tau_effect.Rmd

---
title: "The rol of $\\tau$"
date: "Created: 05-07-2024. Last modified: `r format(Sys.time(), '%d-%m-%Y.')`"
output:
  html_document:
    mathjax: "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
    highlight: pygments
    theme: flatly
    code_folding: show # class.source = "fold-hide" to hide code and add a button to show it
    # df_print: paged
    toc: true
    toc_float:
      collapsed: true
      smooth_scroll: true
    number_sections: false
    fig_caption: true
    code_download: true
always_allow_html: true
bibliography: 
  - references.bib
  - grateful-refs.bib
header-includes:
  - \newcommand{\ar}{\mathbb{R}}
  - \newcommand{\llav}[1]{\left\{#1\right\}}
  - \newcommand{\pare}[1]{\left(#1\right)}
  - \newcommand{\Ncal}{\mathcal{N}}
  - \newcommand{\Vcal}{\mathcal{V}}
  - \newcommand{\Ecal}{\mathcal{E}}
  - \newcommand{\Wcal}{\mathcal{W}}
---

```{r xaringanExtra-clipboard, echo = FALSE}
htmltools::tagList(
  xaringanExtra::use_clipboard(
    button_text = "<i class=\"fa-solid fa-clipboard\" style=\"color: #00008B\"></i>",
    success_text = "<i class=\"fa fa-check\" style=\"color: #90BE6D\"></i>",
    error_text = "<i class=\"fa fa-times-circle\" style=\"color: #F94144\"></i>"
  ),
  rmarkdown::html_dependency_font_awesome()
)
```


```{css, echo = FALSE}
body .main-container {
  max-width: 100% !important;
  width: 100% !important;
}
body {
  max-width: 100% !important;
}

body, td {
   font-size: 16px;
}
code.r{
  font-size: 14px;
}
pre {
  font-size: 14px
}
.custom-box {
  background-color: #f5f7fa; /* Light grey-blue background */
  border-color: #e1e8ed; /* Light border color */
  color: #2c3e50; /* Dark text color */
  padding: 15px; /* Padding inside the box */
  border-radius: 5px; /* Rounded corners */
  margin-bottom: 20px; /* Spacing below the box */
}
.caption {
  margin: auto;
  text-align: center;
  margin-bottom: 20px; /* Spacing below the box */
}
```


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


```{r}
# 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
}
```


```{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))
}
```


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

```{r}
library(rSPDE)
library(MetricGraph)

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

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



Let's build a small graph to illustrate the effect of $\tau$ on the graph.

```{r}
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)
```

```{r}
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
```


```{r}
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
```


# Plot the functions. They play the role of covariates.

:::: {style="display: grid; grid-template-columns: 400px 400px; grid-column-gap: 1px;"}

::: {}

```{r, out.width = "100%", fig.cap = captioner("Discontinuous covariate.")}
aux_graph$plot_function(aux_f, vertex_size = 0, type = "plotly", line_color = "blue", line_width = 3, continuous = FALSE, interpolate_plot = FALSE) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = "Non-continuous covariate")
```


:::

::: {}

```{r, out.width = "100%", fig.cap = captioner("Continuous covariate.")}
graph$plot_function(g, vertex_size = 0, type = "plotly", line_color = "red", line_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = "Continuous covariate")
```


:::

::::


```{r}
a = 1.8
normal_sample <- rnorm(length(g))
```


# Both covariates are continuous


```{r}
# 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)
```

:::: {style="display: grid; grid-template-columns: 400px 400px 400px; grid-column-gap: 1px;"}

::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\tau$.")}
graph$plot_function(model_for_tau, vertex_size = 0, type = "plotly", line_color = "blue", line_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\tau(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```

:::

::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\kappa$.")}
graph$plot_function(model_for_kappa, vertex_size = 0, type = "plotly", line_color = "red", line_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\kappa(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```


:::

::: {}

