3-6-stl.Rmd

---
title: "3. Time series decomposition"
author: "3.6 STL decomposition"
date: "OTexts.org/fpp3/"
classoption: aspectratio=169
titlepage: fpp3title.png
titlecolor: fpp3red
toc: false
output:
  binb::monash:
    colortheme: monashwhite
    fig_width: 7.5
    fig_height: 3
    keep_tex: no
    includes:
      in_header: fpp3header.tex
---

```{r setup, include=FALSE}
source("setup.R")
library(gganimate)
```

## STL decomposition

\fontsize{13}{14}\sf

  *  STL: "Seasonal and Trend decomposition using Loess"
  *  Very versatile and robust.
  *  Unlike X-12-ARIMA, STL will handle any type of seasonality.
  *  Seasonal component allowed to change over time, and rate of change controlled by user.
  *  Smoothness of trend-cycle also controlled by user.
  *  Robust to outliers
  *  Not trading day or calendar adjustments.
  *  Only additive.
  *  Take logs to get multiplicative decomposition.
  *  Use Box-Cox transformations to get other decompositions.

## STL decomposition

```{r usretail, include=FALSE}
us_retail_employment <- us_employment |>
  filter(year(Month) >= 1990, Title == "Retail Trade") |>
  select(-Series_ID)
us_retail_employment
```

```{r stlwindow9, warning=FALSE, fig.width=8.5, fig.height=3.4}
us_retail_employment |>
  model(STL(Employed ~ season(window = 9), robust = TRUE)) |>
  components() |>
  autoplot() + labs(title = "STL decomposition: US retail employment")
```

## STL decomposition

```{r stlwindowanim, echo=FALSE, warning=FALSE, message=FALSE, fig.show='animate', interval=1/10,  fig.height=5.35, fig.width=8, aniopts='controls,buttonsize=0.3cm,width=11.5cm', eval=TRUE}
s_windows <- seq(5, 55, by = 2)
stl_defs <- purrr::map(s_windows, function(s_window) {
  STL(Employed ~ season(window = s_window), robust = TRUE)
})
names(stl_defs) <- sprintf("season(window=%02d)", s_windows)

us_retail_employment |>
  model(!!!stl_defs) |>
  components() |>
  as_tibble() |>
  pivot_longer(Employed:remainder,
    names_to = "component", names_ptypes = list(component = factor(levels = c("Employed", "trend", "season_year", "remainder"))),
    values_to = "Employed"
  ) |>
  ggplot(aes(x = Month, y = Employed)) +
  geom_line() +
  facet_grid(rows = vars(component), scales = "free_y") +
  labs(
    title = "STL decomposition of US retail employment",
    subtitle = "{closest_state}"
  ) +
  transition_states(.model, wrap=FALSE)
```

\vspace*{10cm}

## STL decomposition

```{r echo = TRUE, results = 'hide'}
us_retail_employment |>
  model(STL(Employed ~ season(window = 5))) |>
  components()

us_retail_employment |>
  model(STL(
    Employed ~ trend(window = 15) +
      season(window = "periodic"),
    robust = TRUE
  )) |>
  components()
```

\fontsize{12}{13}\sf

  *  `trend(window = ?)` controls wiggliness of trend component.
  *  `season(window = ?)` controls variation on seasonal component.
  *  `season(window = 'periodic')` is equivalent to an infinite window.

## STL decomposition

```{r mstl, fig.width=8.5, fig.height=3.4}
us_retail_employment |>
  model(STL(Employed)) |>
  components() |>
  autoplot()
```

\only<2>{\begin{textblock}{7}(8,0.2)\fontsize{11}{11}\sf
\begin{alertblock}{}
\begin{itemize}\tightlist
\item \texttt{STL()} chooses \texttt{season(window=13)} by default
\item Can include transformations.
\end{itemize}
\end{alertblock}
\end{textblock}}

## STL decomposition
\fontsize{13}{14.5}\sf

* Algorithm that updates trend and seasonal components iteratively.
* Starts with $\hat{T}_t=0$
* Uses a mixture of loess and moving averages to successively refine the trend and seasonal estimates.
* The trend window controls loess bandwidth applied to deasonalised values.
* The season window controls loess bandwidth applied to detrended subseries.
* Robustness weights based on remainder.
* Default season: `window = 13`
* Default trend:\mbox{}\hfill\hbox{\texttt{window = nextodd(ceiling((1.5*period)/(1-(1.5/s.window)))}}