chapter_2_lesson_2.qmd

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

```{=html}
<script type="text/javascript">
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    var e = document.getElementById(id);
    e.style.display = (e.style.display == 'block') ? 'none' : 'block';
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```

```{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")
  
  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)
}

```


## Learning Outcomes

{{< include outcomes/_chapter_2_lesson_2_outcomes.qmd >}}




## Preparation

-   Read Sections 2.2.5


## Learning Journal Exchange (10 min)

-   Review another student's journal
-   What would you add to your learning journal after reading your partner's?
-   What would you recommend your partner add to their learning journal?
-   Sign the Learning Journal review sheet for your peer


## Hands-on Exercise -- Exploring Sample Autocorrelation (40 min)

<!-- Compute sample acvf and acf -->

### Comparison of Independent and Autocorrelated Error Terms

In the previous lesson, we computed the sample covariance and sample correlation coefficient between two independent variables. When working with time series, the observations are not independent. There is often a relationship between sequential observations. We will compute the autocovariance function and autocorrelation function for a time series. Note: the prefix "auto" comes from a Greek root meaning "self."

The figure below illustrates the difference between a series of data, where the residuals are independent compared to a series with autocorrelated data.

```{r}
#| echo: false 

# Generate data
set.seed(101) 
x <- 1:100*0.5 + arima.sim(n=100-1, list(ar=0.1, ma=0.2, order=c(1,1,1)))
e <- rnorm(100, 0, 2.5)
y <- 1:100*0.5 + e  

# Create data frame  
df <- data.frame(
  time = 1:100,
  Autocorrelated = x,
  Independent = y
)

# Plot  
ggplot(df, aes(x = time)) +
  geom_line(aes(y = Independent, color = "Independent")) + 
  geom_point(aes(y = Independent, color = "Independent")) + 
  geom_line(aes(y = Autocorrelated, color = "Autocorrelated")) + 
  geom_point(aes(y = Autocorrelated, color = "Autocorrelated")) +
  scale_color_okabeito(
    palette = "full",
    reverse = FALSE,
    order = c(2,4),
    aesthetics = "color"
  ) +
  labs(title="Comparison of Independent and Autocorrelated Error Terms", 
       x="Time",
       y="Values",
       color="Series") +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))
```

::: {.callout-tip icon=false title="Check Your Understanding"}

-   The variances of the residuals for these two series are approximately equal. What characteristics distinguish the two series in the figure above?

:::

### Autocovariance and Autocorrelation 

We will use the following data to explore the concepts of autovariance 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)


cov_df %>%
  dplyr::select(t, x) %>% 
  rename("$$ x_t $$"= x) %>% 
  display_table()
```

You can use this R command to read in the observations.

```{r}
#| echo: false

cat("x <- c(", paste(cov_df$x, collapse = ", "),")")
```

```{r}
#| echo: false

ggplot(cov_df, aes(x = t, y = x)) +
  geom_line(color = "#0072B2") +
  geom_point(color = "#0072B2") +
#   scale_color_okabeito(
#   palette = "full",
#   reverse = TRUE,
#   order = c(1,7,8,5,6,3,4,2,9),
#   aesthetics = "color"
# ) +
  ylim(0,ceiling(max(cov_df$x))) +
  labs(title = "Sample Time Series",
       x = "Time",
       y = "x") +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5)) 
```

We will use the sample mean of these data repeatedly. The value of $\bar x$ is:

$$
  \bar x 
    = \frac{1}{n} \sum\limits_{t=1}^{n} x_t 
    = \frac{1}{`r n`} \cdot `r sum(cov_df$x) %>% round(4)`
    = `r mean(cov_df$x)`
$$


We will be finding the autocovariance and correlation of a time series with itself. First, we start with a lag of 1. With a lag of 1 the corresponding values of the time series that are being compared are shifted by one time unit. Then, we will consider any integer lag: lag $k$.


### Lag $k$ Sample Autocovariance Function (acvf), $c_k$

The **lag** $k$ sample autocovariance function, acvf, denoted $c_k$, is defined as

$$
  c_k = \frac{1}{n} \sum\limits_{t=1}^{n-k}(x_t-\bar x)(x_{t+k}-\bar x)
$$

We denote the lag by the letter $k$, where $k \ge 0$. This is the number of values the data set is shifted to compute the autocovariance.

