8-ets_a.qmd

---
title: "ETC3550/ETC5550 Applied forecasting"
author: "Ch8. Simple Exponential smoothing"
institute: "OTexts.org/fpp3/"
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fig-height: 3
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---

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

## Historical perspective

 * Developed in the 1950s and 1960s as methods (algorithms) to produce point forecasts.
 * Combine a "level", "trend" (slope) and "seasonal" component to describe a time series.
 * The rate of change of the components are controlled by "smoothing parameters": $\alpha$, $\beta$ and $\gamma$ respectively.
  * Need to choose best values for the smoothing parameters (and initial states).
  * Equivalent ETS state space models developed in the 1990s and 2000s.

## Big idea: control the rate of change
\fontsize{13}{14}\sf

$\alpha$ controls the flexibility of the **level**

* If $\alpha = 0$, the level never updates (mean)
* If $\alpha = 1$, the level updates completely (naive)

$\beta$ controls the flexibility of the **trend**

* If $\beta = 0$, the trend is linear
* If $\beta = 1$, the trend changes suddenly every observation

$\gamma$ controls the flexibility of the **seasonality**

* If $\gamma = 0$, the seasonality is fixed (seasonal means)
* If $\gamma = 1$, the seasonality updates completely (seasonal naive)

## Models and methods

### Methods

  * Algorithms that return point forecasts.

### Models

  * Generate same point forecasts but can also generate forecast distributions.
  * A stochastic (or random) data generating process that can generate an entire forecast distribution.
  * Allow for "proper" model selection.

## Simple Exponential Smoothing

\vspace*{0.2cm}
\begin{block}{Iterative form}
\centerline{$\pred{y}{t+1}{t} = \alpha y_t + (1-\alpha) \pred{y}{t}{t-1}$}
\end{block}\pause

\begin{block}{Weighted average form}
\centerline{$\displaystyle\pred{y}{T+1}{T}=\sum_{j=0}^{T-1} \alpha(1-\alpha)^j y_{T-j}+(1-\alpha)^T \ell_{0}$}
\end{block}\pause

\begin{block}{Component form}\vspace*{-0.8cm}
\begin{align*}
\text{Forecast equation}&&\pred{y}{t+h}{t} &= \ell_{t}\\
\text{Smoothing equation}&&\ell_{t} &= \alpha y_{t} + (1 - \alpha)\ell_{t-1}
\end{align*}
\end{block}

## Simple Exponential Smoothing
\fontsize{14}{14}\sf

\vspace*{0.2cm}
\begin{block}{Component form}\vspace*{-0.8cm}
\begin{align*}
\text{Forecast equation}&&\pred{y}{t+h}{t} &= \ell_{t}\\
\text{Smoothing equation}&&\ell_{t} &= \alpha y_{t} + (1 - \alpha)\ell_{t-1}
\end{align*}
\end{block}\pause\vspace*{-0.2cm}

Forecast error: $e_t = y_t - \pred{y}{t}{t-1} = y_t - \ell_{t-1}$.\pause
\begin{block}{Error correction form}\vspace*{-0.8cm}
\begin{align*}
y_t &= \ell_{t-1} + e_t\\
\ell_{t}
         &= \ell_{t-1}+\alpha( y_{t}-\ell_{t-1})\\
         &= \ell_{t-1}+\alpha e_{t}
\end{align*}
\end{block}\pause\vspace*{-0.2cm}

Specify probability distribution: $e_t = \varepsilon_t\sim\text{NID}(0,\sigma^2)$.

## ETS(A,N,N): SES with additive errors

\vspace*{0.2cm}
\begin{block}{ETS(A,N,N) model}\vspace*{-0.8cm}
\begin{align*}
\text{Observation equation}&& y_t &= \ell_{t-1} + \varepsilon_t\\
\text{State equation}&& \ell_t&=\ell_{t-1}+\alpha \varepsilon_t
\end{align*}
\end{block}
where $\varepsilon_t\sim\text{NID}(0,\sigma^2)$.

  * "innovations" or "single source of error" because equations have the same error process, $\varepsilon_t$.
  * Observation equation: relationship between observations and states.
  * State equation(s): evolution of the state(s) through time.

## ETS(M,N,N): SES with multiplicative errors.

  * Specify relative errors  $\varepsilon_t=\frac{y_t-\pred{y}{t}{t-1}}{\pred{y}{t}{t-1}}\sim \text{NID}(0,\sigma^2)$
  * Substituting $\pred{y}{t}{t-1}=\ell_{t-1}$ gives:
    * $y_t = \ell_{t-1}+\ell_{t-1}\varepsilon_t$
    * $e_t = y_t - \pred{y}{t}{t-1} = \ell_{t-1}\varepsilon_t$

\pause
\begin{block}{ETS(M,N,N) model}\vspace*{-0.8cm}
\begin{align*}
\text{Observation equation}&& y_t &= \ell_{t-1}(1 + \varepsilon_t)\\
\text{State equation}&& \ell_t&=\ell_{t-1}(1+\alpha \varepsilon_t)
\end{align*}
\end{block}
\pause\vspace*{-0.4cm}

  * Models with additive and multiplicative errors with the same parameters generate the same point forecasts but different prediction intervals.