Simulation study

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# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
  truncated <- floor(x * 100) / 100
  sprintf("%.2f", truncated)
}

# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
# remotes::install_github("davidbolin/ngme2", ref = "devel")


library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(dplyr)
library(ggplot2)
library(plotly)
library(tidyr)
library(patchwork)
library(here)

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

## [1] "Successfully connected to Slack"

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)
  print(param_spde)
  param_matern <- summary(fit_matern)
  print(param_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))
}

# Build the graph
# graph <- metric_graph$new(edges = pems$edges)

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 = 20)
edge4 <- cbind(sin(theta),1+ cos(theta))
edges = list(edge1, edge2, edge3, edge4)
graph <- metric_graph$new(edges = edges)


# Build the mesh on the graph
graph$build_mesh(h = 0.01)

# Parameters
sigma <- 1.2
range <- 0.2
nu <- 0.8
rspde.order <- 1

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


op <- matern.operators(nu = nu, range = range, sigma = sigma, 
                       parameterization = "matern",
                       m = rspde.order, graph = graph) 
n.rep <- 10
u.rep <- simulate(op, nsim = n.rep)

obs.per.edge <- 50
obs.loc <- NULL
for(i in 1:graph$nE) {
  obs.loc <- rbind(obs.loc,
                   cbind(rep(i,obs.per.edge), runif(obs.per.edge)))
}
n.obs <- obs.per.edge*graph$nE
A <- graph$fem_basis(obs.loc)

sigma.e <- 0.1
Y.rep <- A %*% u.rep + sigma.e * matrix(rnorm(n.obs * n.rep), ncol = n.rep)
y_vec <- 1 + as.vector(Y.rep)
repl <- rep(1:n.rep, each = n.obs)                       

df_data_repl <- data.frame(y = y_vec,
                           edge_number = rep(obs.loc[,1], n.rep),
                           distance_on_edge = rep(obs.loc[,2], n.rep), 
                           repl = repl)

graph$add_observations(data = df_data_repl, normalized = TRUE, 
                       group = "repl", clear_obs = TRUE)

slackr_msg(text = "Data simulated and added", channel = "#research")

#graph$plot(data = "y", group = 2, vertex_size = 0, type = "ggplot") |> ggplotly()

############### Case alpha = 1 ###########################
# 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) +
  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
# Summarize the results
summary(rspde_fit_stat)

slackr_msg(text = "Model nu =0.5 fitted", channel = "#research")

################ Case alpha = 2 ########################### 
# 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) +
  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
# Summarize the results
summary(rspde_fit_stat)

slackr_msg(text = "Model nu =1.5 fitted", channel = "#research")

################ Case alpha estimated ###########################
# Build the model
rspde_model_stat <- rspde.metric_graph(graph,
                                       #nu.upper.bound = 4,
                                       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) +
  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
# Summarize the results
summary(rspde_fit_stat)

slackr_msg(text = "Model nu estimated fitted", channel = "#research")


mean_and_mode_params_statnu0.5
mean_and_mode_params_statnu1.5
mean_and_mode_params_statnuest



################# Define the groups for the simulation with replicates ####################
data <- graph$get_data()
# Define aux data frame to compute the distance matrix
aux <- data |> filter(repl == 1) |>
  rename(distance_on_edge = .distance_on_edge, edge_number = .edge_number) |> # Rename the variables (because graph$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$compute_geodist_PtE(PtE = aux,
                                        normalized = TRUE,
                                        include_vertices = FALSE)
# Define the distance vector
distance = seq(from = 0, to = 1, by = 0.1)/4
distance <- distance[-1] # Remove 0
# Compute the groups for one replicate
GROUPS <- list()
for (j in 1:length(distance)) {
  GROUPS[[j]] = list()
  for (i in 1:nrow(aux)) {
    GROUPS[[j]][[i]] <- which(as.vector(distmatrix[i, ]) > distance[j])
  }
}
# Compute the groups for all replicates, based on the groups of the first replicate
nrowY <- length(unique(data$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

slackr_msg(text = "Groups computed", channel = "#research")

list_models_before_cv <- list(mean_and_mode_params_statnuest = mean_and_mode_params_statnuest,
              rspde_fit_statnu0.5 = rspde_fit_statnu0.5,
              rspde_fit_statnu1.5 = rspde_fit_statnu1.5,
              rspde_fit_statnuest = rspde_fit_statnuest,
              data = data,
              distance = distance,
              GROUPS = GROUPS)
save(list_models_before_cv, file = here::here("data_files/cv_models.RData"))

load(here::here("data_files/cv_models.RData"))
GROUPS <- list_models_before_cv$GROUPS
data <- list_models_before_cv$data
point_of_interest <- 1 # Any number between 1 and nrow(data)
small_neighborhood <- GROUPS[[10]][[point_of_interest]]
large_neighborhood <- GROUPS[[5]][[point_of_interest]]
p <- graph$plot(vertex_size = 0) +
  geom_point(data = data,
             aes(x = .coord_x, y = .coord_y, color = "All points"),
             size = 3) +
  geom_point(data = data[large_neighborhood, ],
             aes(x = .coord_x, y = .coord_y, color = "Small excluded neighborhood"),
             size = 2) +
  geom_point(data = data[small_neighborhood, ],
             aes(x = .coord_x, y = .coord_y, color = "Large excluded 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",
      "Small excluded neighborhood" = "green",
      "Large excluded 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::here("data_files/cv_groups_plot.RData"))

load(here::here("data_files/cv_groups_plot.RData"))
p

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

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

###################### Cross-validation with replicates ##########################

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

# cross-validation for-loop
for (j in 1:length(distance)) {
  print(j)
  slackr_msg(text = paste0("server 3: j =", j), channel = "#research")

  # 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]])
  # 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)
  
  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))
}

# Create data frames
mse_df <- data.frame(
  distance,
  Statnu0.5 = mse.statnu0.5,
  Statnu1.5 = mse.statnu1.5,
  Statnuest = mse.statnuest
)

ls_df <- data.frame(
  distance,
  Statnu0.5 = -ls.statnu0.5,
  Statnu1.5 = -ls.statnu1.5,
  Statnuest = -ls.statnuest
)

# Save the results
list_to_save <- list(mean_and_mode_params_statnu0.5 = mean_and_mode_params_statnu0.5,
                     mean_and_mode_params_statnu1.5 = mean_and_mode_params_statnu1.5,
                     mean_and_mode_params_statnuest = mean_and_mode_params_statnuest,
                     mse_df = mse_df, 
                     ls_df = ls_df)

save(list_to_save, file = here::here("data_files/cv_result.RData"))

load(here::here("data_files/cv_result.RData"))
mse_df <- list_to_save$mse_df
ls_df <- list_to_save$ls_df
mean_and_mode_params_statnuest <- list_to_save$mean_and_mode_params_statnuest
################### Plot the results ##########################
choose_index <- seq(1, nrow(mse_df), by = 1)
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", 
  "Statnu1.5" = "2", 
  "Statnuest" = paste(round(mean_and_mode_params_statnuest[4,1]+0.5, 3), "(est)")
)

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

linetype_mapping <- c(
  "Statnu0.5" = "solid", 
  "Statnu1.5" = "solid", 
  "Statnuest" = "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 = "$\\mbox{Geodesic distance } R$") +
  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 = "$\\mbox{Geodesic distance } R$") +
  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 <- (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::here("data_files/cv_sim.png"), plot = combined_plot, width = 9.22, height = 4.01, dpi = 500)
myggsave(combined_plot, width = 9.22, height = 4.01)

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

Figure 2: MSE and negative Log-Score as functions of distance.

1 References

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