chapter_2_lesson_2_handout.qmd

---
title: "Autocorrelation Concepts"
subtitle: "Chapter 2: Lesson 2"
format: html
editor: source
sidebar: false
---
```{r}
#| include: false
source("common_functions.R")
```

```{r}
#| include: false

get_data_for_cov_table <- function(offset = 1) {
  # set random number generator seed
  set.seed(997)
  
  # set parameters
  n <- 10
  rho <- 0.99
  mu <- 0
  sigma <- 3
  
  # build population correlation matrix
  tmp.r <- matrix(rho, n, n)
  tmp.r <- tmp.r^abs(row(tmp.r)-col(tmp.r))
  
  # simulate correlated normal random data
  x1 <- round(mvrnorm(1, rep(mu,n), sigma^2 * tmp.r),1) 
  
  # build a data frame
  df <- data.frame(t = 1:length(x1),
                   x = x1,
                   y = lead(x1, offset)) %>%
    mutate(
      xx = x - mean(x),
      xx2 = xx^2,
      yy = y - mean(x),
      # yy2 = yy^2,
      xy = xx * yy
    ) %>% 
    dplyr::select(t, x, y, xx, xx2, yy, xy)
  
  return(df)
}

make_cov_table_df <- function(df, offset=1, decimals_1st_order = 5, decimals_2nd_order = 5) {
  # Color vector
  # oi_colors <- c("#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#F5C710", "#CC79A7", "#999999")
  oi_colors <- c("black", "black", "black", "black", "black", "black", "black", "black", "black") # Handout is in black print
  
  df_summary <- df %>% 
    summarize(
      x = sum(x),
      y = sum(y, na.rm = TRUE),
      xx = sum(xx),
      yy = sum(yy, na.rm = TRUE),
      xx2 = sum(xx2),
      # yy2 = sum(yy2, na.rm = TRUE),
      xy = sum(xy, na.rm = TRUE)
    ) %>% 
    # round_df(3) %>% 
    mutate(t = paste0("sum")) %>% 
    mutate(
      t = paste0("sum"),
      x = round_as_text(x, 1),
      y = round_as_text(y, 1),
      xx = round_as_text(xx, decimals_1st_order),
      xx2 = round_as_text(xx2, decimals_2nd_order),
      yy = round_as_text(yy, decimals_1st_order),
      # yy2 = round_as_text(yy2, decimals_2nd_order),
      xy = round_as_text(xy, decimals_2nd_order)
    ) %>% 
    dplyr::select(t, x, y, xx, xx2, yy, xy)
  
  out <- df %>%
    mutate(
      t = as.character(t),
      x = round_as_text(x, 1),
      y = round_as_text(y, 1),
      xx = round_as_text(xx, decimals_1st_order),
      xx2 = round_as_text(xx2, decimals_2nd_order),
      yy = round_as_text(yy, decimals_1st_order),
      # yy2 = round_as_text(yy2, decimals_2nd_order),
      xy = round_as_text(xy, decimals_2nd_order)
    ) %>% 
    mutate(
      x = cell_spec(x, 
                    color = case_when(
                      is.na(x) ~ "#999999",
                      TRUE ~ oi_colors[( row_number() + 0 ) %% 9 + 1]
                    )
      ),
      y = cell_spec(y, 
                    color = case_when(
                      is.na(y) ~ "#999999",
                      TRUE ~ oi_colors[( row_number() + offset ) %% 9 + 1]
                    )
      )
    ) %>% 
    mutate(
      # x = ifelse(row_number() > nrow(.) - offset, paste0("[",x,"]"), x),
      y = ifelse(row_number() > nrow(.) - offset, NA, y),
    ) %>% 
    replace(., is.na(.), "—") %>%
    bind_rows(df_summary) %>% 
    rename(
      "x_t" = x,
      "x_{t+k}" = y,
      # paste0("x_{t+", offset, "}") = y,
      "x_t-mean(x)" = xx, 
      "(x_t-mean(x))^2" = xx2, 
      "x_{t+k}-mean(x)" = yy,
      # "(x_{t+k}-mean(x))^2" = yy2, 
      "(x-mean(x))(x_{t+k}-mean(x))" = xy
    )
  
  return(out)
}

# Eliminates the values in columns 3 onward for a specified row_num
make_blank_row_in_cov_table <- function(df, row_num) {
  for (col_num in 3:ncol(df)) {
    df[row_num, col_num] = ""
  }
  return(df)
}

# Compute summary values
compute_summaries <- function(df, digits = 4) {
  df %>% 
    summarize(
      n = n(),
      mean_x = mean(x),
      mean_y = mean(y, na.rm = TRUE),
      ss_x = sum(xx2),
      # ss_y = sum(yy2, na.rm = TRUE),
      ss_xy = sum(xy, na.rm = TRUE),
      c_0 = sum(xx2) / nrow(.),
      c_k = sum(xy, na.rm = TRUE) / nrow(.),
      r_k = c_k / c_0
    ) %>% 
    round_df(digits)
}

