chapter_3_lesson_2_handout.qmd

---
title: "Exponential Smoothing (EWMA)"
subtitle: "Chapter 3: Lesson 2"
format: html
editor: source
sidebar: true
---

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

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



#### Table 1: Calculation of $a_t$ and $e_t$ for a sample time series

```{r}
#| echo: false

x1 <- get_toy_data()

alpha <- 0.2

ewma_toy_df <- data.frame(t = 1:length(x1), x_t = x1, a_t = x1) |>
  mutate(
    a_t_computation = "",
    e_t = as.numeric(NA),
    e_t_computation = ""
  )

ewma_toy_df$a_t_computation[1] = paste0("$$",ewma_toy_df$x_t[1], "$$")

for (t in 2:length(x1)) {
  ewma_toy_df$a_t[t] <- alpha * ewma_toy_df$x_t[t] + (1 - alpha) * ewma_toy_df$a_t[t-1]
  ewma_toy_df$a_t_computation[t] <- paste("$$", alpha, "\\cdot", ewma_toy_df$x_t[t], "+", "(1 - ",alpha,") \\cdot", round(ewma_toy_df$a_t[t-1],3), "=", round(ewma_toy_df$a_t[t],3), "$$")
  ewma_toy_df$e_t[t] <- ewma_toy_df$x_t[t] - ewma_toy_df$a_t[t-1]
  ewma_toy_df$e_t_computation[t] <- paste("$$", ewma_toy_df$x_t[t], "-", round(ewma_toy_df$a_t[t-1],3), "=", round(ewma_toy_df$e_t[t],3), "$$")
}

ewma_toy_df |>
  dplyr::select(t, x_t, a_t, e_t) |>
  blank_out_cells_in_df() |>
  rename(
    "$$t$$" = t,
    "$$x_t$$" = x_t,
    "$$a_t$$" = a_t,
    "$$e_t$$" = e_t
  ) |>
  display_table("0.5in") |>
  column_spec(3:4, width_min = "2.25in")
```