chapter_4_lesson_1_handout.qmd

---
title: "White Noise and Random Walks: Part 1 -- Handout"
subtitle: "Chapter 4: Lesson 1"
format: html
editor: source
sidebar: false
---

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

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### Coin Toss Experiment

::: {.callout-tip appearance="minimal"}


1.  Start the time series at $x_0 = 0$.
2.  Toss a coin. 
    -   If the coin shows heads, then $x_t = x_{t-1}+1$
    -   If the coin shows tails, then $x_t = x_{t-1}-1$
3.  Plot the new point on the time plot.
4.  Complete steps 2 and 3 a total of $n=60$ times.


```{r}
#| echo: false

set.seed(7)

df <- data.frame(x=0:60) |>
  mutate(w = ifelse(row_number() == 1, 0, sample(c(-1,1), size = 60, replace = TRUE))) |>
  mutate(y = cumsum(w))
# df  
# 
df2 <- expand_grid(x = 0:60, y = -20:20)

df_point <- data.frame(x = 0, y = 0)

ggplot(data=df2, aes(x=x, y=y)) +
  # geom_point(data = df2, aes(x=x, y=y), size = 0.01) +
  # geom_line(data = df, aes(x=x, y=y)) +
  # geom_point(data = df, aes(x=x, y=y), size = 0.5) +
  geom_point(data = df_point, aes(x=x, y=y), size = 1.5) +
  scale_x_continuous(limits = c(0,60),
                     breaks = seq(0, 60, by = 5),
                     minor_breaks = seq(0, 60, 1)) +
  scale_y_continuous(limits = c(-20,20),
                     breaks = seq(-20, 20, by = 5),
                     minor_breaks = seq(-20, 20, 1)) +
  labs(
      x = "Toss Number",
      y = "Value",
      title = "Cumulative Results of Coin Tosses"
  ) +
  theme_minimal() +
  theme(
    panel.grid.major = element_line(colour = "black"),
    # panel.grid.minor = element_line(colour = "black", linetype = "dotted", linewidth = 0.5)
  ) +
  theme(
    plot.title = element_text(hjust = 0.5)
  )
```


:::



<br>
<br>


#### Backward Shift 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>
Evaluate the following. 

-   $\mathbf{B} x_8$
-   $\mathbf{B}^5 x_8$
-   $(\mathbf{B}^5 - \mathbf{B} ) x_8$
-   $( \mathbf{B}^2 - 6 \mathbf{B} + 9 ) x_8$
-   $( (\mathbf{B} - 6 )\mathbf{B} + 9 ) x_8$
-   $( \mathbf{B} - 3 )^2 x_8 =  ( \mathbf{B} - 3 ) \left[ ( \mathbf{B} - 3 ) x_8 \right]$
-   $( 1 - \frac{1}{2} \mathbf{B} - \frac{1}{4} \mathbf{B}^2 - \frac{1}{8} \mathbf{B}^3 ) x_8$