PeMS 2, modeling

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This vignette compares different models for PeMS data. It uses pems_repl1_data.RData, which is a file with a graph and data created in pems_repl1.html.

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knitr::opts_chunk$set(
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# 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) {
  truncated <- floor(x * 100) / 100
  sprintf("%.2f", truncated)
}

Below we load the necessary libraries.

# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)

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

library(ggplot2)
library(cowplot)
library(ggpubr) #annotate_figure()
library(grid) #textGrob()
library(ggmap)

library(viridis)
library(OpenStreetMap)


library(tidyr)
library(sf)

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

Below we define function process_model_results() to extract the summary of the parameters of the model.

process_model_results <- function(fit, model) {
  fit_spde <- rspde.result(fit, "field", model, parameterization = "spde")
  fit_matern <- rspde.result(fit, "field", model, parameterization = "matern")
  df_for_plot_spde <- gg_df(fit_spde)
  df_for_plot_matern <- gg_df(fit_matern)
  param_spde <- summary(fit_spde)
  param_matern <- summary(fit_matern)
  param_fixed <- fit$summary.fixed[,1:6]
  marginal.posterior.sigma_e = inla.tmarginal(
    fun = function(x) exp(-x/2), 
    marginal = fit[["internal.marginals.hyperpar"]][["Log precision for the Gaussian observations"]])
  quant.sigma_e <- capture.output({result_tmp <- inla.zmarginal(marginal.posterior.sigma_e)}, file = "/dev/null") 
  quant.sigma_e <- result_tmp
  statistics.sigma_e <- unlist(quant.sigma_e)[c(1,2,3,5,7)]
  mode.sigma_e <- inla.mmarginal(marginal.posterior.sigma_e)
  allparams <- rbind(param_fixed, param_spde, param_matern, c(statistics.sigma_e, mode.sigma_e))
  rownames(allparams)[nrow(allparams)] <- "sigma_e"
  return(list(allparams = allparams, df_for_plot_spde = df_for_plot_spde, df_for_plot_matern = df_for_plot_matern))
}

We first load the data in the file pems_repl1_data.RData and extract the data from the graph.

# Load the data
load(here("data_files/pems_repl1_data.RData"))
# Extract the data from the graph
data <- graph$get_data()

Below we extract the locations to compute the distance matrix. Using this matrix, we define the groups for cross-validation. Observe that we only compute the distance matrix for the first replicate and compute the groups for it. As all replicates share the same locations, we can use the groups structure from the first replicate for all replicates.

# Define the distance vector
distance = seq(from = 0, to = 10, by = 0.1)

# Define aux data frame to compute the distance matrix
aux <- data |> filter(repl == 1) |>
  rename(distance_on_edge = .distance_on_edge, edge_number = .edge_number) |> # Rename the variables (because graph$compute_geodist_PtE() requires so)
  as.data.frame() |> # Transform to a data frame (i.e., remove the metric_graph class)
  dplyr::select(edge_number, distance_on_edge)

# Compute the distance matrix
distmatrix <- graph$compute_geodist_PtE(PtE = aux,
                                             normalized = TRUE,
                                             include_vertices = FALSE)

# Compute the groups for one replicate
GROUPS <- list()
for (j in 1:length(distance)) {
  GROUPS[[j]] = list()
  for (i in 1:nrow(aux)) {
    GROUPS[[j]][[i]] <- which(as.vector(distmatrix[i, ]) <= distance[j])
  }
}
# Compute the groups for all replicates, based on the groups of the first replicate
nrowY <- length(unique(data$repl))
ncolY <- nrow(filter(data, repl == 1))
NEW_GROUPS <- list()
for (j in 1:length(distance)) {
  my_list <- GROUPS[[j]]
  aux_list <- list()
  for (i in 0:(nrowY - 1)) {
  added_vectors <- lapply(my_list, function(vec) vec + i*ncolY)
  aux_list <- c(aux_list, added_vectors)
  }
  NEW_GROUPS[[j]] <- aux_list
}

GROUPS <- NEW_GROUPS
save(GROUPS, file = here("data_files/groups_for_cv.RData"))

Below we plot to check that the groups are correctly defined.

point_of_interest <- 3 # Any number between 1 and nrow(data)
small_neighborhood <- GROUPS[[20]][[point_of_interest]]
large_neighborhood <- GROUPS[[50]][[point_of_interest]]
p <- graph$plot(vertex_size = 0) +
  geom_point(data = data,
             aes(x = .coord_x, y = .coord_y, color = "All points"),
             size = 2) +
  geom_point(data = data[large_neighborhood, ],
             aes(x = .coord_x, y = .coord_y, color = "Large neighborhood"),
             size = 1.5) +
  geom_point(data = data[small_neighborhood, ],
             aes(x = .coord_x, y = .coord_y, color = "Small neighborhood"),
             size = 1) +
  geom_point(data = data[point_of_interest, ],
             aes(x = .coord_x, y = .coord_y, color = "Point of interest"),
             size = 0.5) +
  scale_color_manual(
    values = c(
      "All points" = "darkviolet",
      "Large neighborhood" = "green",
      "Small neighborhood" = "blue",
      "Point of interest" = "red"
    ),
    name = ""
  ) +
  ggtitle("Groups") + 
  theme_minimal() + 
  theme(text = element_text(family = "Palatino"),
        plot.title = element_text(hjust = 0.5)) +
  coord_fixed()

p <- plotly::ggplotly(p)
save(p, file = here("data_files/plotly_groups_for_cv.RData"))

load(here("data_files/plotly_groups_for_cv.RData"))
p

Figure 1: Illustrations of groups for cross-validation based on the distance matrix.

