
  layout(font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

save(kkk7, file = here::here("data_files/kkk7.Rdata"))
save(kkk8, file = here::here("data_files/kkk8.Rdata"))
save(kkk9, file = here::here("data_files/kkk9.Rdata"))

5 The covariate for \(\tau\) is continuous and the covariate for \(\kappa\) is discontinuous

We now consider the case where the covariate for \(\tau\) is continuous and the covariate for \(\kappa\) is discontinuous, that is \(\tau(s) = \exp(g(s))\) and \(\kappa(s) = \exp(f(s))\). Using these, we simulate \(u(s)\). See Figures 9, 10, and 11.

# Define the matrices B.tau and B.kappa
B.tau =   cbind(0, 1, 0, g, 0) #rep(1, length(f))
B.kappa = cbind(0, 0, 1, 0, f)
# Log-regression coefficients
theta <- c(0, 0, 1, 1)
# Compute the operator
op <- rSPDE::spde.matern.operators(graph = graph,
                                      B.tau = B.tau,
                                      B.kappa =  B.kappa,
                                      parameterization = "spde",
                                      theta = theta,
                                      alpha = alpha)
# Simulate the non-stationary field
Sigma <- precision(op)
R <- chol(Sigma)
u = solve(R, normal_sample)

aux_B.kappa = cbind(0, 0, 1, 0, aux_f)
model_for_tau <- exp(B.tau[,-1]%*%theta)
model_for_kappa <- exp(aux_B.kappa[,-1]%*%theta)

kkk10 <- graph$plot_function(model_for_tau, vertex_size = 1, type = "plotly", line_color = "#0000C8", line_width = gsw, edge_color = "black", edge_width = gsw) %>%
  config(mathjax = 'cdn') %>%
  layout(font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk11 <- aux_graph$plot_function(model_for_kappa, vertex_size = 1, type = "plotly", line_color = "red", line_width = gsw, edge_color = "black", edge_width = gsw) %>%
  config(mathjax = 'cdn') %>%
  layout(font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

kkk12 <- graph$plot_function(X = u, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = gsw, edge_color = "black", edge_width = gsw) %>%
  config(mathjax = 'cdn') %>%
  layout(font = list(family = "Palatino"),
         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 = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))

save(kkk10, file = here::here("data_files/kkk10.Rdata"))
save(kkk11, file = here::here("data_files/kkk11.Rdata"))
save(kkk12, file = here::here("data_files/kkk12.Rdata"))

load(here::here("data_files/kkk7.Rdata"))
kkk7

load(here::here("data_files/kkk8.Rdata"))
kkk8

load(here::here("data_files/kkk9.Rdata"))
kkk9

6 References

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

We used R version 4.5.2 (R Core Team 2025a) and the following R packages: cowplot v. 1.2.0 (Wilke 2025), ggmap v. 4.0.2 (Kahle and Wickham 2013), ggpubr v. 0.6.3 (Kassambara 2026), ggtext v. 0.1.2 (Wilke and Wiernik 2022), glue v. 1.8.0 (Hester and Bryan 2024), grid v. 4.5.2 (R Core Team 2025b), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 25.11.22 (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. 2.13.0 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.50 (Xie 2014, 2015, 2025), latex2exp v. 0.9.8 (Meschiari 2026), Matrix v. 1.7.3 (Bates, Maechler, and Jagan 2025), MetricGraph v. 1.5.0.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), OpenStreetMap v. 0.4.1 (Fellows and Stotz 2025), patchwork v. 1.3.1 (Pedersen 2025), plotly v. 4.11.0 (Sievert 2020), plotrix v. 3.8.14 (J 2006), renv v. 1.1.7 (Ushey and Wickham 2026), reshape2 v. 1.4.4 (Wickham 2007), reticulate v. 1.44.1 (Ushey, Allaire, and Tang 2025), rmarkdown v. 2.30 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2025), rSPDE v. 2.5.2.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.4.0 (Wickham, Pedersen, and Seidel 2025), sf v. 1.1.0 (E. Pebesma 2018; E. Pebesma and Bivand 2023), slackr v. 3.4.0 (Kaye et al. 2025), sp v. 2.2.1 (E. J. Pebesma and Bivand 2005; Bivand, Pebesma, and Gomez-Rubio 2013), tidyverse v. 2.0.0 (Wickham et al. 2019), tikzDevice v. 0.12.6 (Sharpsteen and Bracken 2023), viridis v. 0.6.5 (Garnier et al. 2024), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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