```{r, out.width = "100%", fig.cap = captioner("Simulated field.")}
graph$plot_function(X = u, vertex_size = 0, type = "plotly", line_color = "darkgreen", line_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("u(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```

:::

::::


# Both covariates are discontinuos


```{r}
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)
```




```{r}
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)
```


:::: {style="display: grid; grid-template-columns: 400px 400px 400px; grid-column-gap: 1px;"}


::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\tau$.")}
aux_graph$plot_function(model_for_tau, vertex_size = 0, type = "plotly", line_color = "blue", line_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\tau(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```

:::


::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\kappa$.")}
aux_graph$plot_function(model_for_kappa, vertex_size = 0, type = "plotly", line_color = "red", line_width = 3) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\kappa(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```


:::



::: {}


```{r, out.width = "100%", fig.cap = captioner("Simulated field.")}
aux_graph$plot_function(data = "f", newdata= prod, vertex_size = 0, type = "plotly", line_color = "darkgreen", line_width = 3, continuous = FALSE) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("u(s)"), 
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```


:::


::::



# The covariate for $\tau$ is discontinuous and the covariate for $\kappa$ is continuous

```{r}
# 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)
```

```{r}
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)
```

:::: {style="display: grid; grid-template-columns: 400px 400px 400px; grid-column-gap: 1px;"}

::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\tau$.")}
aux_graph$plot_function(model_for_tau, vertex_size = 1, type = "plotly", line_color = "blue", line_width = 3, edge_color = "black", edge_width = 3) %>%
  config(mathjax = 'cdn') %>%
  layout(title = TeX("\\tau(s)"),
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```

:::


::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\kappa$.")}
graph$plot_function(model_for_kappa, vertex_size = 1, type = "plotly", line_color = "red", line_width = 3, edge_color = "black", edge_width = 3) %>%
  config(mathjax = 'cdn') %>%
  layout(title = TeX("\\kappa(s)"),
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```


:::



::: {}


```{r, out.width = "100%", fig.cap = captioner("Simulated field.")}
aux_graph$plot_function(data = "f", newdata= prod, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = 3, continuous = FALSE, interpolate_plot = FALSE, edge_color = "black", edge_width = 3) %>%
  config(mathjax = 'cdn') %>%
  layout(title = TeX("u(s)"),
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```


:::



::::




# The covariate for $\tau$ is continuous and the covariate for $\kappa$ is discontinuous

```{r}
# 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)
```


:::: {style="display: grid; grid-template-columns: 400px 400px 400px; grid-column-gap: 1px;"}

::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\tau$.")}
graph$plot_function(model_for_tau, vertex_size = 1, type = "plotly", line_color = "blue", line_width = 3, edge_color = "black", edge_width = 3) %>%
  config(mathjax = 'cdn') %>%
  layout(title = TeX("\\tau(s)"),
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```

:::

::: {}

```{r, out.width = "100%", fig.cap = captioner("Model for $\\kappa$.")}
aux_graph$plot_function(model_for_kappa, vertex_size = 1, type = "plotly", line_color = "red", line_width = 3, edge_color = "black", edge_width = 3) %>%
  config(mathjax = 'cdn') %>%
  layout(title = TeX("\\kappa(s)"),
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```


:::

::: {}

```{r, out.width = "100%", fig.cap = captioner("Simulated field.")}
graph$plot_function(X = u, vertex_size = 1, type = "plotly", line_color = "darkgreen", line_width = 3, edge_color = "black", edge_width = 3) %>%
  config(mathjax = 'cdn') %>%
  layout(title = TeX("u(s)"),
font = list(family = "Palatino"),
         showlegend = FALSE,
         scene = list(
           aspectratio = list(x = 1/2, y = 1, z = 1),
camera = list(
      eye = list(x = -1*a, y = 0.5*a, z = 0.7*a))))
```

:::

::::


# References

```{r}
cite_packages(output = "paragraph", out.dir = ".")
```