Below we define the non-stationary parameters.

# Non-stationary parameters
B.tau = cbind(0, 1, 0, cov, 0)
B.kappa = cbind(0, 0, 1, 0, cov)

We now model the speed records \(y_i\) as 13 independent replicates satisfying \[\begin{equation} \label{applimodel} y_i|u(\cdot)\sim N(\beta_0 + \beta_1\text{mean.cov}(s_i) + u(s_i),\sigma_\epsilon^2),\;i = 1,\dots, 314, \end{equation}\] where \(u(\cdot)\) is a Gaussian process on the highway network. We consider stationary models with \(\kappa,\tau>0\) and non-stationary models where \(\tau\) and \(\kappa\) are given by \[\begin{equation} \label{logregressions} \begin{aligned} \log(\tau(s)) &= \theta_1 + \theta_3 \text{std.cov}(s),\\ \log(\kappa(s)) &= \theta_2 + \theta_4 \text{std.cov}(s). \end{aligned} \end{equation}\]

For each of the two classes of models, we consider three cases: when (1) \(\nu\) is fixed to 0.5 or (2) 1.5, and (3) \(\nu\) is estimated from the data.

Below cov refers to \(\text{std.cov}(s)\) and mean_value refers to \(\text{mean.cov}(s)\).

1 Case \(\nu = 0.5\)

We first consider the stationary model.

# Build the model
rspde_model_stat <- rspde.metric_graph(graph,
                                       parameterization = "spde",
                                       nu = 0.5)
# Prepare the data for fitting
data_rspde_bru_stat <- graph_data_rspde(rspde_model_stat,
                                        repl = ".all",
                                        bru = TRUE,
                                        repl_col = "repl")
# Define the component
cmp_stat <- y ~ -1 +
  Intercept(1) +
  mean_value +
  field(cbind(.edge_number, .distance_on_edge), 
        model = rspde_model_stat,
        replicate = repl)
# Fit the model
rspde_fit_stat <-
  bru(cmp_stat,
      data = data_rspde_bru_stat[["data"]],
      family = "gaussian",
      options = list(verbose = FALSE)
  )

output_from_models <-process_model_results(rspde_fit_stat, rspde_model_stat)
parameters_statistics <- output_from_models$allparams
mean_and_mode_params_statnu0.5 <- parameters_statistics[, c(1,6)]
rspde_fit_statnu0.5 <- rspde_fit_stat

We now fit the non-stationary model.

# Build the model
rspde_model_nonstat <- rspde.metric_graph(graph,
                                          B.tau = B.tau,
                                          B.kappa =  B.kappa,
                                          parameterization = "spde",
                                          nu = 0.5)
# Prepare the data for fitting
data_rspde_bru_nonstat <- graph_data_rspde(rspde_model_nonstat,
                                        repl = ".all",
                                        bru = TRUE,
                                        repl_col = "repl")
# Define the component
cmp_nonstat <- y ~ -1 +
  Intercept(1) +
  mean_value +
  field(cbind(.edge_number, .distance_on_edge), 
        model = rspde_model_nonstat,
        replicate = repl)
# Fit the model
rspde_fit_nonstat <-
  bru(cmp_nonstat,
      data = data_rspde_bru_nonstat[["data"]],
      family = "gaussian",
      options = list(verbose = FALSE)
  )

output_from_models <- process_model_results(rspde_fit_nonstat, rspde_model_nonstat)
parameters_statistics <- output_from_models$allparams
mean_and_mode_params_nonstatnu0.5 <- parameters_statistics[, c(1,6)]
rspde_fit_nonstatnu0.5 <- rspde_fit_nonstat

Below we consider the prediction of replicate 14.