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Explain the equation for $c_k$ to your partner.
-   What is the equation for $c_0$, the value of the autocovariance function with lag $k=0$?
    -   This expression is very similar to a definition we have encountered previously. What is it?

:::


#### Lag $k=1$ Sample Autocovariance Function, $c_1$

We will now find the autocovariance between the values in a time series ($x = x_t$) and the same values, shifted by one unit of time ($y = x_{t+1}$).

```{r}
#| echo: false
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) 
summarized <- compute_summaries(cov_df)
cov_table %>%
  rename(
    "$$ x_t $$" = x_t,
    "$$ x_{t+1} $$" = `x_{t+k}`,
    "$$ x_t-\\bar x $$" = `x_t-mean(x)`,
    "$$ (x_t-\\bar x)^2 $$" = `(x_t-mean(x))^2`,
    "$$ x_{t+1}-\\bar x$$"= `x_{t+k}-mean(x)`,
    "$$ (x-\\bar x)(x_{t+1}-\\bar x) $$"= `(x-mean(x))(x_{t+k}-mean(x))`
  ) %>% 
  display_table()
```

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Working with your assigned partner, compute each of the values in row 1 by hand.  Recall that $\bar x = `r mean(cov_df$x)`$.
-   With your partner, add up the values in the last column to verify that the sum is `r sum(cov_df$xy, na.rm = TRUE) %>% round(10)`.

:::

The scatterplot below illustrates the relationship between the observed data ($x_t$) and the next observation ($x_{t+1}$).

```{r}
#| echo: false

ggplot(cov_df %>% na.omit(), aes(x = x, y = y)) +
  geom_point(size = 2, colour= "#0072B2") +
  xlab(expression(x[t])) +
  ylab(expression(x[t+1])) +
  theme_bw() +  
  ggtitle(paste0("Data pairs with a lag of k=",offset_value)) + 
  theme(plot.title = element_text(hjust = 0.5)) 
```

In this example, the second variable is $x_{t+1}$, where $t>1$. the autocovariance of $x_t$ and $x_{t+1}$ is:

$$
  c_1 
    = \frac{1}{n} \sum\limits_{t=1}^{n-1}(x_t-\bar x)(x_{t+1}-\bar x)
    = \frac{1}{`r n`} \sum\limits_{t=1}^{`r n - offset_value`}(x_t-\bar x)(x_{t+1}-\bar x)
  = \frac{1}{`r n`} \cdot `r summarized$ss_xy`
  = `r summarized$c_k`
$$

This is the (auto)covariance of $x$ with itself, but with a lag of 1 time unit. This is the value of the **lag** $k=1$ autocovariance function, acvf_1.

::: {.callout-tip icon=false title="Check Your Understanding"}

-   What does the lag 1 autocovariance measure?

:::

### Lag $k$ Sample Autocorrelation Function (acf), $r_k$

The **sample autocorrelation function, acf**, denoted $r_k$, where $k$ is the lag, is defined as

$$
  r_k 
    = \frac{c_k}{c_0} 
    = \frac{ \frac{1}{n} \sum\limits_{t=1}^{n-k}(x_t-\bar x)(x_{t+k}-\bar x) }{ \frac{1}{n} \sum\limits_{t=1}^{n}(x_t-\bar x)^2 }
    = \frac{ \sum\limits_{t=1}^{n-k}(x_t-\bar x)(x_{t+k}-\bar x) }{ \sum\limits_{t=1}^{n}(x_t-\bar x)^2 }
$$

Note that $c_0$ is the variance of $x$, but computed by dividing by $n$, instead of $n-1$.

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Interpret the components of the numerator and the denominator of the expression for $r_k$ to your partner.

:::

#### Lag $k=1$ Sample Autocorrelation Function, $r_1$

We can compute the **lag 1 autocorrelation** or the **autocorrelation of** $x$ with lag 1 as the quotient $r_1 = \frac{c_1}{c_0}$. We have already determined that 
$c_1 = `r summarized$c_k %>% round(4)`$. We now compute $c_0$:

$$ 
  c_0 = \frac{1}{n} \sum\limits_{t=1}^{n-0} (x_t-\bar x)(x_{t+0}-\bar x) 
    = \frac{1}{n} \sum\limits_{t=1}^{n} (x_t-\bar x)^2
    = \frac{1}{`r n`} \cdot `r summarized$ss_x %>% round(4)`
    = `r summarized$c_0 %>% round(4)`
$$