```


<!-- ### Autocovariance and Autocorrelation  -->

```{r}
#| echo: false

offset_value <- 1
cov_df <- get_data_for_cov_table(offset = offset_value)

# Obtain the number of data values.
n <- nrow(cov_df)
```




#### Lag $k=1$

```{r}
#| echo: false

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# I could not figure out how to fit the first two tables on one page, so I punted and only displayed part of the table. -- Craig
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offset_value <- 1
cov_df <- get_data_for_cov_table(offset = offset_value)
cov_table <- cov_df %>%
  make_cov_table_df(offset = offset_value)#, decimals_1st_order = 1, decimals_2nd_order = 2) 
cov_table %>%
  display_partial_table(4,3) # Partial table


summarized <- compute_summaries(cov_df)
```


```{r}
#| echo: false

# Initialize table
auto_cov_cor_table <- data.frame(lag = integer(),
                  autocovariance = double(),
                  autocorrelation = double())
```

```{r}
#| echo: false

# Add rows to the table
auto_cov_cor_table <- auto_cov_cor_table %>% 
  bind_rows(c(
    lag = offset_value, 
    autocovariance = summarized$c_k, 
    autocorrelation = summarized$r_k
    )
  )

# # Display the table, except row 2
# auto_cov_cor_table %>%
#   mutate( ####################################### Suppresses row 2
#     autocovariance = ifelse(row_number() == 2,"", autocovariance),
#     autocorrelation = ifelse(row_number() == 2,"", autocorrelation)
#   ) %>% 
#   knitr::kable(format = "html", align='ccc', escape = FALSE, width = NA) %>%
#   kable_styling(full_width = FALSE, "striped") 
```

#### Lag $k = 2$

```{r}
#| echo: false
offset_value <- 2
cov_df <- get_data_for_cov_table(offset = offset_value)
cov_table <- cov_df %>%
  make_cov_table_df(offset = offset_value)#, decimals_1st_order = 1, decimals_2nd_order = 2) 
cov_table %>%
  ############ Let students fill in these rows ##########
  make_blank_row_in_cov_table(4) %>% 
  make_blank_row_in_cov_table(5) %>% 
  make_blank_row_in_cov_table(6) %>%
  make_blank_row_in_cov_table(11) %>% 
  display_table()

summarized <- compute_summaries(cov_df)
```


<br>
<br>
<br>

#### Lag $k = 3$

```{r}
#| echo: false
offset_value <- 3
cov_df <- get_data_for_cov_table(offset = offset_value)
cov_table <- cov_df %>%
  make_cov_table_df(offset = offset_value)#, decimals_1st_order = 1, decimals_2nd_order = 2) 
cov_table %>%
  display_table()

summarized <- compute_summaries(cov_df)
```


```{r}
#| echo: false

# Add rows to the table
auto_cov_cor_table <- auto_cov_cor_table %>% 
  bind_rows(c(
    lag = offset_value, 
    autocovariance = summarized$c_k, 
    autocorrelation = summarized$r_k
    )
  )

# # Display the table, except row 2
# auto_cov_cor_table %>%
#   mutate( ####################################### Suppresses row 2
#     autocovariance = ifelse(row_number() == 2,"", autocovariance),
#     autocorrelation = ifelse(row_number() == 2,"", autocorrelation)
#   ) %>% 
#   knitr::kable(format = "html", align='ccc', escape = FALSE, width = NA) %>%
#   kable_styling(full_width = FALSE, "striped") 
```

#### Lag $k = 4$


```{r}
#| echo: false
offset_value <- 4
cov_df <- get_data_for_cov_table(offset = offset_value)
cov_table <- cov_df %>%
  make_cov_table_df(offset = offset_value)#, decimals_1st_order = 1, decimals_2nd_order = 2) 
cov_table %>%
  make_blank_row_in_cov_table(1) %>% 
  make_blank_row_in_cov_table(2) %>%  
  make_blank_row_in_cov_table(3) %>%  
  make_blank_row_in_cov_table(4) %>%  
  make_blank_row_in_cov_table(5) %>%  
  make_blank_row_in_cov_table(6) %>%  
  make_blank_row_in_cov_table(7) %>%  
  make_blank_row_in_cov_table(8) %>%  
  make_blank_row_in_cov_table(9) %>%  
  make_blank_row_in_cov_table(10) %>%  
  make_blank_row_in_cov_table(11) %>% 
  display_table()

summarized <- compute_summaries(cov_df)
```