# Load the maps p12 and p13 from pems_repl1 vignette
load(here("data_files/maps_zoom12and13from_stadia.RData"))
# We consider replicate 14
replicate.number <- 1
# Prepare the data for prediction
data_prd_list_for_rep <- data_prd_list_mesh
data_prd_list_for_rep[["mean_value"]] <- cov_for_mean_to_plot
data_prd_list_for_rep[["repl"]] <- rep(replicate.number, nrow(data_prd_list_mesh))
# Perform the prediction
repl1_pred_full <- predict(rspde_fit_nonstat, newdata = data_prd_list_for_rep, ~Intercept + mean_value + field_eval(cbind(.edge_number, .distance_on_edge), replicate = repl))
repl1_pred_mean <- repl1_pred_full$mean
# Extract the Euclidean coordinates of the mesh points
xypoints <- graph$mesh$V
# Extract the range of the coordinates 
x_left <- range(xypoints[,1])[1]
x_right <- range(xypoints[,1])[2]
y_bottom <- range(xypoints[,2])[1]
y_top <- range(xypoints[,2])[2]
# Define coordinates for small windows
coordx_lwr1 <- -121.878
coordx_upr1 <- -121.828
coordy_lwr1 <- 37.315
coordy_upr1 <- 37.365

coordx_lwr2<- -122.075
coordx_upr2 <- -122.025
coordy_lwr2 <- 37.365
coordy_upr2 <- 37.415
# Define the colors for the windows
lower_color <- "darkred"   # Dark purple
upper_color <- "darkblue"  # Yellow
# Plot the field on top of the map
f12 <- graph$plot_function(X = repl1_pred_mean, 
                          vertex_size = 0, 
                          p = p12,
                          edge_width = 0.5) + 
  theme_minimal() + 
  theme(text = element_text(family = "Palatino"), 
        axis.text = element_text(size = 8),
        legend.text = element_text(size = 8),
        plot.margin = unit(-0.4*c(1,0,1,1), "cm")
        ) +
  labs(color = "", x = "", y = "") +
  xlim(x_left, x_right) + 
  ylim(y_bottom, y_top)
# Plot the field on top of the map
f13 <- graph$plot_function(X = repl1_pred_mean, 
                          vertex_size = 0, 
                          p = p13,
                          edge_width = 0.5) + 
  theme_minimal() + 
  theme(text = element_text(family = "Palatino"), 
        axis.text = element_text(size = 8),
        legend.text = element_text(size = 8),
        plot.margin = unit(-0.4*c(1,0,1,1), "cm")
        ) +
  labs(color = "", x = "", y = "") +
  xlim(x_left, x_right) + 
  ylim(y_bottom, y_top)

g12 <- graph$plot(data = "y", group = 1, vertex_size = 0, p = f12, edge_width = 0, data_size = 1) + 
  labs(color = "", x = "", y = "") + 
  annotate("segment", x = coordx_lwr1, y = coordy_lwr1, xend = coordx_upr1, yend = coordy_lwr1, 
           linewidth = 0.4, color = upper_color) +  # Bottom line
  annotate("segment", x = coordx_lwr1, y = coordy_upr1, xend = coordx_upr1, yend = coordy_upr1, 
           linewidth = 0.4, color = upper_color) +  # Top line
  annotate("segment", x = coordx_lwr1, y = coordy_lwr1, xend = coordx_lwr1, yend = coordy_upr1, 
           linewidth = 0.4, color = upper_color) +  # Left line
  annotate("segment", x = coordx_upr1, y = coordy_lwr1, xend = coordx_upr1, yend = coordy_upr1, 
           linewidth = 0.4, color = upper_color) +  # Right line
  annotate("segment", x = coordx_lwr2, y = coordy_lwr2, xend = coordx_upr2, yend = coordy_lwr2, 
           linewidth = 0.4, color = lower_color) +  
  annotate("segment", x = coordx_lwr2, y = coordy_upr2, xend = coordx_upr2, yend = coordy_upr2, 
           linewidth = 0.4, color = lower_color) +  
  annotate("segment", x = coordx_lwr2, y = coordy_lwr2, xend = coordx_lwr2, yend = coordy_upr2, 
           linewidth = 0.4, color = lower_color) +  
  annotate("segment", x = coordx_upr2, y = coordy_lwr2, xend = coordx_upr2, yend = coordy_upr2, 
           linewidth = 0.4, color = lower_color)
g13 <- graph$plot(data = "y", group = 1, vertex_size = 0, p = f13, edge_width = 0, data_size = 1) + 
  labs(color = "", x = "", y = "") + 
  annotate("segment", x = coordx_lwr1, y = coordy_lwr1, xend = coordx_upr1, yend = coordy_lwr1, 
           linewidth = 0.4, color = upper_color) +  # Bottom line
  annotate("segment", x = coordx_lwr1, y = coordy_upr1, xend = coordx_upr1, yend = coordy_upr1, 
           linewidth = 0.4, color = upper_color) +  # Top line
  annotate("segment", x = coordx_lwr1, y = coordy_lwr1, xend = coordx_lwr1, yend = coordy_upr1, 
           linewidth = 0.4, color = upper_color) +  # Left line
  annotate("segment", x = coordx_upr1, y = coordy_lwr1, xend = coordx_upr1, yend = coordy_upr1, 
           linewidth = 0.4, color = upper_color) +  # Right line
  annotate("segment", x = coordx_lwr2, y = coordy_lwr2, xend = coordx_upr2, yend = coordy_lwr2, 
           linewidth = 0.4, color = lower_color) +  
  annotate("segment", x = coordx_lwr2, y = coordy_upr2, xend = coordx_upr2, yend = coordy_upr2, 
           linewidth = 0.4, color = lower_color) +  
  annotate("segment", x = coordx_lwr2, y = coordy_lwr2, xend = coordx_lwr2, yend = coordy_upr2, 
           linewidth = 0.4, color = lower_color) +  
  annotate("segment", x = coordx_upr2, y = coordy_lwr2, xend = coordx_upr2, yend = coordy_upr2, 
           linewidth = 0.4, color = lower_color) 

