chapter_4_lesson_2_handout.qmd

---
title: "White Noise and Random Walks: Part 2 -- Handout"
subtitle: "Chapter 4: Lesson 2"
format: html
editor: source
sidebar: false
---

```{r}
#| include: false
source("common_functions.R")
```

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#### Difference Operator

Let $\{x_t\}$ be a time series with the following values. 

<center>
```{r, results='asis'}
#| echo: false

set.seed(6)
n <- 8
d_operator <- data.frame(t = c(1:n), x = sample(1:15, n, replace = FALSE)) |>
  mutate(diff = t - n)

#cat( paste( paste0("$x_{t", ifelse(d_operator$t==n,"",d_operator$t-n), "} = ", d_operator$x, "$"), collapse = ",$~$ " ) ) 

cat( paste( paste0("$x_{", d_operator$t, "} = ", d_operator$x, "$"), collapse = ",$~$ " ) ) 

# Computes the value of the "power_on_d"^th difference from x_n
d_value <- function(power_on_d = 0) {
  out <- d_operator |> #### Note the use of this global variable
    filter(diff == -power_on_d) |>
    dplyr::select(x) |>
    pull()
  
  return(out)
}

ts_val <- function(t_value) {
  out <- d_operator |> #### Note the use of this global variable
    filter(t == t_value) |>
    dplyr::select(x) |>
    pull()
  
  return(out)
}
```
</center>

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-   Find the first differences, $\nabla x_t$
-   Find the second differences, $\nabla^2 x_t$.
-   Fill in the missing steps:
\begin{align*}
\nabla^2 x_8 &= (1-\mathbf{B} )^2 x_8 \\
  &= (1-\mathbf{B} ) \left[ (1-\mathbf{B} ) x_8 \right] \\
  &  ~~~~~~~~~~~~~~~~~~~~~~ ⋮ \\
  &= (x_8-x_7)-(x_7-x_6)
\end{align*}
and check that this is equal to the last term in the sequence of second differences.

:::

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```{r}
#| echo: false

d_operator |>
  dplyr::select(-diff) |>
  mutate(diff1 = x - lag(x)) |>
  mutate(diff2 = diff1 - lag(diff1)) |>
  rename(
    "$$t$$" = t,
    "$$x_t$$" = x,
    "$$\\nabla x_t$$" = diff1,
    "$$\\nabla^2 x_t$$" = diff2
  ) |>
  replace_na_with_char("") |>
  replace_cells_with_char(rows = 1:8, cols = 3:4) |>
  display_table("0.5in")
```

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#### Computing Differences

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<!-- ##### Linear Function -->

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```{r}
#| echo: false
#| fig-asp: 1
#| fig-width: 3

linear_fcn <- function(t) { return(5 + 2.5 * t) }

df_linear <- data.frame(t = 1:9) |>
  mutate(x = linear_fcn(t)) |>
  mutate(diff1 = x - lag(x)) |>
  mutate(diff2 = diff1 - lag(diff1)) |>
  mutate(diff3 = diff2 - lag(diff2)) 

df_linear |>
  ggplot(aes(x = t, y = x)) +
  geom_function(fun = linear_fcn, color = "#D55E00", xlim=c(0.8, 9.2)) +
  geom_point() +
  scale_x_continuous(breaks=seq(1, 9, by = 1)) +
  scale_y_continuous(breaks=seq(0, 30, by = 5)) +
  coord_cartesian(ylim = c(0,30)) +
  theme_bw() +
  theme(panel.grid.minor = element_blank()) +
  labs(
    title = "Linear Function",
    subtitle = "x = 2.5 t + 5"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5),
    plot.subtitle = element_text(hjust = 0.5)
  )
```

:::

::: {.column width="10%"}
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:::