We use $c_0$ and $c_1$ to compute $r_1$. Here are two ways we can compute this value:

\begin{align*}
  r_1
    &= \frac{c_1}{c_0} 
    =
    \frac{ \frac{1}{n} \sum\limits_{t=1}^{`r n - offset_value`}(x_t-\bar x)(x_{t+1}-\bar x)
        }{
            \frac{1}{n} \sum\limits_{t=1}^{`r n`}(x_t-\bar x)^2
        }  
    =
    \frac{
        \frac{1}{`r summarized$n`} \cdot `r summarized$ss_xy`
      }{
        \frac{1}{`r summarized$n`} \cdot `r summarized$ss_x`
      }
    = \frac{`r summarized$c_k`}{`r summarized$c_0`}
    = `r summarized$r_k %>% round(4)`
    \\
    &=
    \frac{ \sum\limits_{t=1}^{`r n - offset_value`}(x_t-\bar x)(x_{t+1}-\bar x)
        }{
           \sum\limits_{t=1}^{`r n`}(x_t-\bar x)^2
        }  
    = \frac{`r summarized$ss_xy`}{`r summarized$ss_x`}
    = `r summarized$r_k %>% round(4)`
    
\end{align*}

-   What does the lag 1 autocorrelation, $c_1$, measure?

```{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 $k = 2$

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Working with your assigned partner, fill in the blanks in the following table. Use the results to compute $c_2$ and $r_2$.

<a href="https://github.com/TBrost/BYUI-Timeseries-Drafts/raw/master/handouts/chapter_2_2_handout.xlsx" download="chapter_2_2_handout.xlsx"> Tables-Handout-Excel </a>

:::

```{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) 
summarized <- compute_summaries(cov_df)
cov_table %>%
    rename(
    "$$ x_t $$" = x_t,
    "$$ x_{t+k} $$" = `x_{t+k}`,
    "$$ x_t-\\bar x $$" = `x_t-mean(x)`,
    "$$ (x_t-\\bar x)^2 $$" = `(x_t-mean(x))^2`,
    "$$ x_{t+k}-\\bar x$$"= `x_{t+k}-mean(x)`,
    "$$ (x-\\bar x)(x_{t+k}-\\bar x) $$"= `(x-mean(x))(x_{t+k}-mean(x))`
  ) %>%
  ############ 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()
```

The figure below illustrates the relationship between $x_t$ and $x_{t+2}$.

```{r}
#| echo: false

ggplot(cov_df %>% na.omit(), aes(x = x, y = y)) +
  geom_point(size = 2, colour= "#0072B2") +
  xlab(expression(x[t])) +
  ylab(expression(x[t+2])) +
  theme_bw() +
  ggtitle(paste0("Data pairs with a lag of k=",offset_value)) +
  theme(plot.title = element_text(hjust = 0.5))
```

```{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
    )
  )
```

#### 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) 
summarized <- compute_summaries(cov_df)
cov_table %>%
    rename(
    "$$ x_t $$" = x_t,
    "$$ x_{t+k} $$" = `x_{t+k}`,
    "$$ x_t-\\bar x $$" = `x_t-mean(x)`,
    "$$ (x_t-\\bar x)^2 $$" = `(x_t-mean(x))^2`,
    "$$ x_{t+k}-\\bar x$$"= `x_{t+k}-mean(x)`,
    "$$ (x-\\bar x)(x_{t+k}-\\bar x) $$"= `(x-mean(x))(x_{t+k}-mean(x))`
  ) %>%
  display_table()
```

The figure below illustrates the correlations between $x_t$ and $x_{t+3}$. Note that $c_3 = \dfrac{`r summarized$ss_xy`}{`r summarized$n`} = `r summarized$c_k`$ and  $r_3 = \dfrac{`r summarized$c_k`}{`r summarized$c_0`} = `r summarized$r_k`$.

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Does the value of $r_3 = `r summarized$r_k`$ seem reasonable, given the pattern in this plot?

:::

```{r}
#| echo: false

ggplot(cov_df %>% na.omit(), aes(x = x, y = y)) +
  geom_point(size = 2, colour= "#0072B2") +
  xlab(expression(x[t])) +
  ylab(expression(x[t+3])) +
  theme_bw() +  
  ggtitle(paste0("Data pairs with a lag of k=",offset_value)) + 
  theme(plot.title = element_text(hjust = 0.5)) 
```

```{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
    )
  )
```

#### Lag $k = 4$

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Compute $c_4$ and $r_4$ using R (but not automated functions), Excel, or hand calculations.