r1 <- g13 + xlim(coordx_lwr1, coordx_upr1) + 
                    ylim(coordy_lwr1, coordy_upr1) + 
  theme(legend.position = "none", 
        plot.margin = unit(-0.2*c(1,1,1,1), "cm"))

r2 <- g13 + xlim(coordx_lwr2, coordx_upr2) + 
                    ylim(coordy_lwr2, coordy_upr2) + 
  theme(legend.position = "none", 
        plot.margin = unit(-0.2*c(1,1,1,1), "cm"))

# Arrange p2 and p3 horizontally
left_col <- plot_grid(r2, r1, labels = NULL, ncol = 1, nrow = 2, rel_heights = c(1,1))

# Combine the top row with p1 in a grid
combined_plot <- plot_grid(left_col, g12, labels = NULL, ncol = 2, rel_widths = c(1,2)) 
final_plot <- annotate_figure(combined_plot, left = textGrob("Latitude", rot = 90, vjust = 1, gp = gpar(cex = 0.8)),
                              bottom = textGrob("Longitude", vjust = -0.5, gp = gpar(cex = 0.8)))
ggsave(here("data_files/replicate14_3_with_prediction.png"), width = 11.2, height = 5.43, plot = final_plot, dpi = 500)

knitr::include_graphics(here("data_files/replicate14_3_with_prediction.png"))

Figure 2: Speed observations (in mph) on the highway network of the city of San Jose in California, recorded on April 3, 2017. The left panels are zoomed-in areas of the panel to the right.

2 Case \(\nu = 1.5\)

We first consider the stationary model.

# Build the model
rspde_model_stat <- rspde.metric_graph(graph,
                                       parameterization = "spde",
                                       nu = 1.5)
# Prepare the data for fitting
data_rspde_bru_stat <- graph_data_rspde(rspde_model_stat,
                                        repl = ".all",
                                        bru = TRUE,
                                        repl_col = "repl")
# Define the component
cmp_stat <- y ~ -1 +
  Intercept(1) +
  mean_value +
  field(cbind(.edge_number, .distance_on_edge), 
        model = rspde_model_stat,
        replicate = repl)
# Fit the model
rspde_fit_stat <-
  bru(cmp_stat,
      data = data_rspde_bru_stat[["data"]],
      family = "gaussian",
      options = list(verbose = FALSE)
  )

output_from_models <-process_model_results(rspde_fit_stat, rspde_model_stat)
parameters_statistics <- output_from_models$allparams
mean_and_mode_params_statnu1.5 <- parameters_statistics[, c(1,6)]
rspde_fit_statnu1.5 <- rspde_fit_stat

We now fit the non-stationary model.

# Build the model
rspde_model_nonstat <- rspde.metric_graph(graph,
                                          B.tau = B.tau,
                                          B.kappa =  B.kappa,
                                          parameterization = "spde",
                                          nu = 1.5)
# Prepare the data for fitting
data_rspde_bru_nonstat <- graph_data_rspde(rspde_model_nonstat,
                                        repl = ".all",
                                        bru = TRUE,
                                        repl_col = "repl")
# Define the component
cmp_nonstat <- y ~ -1 +
  Intercept(1) +
  mean_value +
  field(cbind(.edge_number, .distance_on_edge), 
        model = rspde_model_nonstat,
        replicate = repl)
# Fit the model
rspde_fit_nonstat <-
  bru(cmp_nonstat,
      data = data_rspde_bru_nonstat[["data"]],
      family = "gaussian",
      options = list(verbose = FALSE)
  )

output_from_models <- process_model_results(rspde_fit_nonstat, rspde_model_nonstat)
parameters_statistics <- output_from_models$allparams
mean_and_mode_params_nonstatnu1.5 <- parameters_statistics[, c(1,6)]
rspde_fit_nonstatnu1.5 <- rspde_fit_nonstat

3 Case \(\nu\) estimated

We first consider the stationary model.