::: {.column width="45%"}

```{r}
#| echo: false

df_linear |>
  rename(
    "$$t$$" = t,
    "$$x_t$$" = x,
    "$$\\nabla x_t$$" = diff1,
    " " = diff2, 
    "  " = diff3
  ) |>
  replace_na_with_char("") |>
  replace_cells_with_char(rows = 1:9, cols = 3:5, new_char = "") |>
  display_table("0.5in")
```

:::
:::
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<!-- ##### Quadratic Function -->


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::: columns
::: {.column width="45%"}

```{r}
#| echo: false
#| fig-asp: 1
#| fig-width: 3

quadratic_fcn <- function(t) { return(2 * (t - 4.75)^2 - 13.125) }

df_quadratic <- data.frame(t = 1:9) |>
  mutate(x = quadratic_fcn(t)) |>
  mutate(diff1 = x - lag(x)) |>
  mutate(diff2 = diff1 - lag(diff1)) |>
  mutate(diff3 = diff2 - lag(diff2))

df_quadratic |>
  ggplot(aes(x = t, y = x)) +
  geom_function(fun = quadratic_fcn, color = "#D55E00", xlim=c(0.8, 9.2)) +
  geom_point() +
  scale_x_continuous(breaks=seq(1, 9, by = 1)) +
  scale_y_continuous(breaks=seq(-15, 30, by = 5)) +
  coord_cartesian(ylim = c(-14,26)) +
  theme_bw() +
  theme(panel.grid.minor = element_blank()) +
  labs(
    title = "Quadratic Function",
    subtitle = "x = 2 (t - 4.75)^2 - 13.125"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5),
    plot.subtitle = element_text(hjust = 0.5)
  )
```

:::

::: {.column width="10%"}
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:::

::: {.column width="45%"}

```{r}
#| echo: false

df_quadratic |>
  rename(
    "$$t$$" = t,
    "$$x_t$$" = x,
    "$$\\nabla x_t$$" = diff1,
    "$$\\nabla^2 x_t$$" = diff2,
    "  " = diff3
  ) |>
  replace_na_with_char("") |>
  replace_cells_with_char(rows = 1:9, cols = 3:5, new_char = "") |>
  display_table("0.5in")
```

:::
:::
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<!-- ##### Cubic Function -->


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::: {.column width="45%"}

```{r}
#| echo: false
#| fig-asp: 1
#| fig-width: 3

cubic_fcn <- function(t) { return((t - 5.5) * (t - 2) * (t - 7) / 10) }

df_cubic <- data.frame(t = 1:9) |>
  mutate(x = cubic_fcn(t)) |>
  mutate(diff1 = x - lag(x)) |>
  mutate(diff2 = diff1 - lag(diff1)) |>
  mutate(diff3 = diff2 - lag(diff2)) 

df_cubic |>
  ggplot(aes(x = t, y = x)) +
  geom_function(fun = cubic_fcn, color = "#D55E00", xlim=c(0.8, 9.2)) +
  geom_point() +
  scale_x_continuous(breaks=seq(1, 9, by = 1)) +
  scale_y_continuous(breaks=seq(-5, 8, by = 1)) +
  theme_bw() +
  theme(panel.grid.minor = element_blank()) +
  labs(
    title = "Cubic Function",
    subtitle = "x = (t - 2)(t - 7)(t - 5.5)/10"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5),
    plot.subtitle = element_text(hjust = 0.5)
  )
```

:::

::: {.column width="10%"}
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:::

::: {.column width="45%"}

```{r}
#| echo: false

df_cubic |>
  rename(
    "$$t$$" = t,
    "$$x_t$$" = x,
    "$$\\nabla x_t$$" = diff1,
    "$$\\nabla^2 x_t$$" = diff2,
    "$$\\nabla^3 x_t$$" = diff3
  ) |>
  replace_na_with_char("") |>
  replace_cells_with_char(rows = 1:9, cols = 3:5, new_char = "") |>
  display_table("0.5in")
```

:::
:::
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:::
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