:::

```{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) 
summarized <- compute_summaries(cov_df)
cov_table %>%
    rename(
    "$$ x_t $$" = x_t,
    "$$ x_{t+k} $$" = `x_{t+k}`,
    "$$ x_t-\\bar x $$" = `x_t-mean(x)`,
    "$$ (x_t-\\bar x)^2 $$" = `(x_t-mean(x))^2`,
    "$$ x_{t+k}-\\bar x$$"= `x_{t+k}-mean(x)`,
    "$$ (x-\\bar x)(x_{t+k}-\\bar x) $$"= `(x-mean(x))(x_{t+k}-mean(x))`
  ) %>%
  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()
```


The figure below illustrates the correlations between $x_t$ and $x_{t+4}$.

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Does the value of $r_4$ you computed seem reasonable, given the pattern in this plot?

:::

```{r}
#| echo: false

ggplot(cov_df %>% na.omit(), aes(x = x, y = y)) +
  geom_point(size = 2, colour= "#0072B2") +
  xlab(expression(x[t])) +
  ylab(expression(x[t+4])) +
  theme_bw() +  
  ggtitle(paste0("Data pairs with a lag of k=",offset_value)) + 
  theme(plot.title = element_text(hjust = 0.5)) 
```

## Class Activity: Using R to compute the acvf and acf (5 min)

We will continue to use the following sample data.

```{r}
#| echo: false

# simulate correlated normal random data
x1 <- get_toy_data()
df <- data.frame(x = x1)

cat(" x <- c(",paste(x1, collapse = ", "),") \n df <- data.frame(x = x)")
```

### acvf

This code gives the values of the acvf.

```{r}
acf(df$x, plot=FALSE, type = "covariance")
```


### acf

We can obtain the acf by changing the argument for the paramter `type` to `"correlation"`.

```{r}
acf(df$x, plot=FALSE, type = "correlation")
```




## Homework Preview (5 min)

-   Review upcoming homework assignment
-   Clarify questions



## Homework

::: {.callout-note icon=false}

## Download Homework

<a href="https://byuistats.github.io/timeseries/homework/homework_2_2.qmd" download="homework_2_2.qmd"> homework_2_2.qmd </a>

:::

<a href="javascript:showhide('Solutions')"
style="font-size:.8em;">Class Activity: k=2</a>

::: {#Solutions style="display:none;"}

<a href="https://github.com/TBrost/BYUI-Timeseries-Drafts/raw/master/handouts/chapter_2_2_handout_key.xlsx" download="chapter_2_2_handout_key.xlsx"> Tables-Handout-Excel-key </a>

Solutions to Class Activity: $k=2$

```{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) 
summarized <- compute_summaries(cov_df)
cov_table %>%
  display_table()
```

\begin{align*}
  c_2
    &= \frac{1}{n} \sum\limits_{t=1}^{n-1}(x_t-\bar x)(x_{t+2}-\bar x)
    = \frac{1}{`r n`} \cdot `r summarized$ss_xy`
    = `r summarized$c_k |> round(3)` \\
  r_2 
    &= \frac{c_2}{c_0} 
    = \frac{`r summarized$c_k |> round(3)`}{`r summarized$c_0 |> round(3)`} 
    = `r summarized$r_k |> round(3)`
\end{align*}

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<a href="javascript:showhide('Solutions_k4')"
style="font-size:.8em;">Class Activity: k=4</a>

::: {#Solutions_k4 style="display:none;"}

Solutions to Class Activity: $k=4$


```{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) 
summarized <- compute_summaries(cov_df)
cov_table %>%
  display_table()
```

\begin{align*}
  c_4
    &= \frac{1}{n} \sum\limits_{t=1}^{n-1}(x_t-\bar x)(x_{t+4}-\bar x)
    = \frac{1}{`r n`} \cdot `r summarized$ss_xy`
    = `r summarized$c_k |> round(3)` \\
  r_4 
    &= \frac{c_4}{c_0} = \frac{`r summarized$c_k |> round(3)`}{`r summarized$c_0 |> round(3)`} 
    = `r summarized$r_k |> round(3)`
\end{align*}

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