# Build the model
rspde_model_stat <- rspde.metric_graph(graph,
                                       parameterization = "spde")
# Prepare the data for fitting
data_rspde_bru_stat <- graph_data_rspde(rspde_model_stat,
                                        repl = ".all",
                                        bru = TRUE,
                                        repl_col = "repl")
# Define the component
cmp_stat <- y ~ -1 +
  Intercept(1) +
  mean_value +
  field(cbind(.edge_number, .distance_on_edge), 
        model = rspde_model_stat,
        replicate = repl)
# Fit the model
rspde_fit_stat <-
  bru(cmp_stat,
      data = data_rspde_bru_stat[["data"]],
      family = "gaussian",
      options = list(verbose = FALSE)
  )

output_from_models <-process_model_results(rspde_fit_stat, rspde_model_stat)
parameters_statistics <- output_from_models$allparams
mean_and_mode_params_statnuest <- parameters_statistics[, c(1,6)]
rspde_fit_statnuest <- rspde_fit_stat

We now fit the non-stationary model.

# Build the model
rspde_model_nonstat <- rspde.metric_graph(graph,
                                          B.tau = B.tau,
                                          B.kappa =  B.kappa,
                                          parameterization = "spde")
# Prepare the data for fitting
data_rspde_bru_nonstat <- graph_data_rspde(rspde_model_nonstat,
                                        repl = ".all",
                                        bru = TRUE,
                                        repl_col = "repl")
# Define the component
cmp_nonstat <- y ~ -1 +
  Intercept(1) +
  mean_value +
  field(cbind(.edge_number, .distance_on_edge), 
        model = rspde_model_nonstat,
        replicate = repl)
# Fit the model
rspde_fit_nonstat <-
  bru(cmp_nonstat,
      data = data_rspde_bru_nonstat[["data"]],
      family = "gaussian",
      options = list(verbose = FALSE)
  )

output_from_models <- process_model_results(rspde_fit_nonstat, rspde_model_nonstat)
parameters_statistics <- output_from_models$allparams
mean_and_mode_params_nonstatnuest <- parameters_statistics[, c(1,6)]
rspde_fit_nonstatnuest <- rspde_fit_nonstat

4 Crossvalidation study

Below we perform leave-group-out pseudo cross-validation (Liu, Van Niekerk, and Rue 2025) following the strategy from (Bolin, Simas, and Xiong 2024a).

mse.statnu0.5 <- mse.nonstatnu0.5 <- ls.statnu0.5 <- ls.nonstatnu0.5 <- rep(0,length(distance))
mse.statnu1.5 <- mse.nonstatnu1.5 <- ls.statnu1.5 <- ls.nonstatnu1.5 <- rep(0,length(distance))
mse.statnuest <- mse.nonstatnuest <- ls.statnuest <- ls.nonstatnuest <- rep(0,length(distance))

# cross-validation for-loop
for (j in 1:length(distance)) {
  print(j)
  # cross-validation of the stationary model
  cv.statnu0.5 <- inla.group.cv(rspde_fit_statnu0.5, groups = GROUPS[[j]])
  cv.statnu1.5 <- inla.group.cv(rspde_fit_statnu1.5, groups = GROUPS[[j]])
  cv.statnuest <- inla.group.cv(rspde_fit_statnuest, groups = GROUPS[[j]])
  # cross-validation of the nonstationary model
  cv.nonstatnu0.5 <- inla.group.cv(rspde_fit_nonstatnu0.5, groups = GROUPS[[j]])
  cv.nonstatnu1.5 <- inla.group.cv(rspde_fit_nonstatnu1.5, groups = GROUPS[[j]])
  cv.nonstatnuest <- inla.group.cv(rspde_fit_nonstatnuest, groups = GROUPS[[j]])
  # obtain MSE and LS
  mse.statnu0.5[j] <- mean((cv.statnu0.5$mean - data$y)^2)
  mse.statnu1.5[j] <- mean((cv.statnu1.5$mean - data$y)^2)
  mse.statnuest[j] <- mean((cv.statnuest$mean - data$y)^2)
  
  
  mse.nonstatnu0.5[j] <- mean((cv.nonstatnu0.5$mean - data$y)^2)
  mse.nonstatnu1.5[j] <- mean((cv.nonstatnu1.5$mean - data$y)^2)
  mse.nonstatnuest[j] <- mean((cv.nonstatnuest$mean - data$y)^2)
  
  
  ls.statnu0.5[j] <- mean(log(cv.statnu0.5$cv))
  ls.statnu1.5[j] <- mean(log(cv.statnu1.5$cv))
  ls.statnuest[j] <- mean(log(cv.statnuest$cv))
  
  ls.nonstatnu0.5[j] <- mean(log(cv.nonstatnu0.5$cv))
  ls.nonstatnu1.5[j] <- mean(log(cv.nonstatnu1.5$cv))
  ls.nonstatnuest[j] <- mean(log(cv.nonstatnuest$cv))
}

# Create data frames
mse_df <- data.frame(
  distance,
  Statnu0.5 = mse.statnu0.5,
  Nonstatnu0.5 = mse.nonstatnu0.5,
  Statnu1.5 = mse.statnu1.5,
  Nonstatnu1.5 = mse.nonstatnu1.5,
  Statnuest = mse.statnuest,
  Nonstatnuest = mse.nonstatnuest
)

ls_df <- data.frame(
  distance,
  Statnu0.5 = -ls.statnu0.5,
  Nonstatnu0.5 = -ls.nonstatnu0.5,
  Statnu1.5 = -ls.statnu1.5,
  Nonstatnu1.5 = -ls.nonstatnu1.5,
  Statnuest = -ls.statnuest,
  Nonstatnuest = -ls.nonstatnuest
)

Save some of the objects to be used in the next vignette.

# Save the results
list_to_save <- list(mean_and_mode_params_statnu0.5 = mean_and_mode_params_statnu0.5,
                     mean_and_mode_params_nonstatnu0.5 = mean_and_mode_params_nonstatnu0.5,
                     mean_and_mode_params_statnu1.5 = mean_and_mode_params_statnu1.5,
                     mean_and_mode_params_nonstatnu1.5 = mean_and_mode_params_nonstatnu1.5, 
                     mean_and_mode_params_statnuest = mean_and_mode_params_statnuest,
                     mean_and_mode_params_nonstatnuest = mean_and_mode_params_nonstatnuest, 
                     mse_df = mse_df, 
                     ls_df = ls_df, 
                     B.tau = B.tau, 
                     B.kappa = B.kappa, 
                     graph = graph)
save(list_to_save, file = here("data_files/pems_repl2_results.RData"))

load(here::here("data_files/pems_repl2_results.RData"))
mean_and_mode_params_statnu0.5 <- list_to_save$mean_and_mode_params_statnu0.5
mean_and_mode_params_nonstatnu0.5 <- list_to_save$mean_and_mode_params_nonstatnu0.5
mean_and_mode_params_statnu1.5 <- list_to_save$mean_and_mode_params_statnu1.5
mean_and_mode_params_nonstatnu1.5 <- list_to_save$mean_and_mode_params_nonstatnu1.5
mean_and_mode_params_statnuest <- list_to_save$mean_and_mode_params_statnuest
mean_and_mode_params_nonstatnuest <- list_to_save$mean_and_mode_params_nonstatnuest
mse_df <- list_to_save$mse_df
ls_df <- list_to_save$ls_df

distance = seq(from = 0, to = 10, by = 0.1)

Below we plot the cross-validation results.

choose_index <- seq(2, nrow(mse_df), by = 3)
mse_df_red <- mse_df[choose_index,]
ls_df_red <- ls_df[choose_index,]
# Convert to long format
mse_long <- mse_df_red %>%
  pivot_longer(cols = -distance, names_to = "nu", values_to = "MSE")

ls_long <- ls_df_red %>%
  pivot_longer(cols = -distance, names_to = "nu", values_to = "LogScore")


# Update the label mappings with the new legend title
label_mapping <- c(
  "Statnu0.5" = "1", 
  "Nonstatnu0.5" = "1", 
  "Statnu1.5" = "2", 
  "Nonstatnu1.5" = "2", 
  "Statnuest" = paste(round(mean_and_mode_params_statnuest[5,1]+0.5, 3), "(est)"), 
  "Nonstatnuest" = paste(round(mean_and_mode_params_nonstatnuest[7,1]+0.5, 3), "(est)")
)

# Define color and linetype mapping
color_mapping <- c(
  "Statnu0.5" = "blue", 
  "Nonstatnu0.5" = "blue", 
  "Statnu1.5" = "black", 
  "Nonstatnu1.5" = "black", 
  "Statnuest" = "red", 
  "Nonstatnuest" = "red"
)

linetype_mapping <- c(
  "Statnu0.5" = "dotdash", 
  "Nonstatnu0.5" = "solid", 
  "Statnu1.5" = "dotdash", 
  "Nonstatnu1.5" = "solid", 
  "Statnuest" = "dotdash", 
  "Nonstatnuest" = "solid"
)

# Plot MSE
mse_plot <- ggplot(mse_long, aes(x = distance, y = MSE, color = nu, linetype = nu)) +
  geom_line(linewidth = 2) +
  labs(y = "MSE", x = "$\\mbox{Geodesic distance } R\\mbox{ }(\\mbox{km})$") +
  scale_color_manual(values = color_mapping, labels = label_mapping, name = "$\\alpha$") +
  scale_linetype_manual(values = linetype_mapping, labels = label_mapping, name = "$\\alpha$") +
  theme_minimal() +
  theme(text = element_text(family = "Palatino"))

# Plot negative log-score
ls_plot <- ggplot(ls_long, aes(x = distance, y = LogScore, color = nu, linetype = nu)) +
  geom_line(linewidth = 2) +
  labs(y = "Negative Log-Score", x = "$\\mbox{Geodesic distance } R\\mbox{ }(\\mbox{km})$") +
  scale_color_manual(values = color_mapping, labels = label_mapping, name = "$\\alpha$") +
  scale_linetype_manual(values = linetype_mapping, labels = label_mapping, name = "$\\alpha$") +
  theme_minimal() +
  theme(text = element_text(family = "Palatino"))

# Combine plots with a shared legend at the top in a single line
combined_plot_pems <- mse_plot + ls_plot + 
  plot_layout(guides = 'collect') & 
  theme(legend.position = 'right') & 
  guides(color = guide_legend(ncol = 1), linetype = guide_legend(nrow = 1))

# Save combined plot
# ggsave(here("data_files/crossval_pems.png"), plot = combined_plot_pems, width = 9.22, height = 4.01, dpi = 500)
myggsave(combined_plot_pems, width = 9.22, height = 4.01)

knitr::include_graphics(here("data_files/tikzpic/combined_plot_pems.pdf"))

Figure 3: MSE and negative Log-Score as functions of distance (in km) for the stationary (dotdash line, \(\boldsymbol{\cdot-\cdot}\)) and non-stationary (solid line, \(\boldsymbol{-\!\!\!-\!\!\!-}\)) cases with \(\nu = 0.5\), \(\nu = 1.5\), and \(\nu\) estimated (est).

5 Estimated values

5.0.1 Estimated parameters for the stationary model with \(\nu = 0.5\)

mean_and_mode_params_statnu0.5

5.0.2 Estimated parameters for the non-stationary model with \(\nu = 0.5\)

mean_and_mode_params_nonstatnu0.5

5.0.3 Estimated parameters for the stationary model with \(\nu = 1.5\)

mean_and_mode_params_statnu1.5

5.0.4 Estimated parameters for the non-stationary model with \(\nu = 1.5\)

mean_and_mode_params_nonstatnu1.5

5.0.5 Estimated parameters for the stationary model with \(\nu\) estimated

mean_and_mode_params_statnuest

5.0.6 Estimated parameters for the non-stationary model with \(\nu\) estimated

mean_and_mode_params_nonstatnuest

6 Linear regression and kNN regression

load(here::here("data_files/Y_mean.RData")) # was created in pems1.Rmd

Y_mu <- apply(Y_raw[1:13,], 2, mean)
load(here::here("data_files/pems_repl1_data.RData"))
df_isocov <- data.frame(y = Y_mu, 
                        edge_number = PtE_raw[,1], 
                        distance_on_edge = PtE_raw[,2])

graph$add_observations(data = df_isocov,
                       edge_number = "edge_number",
                       distance_on_edge = "distance_on_edge",
                       data_coords = "PtE",
                       normalized = TRUE, 
                       clear_obs = TRUE)
# graph$check_euclidean()
# graph
res_exp <- graph_lme(y ~ 1, graph = graph, model = list(type = "isoCov"))
summary(res_exp)
u_est_exp_mean <- predict(res_exp, df_isocov, normalized = TRUE)$mean

plot(Y_mu, type = "l", col = "darkblue")
lines(u_est_exp_mean, col = "darkred")

Y2part <- Y_raw[14:26,]
DF_ISOCOV <- lapply(1:nrow(Y2part), function(i){data.frame(y = Y2part[i,],
                                                   mean_value = u_est_exp_mean,
                                                   edge_number = PtE_raw[,1],
                                                   distance_on_edge = PtE_raw[,2],
                                                   repl = i)})
DF_ISOCOV <- do.call(rbind, DF_ISOCOV)

graph$add_observations(data = DF_ISOCOV, 
                       edge_number = "edge_number",
                       distance_on_edge = "distance_on_edge",
                       data_coords = "PtE",
                       normalized = TRUE, 
                       clear_obs = TRUE, 
                       group = "repl")

RES_EXP <- graph_lme(y ~ mean_value, graph = graph, which_repl = 1:13, model = list(type = "isoCov"))
summary(RES_EXP)
POST <- posterior_crossvalidation_loo(object = RES_EXP, which_repl = 1:13)
MSE_ISOCOV <- POST$scores$rmse^2

load(here::here("data_files/pems_repl1_data.RData"))
data <- graph$get_data()
rspde_model_stat <- rspde.metric_graph(graph,
                                       parameterization = "spde",
                                       nu = 0.5)
# Prepare the data for fitting
data_rspde_bru_stat <- graph_data_rspde(rspde_model_stat,
                                        repl = ".all",
                                        bru = TRUE,
                                        repl_col = "repl")
# Define the component
cmp_stat <- y ~ -1 +
  Intercept(1) +
  mean_value +
  field(cbind(.edge_number, .distance_on_edge), 
        model = rspde_model_stat,
        replicate = repl)
# Fit the model
rspde_fit_stat <-
  bru(cmp_stat,
      data = data_rspde_bru_stat[["data"]],
      family = "gaussian",
      options = list(verbose = FALSE)
  )

load(here::here("data_files/groups_for_cv.RData"))

my_group <- GROUPS[[1]]

cv.statnu0.5 <- inla.group.cv(rspde_fit_stat, groups = my_group)$mean

new_mse <- mean((cv.statnu0.5 - data$y)^2)

# Load the data

load(here::here("data_files/Y_mean.RData")) # was created in pems1.Rmd

Y_mu <- apply(Y_raw[1:13,], 2, mean)

data_simple <- data.frame(y = c(t(Y_raw[14:26,])), 
                          mean_value = rep(Y_mu, times = 13), 
                          repl = rep(1:13, each = 314))

data_simple$repl <- factor(data_simple$repl)

library(lme4)
n <- nrow(data_simple)

pred_loocv <- numeric(n)
#pred_loocv_repl <- numeric(n)

for(i in 1:n){
  train_data <- data_simple[-i, ]
  test_data  <- data_simple[i, , drop = FALSE]
  
  model <- lm(y ~ mean_value, data = train_data)
  pred_loocv[i] <- predict(model, newdata = test_data)
  
  # model <- lmer(y ~ mean_value + (1 | repl), data = train_data)
  # pred_loocv_repl[i] <- predict(model, newdata = test_data, re.form = NULL)
  print(paste("Processed observation", i, "out of", n))
}

mse_loocv_lm <- mean((data_simple$y - pred_loocv)^2)
mse_loocv_lm

# mse_loocv_lmer <- mean((data_simple$y - pred_loocv_repl)^2)
# mse_loocv_lmer

# Load the data
load(here::here("data_files/pems_repl1_data.RData"))
# Extract the data from the graph

initial_data <- graph$get_data()
data <- initial_data |> as.data.frame() |> select(y, mean_value, repl)

n <- length(data |> filter(repl == 1) |> pull(y))
data$repl <- factor(rep(1:13, each = n))

library(FNN)

aux <- initial_data |> filter(repl == 1) |>
  rename(distance_on_edge = .distance_on_edge, 
         edge_number = .edge_number) |> 
  as.data.frame() |> 
  dplyr::select(edge_number, 
                distance_on_edge)

D <- graph$compute_geodist_PtE(
  PtE = aux,
  normalized = TRUE,
  include_vertices = FALSE)

# -----------------------------------------------------------
# 0. Setup
# -----------------------------------------------------------
n_loc  <- 314
n_repl <- 13
n_obs  <- nrow(data)  # 4082

# Location index for each observation in data
# (assumes data is ordered: all 314 locations for repl 1, then repl 2, etc.)
loc_idx <- rep(1:n_loc, times = n_repl)

# -----------------------------------------------------------
# 1. Build normalized distance matrices
# -----------------------------------------------------------

# Spatial: expand 314x314 -> 4082x4082 using location indices
D_space_full <- D[loc_idx, loc_idx]
D_space_norm <- D_space_full / max(D_space_full)

# Covariate: pairwise distances on mean_value across all 4082 observations
D_cov_full <- as.matrix(dist(scale(data$mean_value)))
D_cov_norm <- D_cov_full / max(D_cov_full)

# -----------------------------------------------------------
# 2. Combined distance (precomputed, outside all loops)
# -----------------------------------------------------------
alpha <- 0  # 0 = pure spatial, 1 = pure covariate
D_combined <- alpha * D_cov_norm + (1 - alpha) * D_space_norm

# -----------------------------------------------------------
# 3. LOO cross-validation over k
# -----------------------------------------------------------
k_values <- 1:30

loo_mse <- sapply(k_values, function(k) {
  pred <- numeric(n_obs)
  
  for (i in 1:n_obs) {
    neighbors <- order(D_combined[i, -i])[1:k]
    pred[i]   <- mean(data$y[-i][neighbors])
  }
  
  mean((data$y - pred)^2)
})

# -----------------------------------------------------------
# 4. Results
# -----------------------------------------------------------

best_k    <- k_values[which.min(loo_mse)]
best_KNN_mse <- loo_mse[which.min(loo_mse)]


# Another approach
# 
# k_values <- 1:30
# 
# loo_mse <- sapply(k_values, function(k) {
#   n <- nrow(data)
#   pred <- numeric(n)
#   
#   for (i in 1:n) {
#     pred[i] <- knn.reg(
#       train = matrix(data$mean_value[-i]),
#       y     = data$y[-i],
#       test  = matrix(data$mean_value[i]),
#       k     = k
#     )$pred
#   }
#   
#   mean((data$y - pred)^2)
# })
# 
# plot(k_values, loo_mse, type = "b", xlab = "k", ylab = "LOO RMSE")
# 
# best_k    <- k_values[which.min(loo_mse)]
# best_knn_mse <- loo_mse[which.min(loo_mse)]


save(mse_loocv_lm, best_KNN_mse, MSE_ISOCOV, file = here("data_files/simple_linear_regression_results.RData"))

Below we show the MSE results for the simple linear regression mse_loocv_lm and the kNN regression best_KNN_mse.

load(here("data_files/simple_linear_regression_results.RData"))
mse_df[1,-1]

data.frame(LM = mse_loocv_lm, 
           kNNdistMAT = best_KNN_mse,
           MSE_ISOCOV = MSE_ISOCOV)

7 References

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