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8-ets_b.qmd
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---
title: "ETC3550/ETC5550 Applied forecasting"
author: "Ch8. Exponential smoothing"
institute: "OTexts.org/fpp3/"
pdf-engine: pdflatex
fig-width: 7.5
fig-height: 3
format:
beamer:
theme: monash
aspectratio: 169
fontsize: 14pt
section-titles: false
knitr:
opts_chunk:
dev: "cairo_pdf"
include-in-header: header.tex
execute:
echo: false
message: false
warning: false
---
```{r setup, include=FALSE}
source("setup.R")
library(patchwork)
library(gganimate)
library(purrr)
library(rlang)
library(magick)
```
## ETS models
\begin{block}{}
\hspace*{-0.25cm}\begin{tabular}{l@{}p{2.3cm}@{}c@{}l}
\alert{General n\rlap{otation}}
& & ~E T S~ & ~:\hspace*{0.3cm}\textbf{E}xponen\textbf{T}ial \textbf{S}moothing \\ [-0.2cm]
& \hfill{$\nearrow$\hspace*{-0.1cm}} & {$\uparrow$} & {\hspace*{-0.2cm}$\nwarrow$} \\
& \hfill{\textbf{E}rror\hspace*{0.2cm}} & {\textbf{T}rend} & {\hspace*{0.2cm}\textbf{S}eason}
\end{tabular}
\end{block}
\alert{\textbf{E}rror:} Additive (`"A"`) or multiplicative (`"M"`)
\pause
\alert{\textbf{T}rend:} None (`"N"`), additive (`"A"`), multiplicative (`"M"`), or damped (`"Ad"` or `"Md"`).
\pause
\alert{\textbf{S}easonality:} None (`"N"`), additive (`"A"`) or multiplicative (`"M"`)
## ETS(A,N,N): SES with additive errors
\fontsize{14}{16}\sf\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, $\varepsilon_t$.
* Measurement equation: relationship between observations and states.
* State equation(s): evolution of the state(s) through time.
## ETS(A,A,N)
Holt's methods method with additive errors.\phantom{p}
\begin{block}{}\vspace*{-0.8cm}
\begin{align*}
\text{Forecast equation} && \hat{y}_{t+h|t} &= \ell_{t} + hb_{t}\\
\text{Observation equation}&& y_t&=\ell_{t-1}+b_{t-1} + \varepsilon_t\\
\text{State equations}&& \ell_t&=\ell_{t-1}+b_{t-1}+\alpha \varepsilon_t\\
&& b_t&=b_{t-1}+\beta \varepsilon_t
\end{align*}
\end{block}
* Forecast errors: $\varepsilon_{t} = y_t - \hat{y}_{t|t-1}$
\vspace*{10cm}
## ETS(A,A,A)
Holt-Winters additive method with additive errors.\phantom{p}
\begin{block}{}\vspace*{-0.8cm}
\begin{align*}
\text{Forecast equation} && \hat{y}_{t+h|t} &= \ell_{t} + hb_{t} + s_{t+h-m(k+1)}\\
\text{Observation equation}&& y_t&=\ell_{t-1}+b_{t-1}+s_{t-m} + \varepsilon_t\\
\text{State equations}&& \ell_t&=\ell_{t-1}+b_{t-1}+\alpha \varepsilon_t\\
&& b_t&=b_{t-1}+\beta \varepsilon_t \\
&&s_t &= s_{t-m} + \gamma\varepsilon_t
\end{align*}
\end{block}
* Forecast errors: $\varepsilon_{t} = y_t - \hat{y}_{t|t-1}$
* $k$ is integer part of $(h-1)/m$.
## ETS(M,A,M)
Holt-Winters multiplicative method with multiplicative errors.
\begin{block}{}\vspace*{-0.8cm}
\begin{align*}
\text{Forecast equation} && \hat{y}_{t+h|t} &= (\ell_{t} + hb_{t}) s_{t+h-m(k+1)}\\
\text{Observation equation}&& y_t&= (\ell_{t-1}+b_{t-1})s_{t-m}(1 + \varepsilon_t)\\
\text{State equations}&& \ell_t&=(\ell_{t-1}+b_{t-1})(1+\alpha \varepsilon_t)\\
&& b_t&=b_{t-1} +\beta(\ell_{t-1}+b_{t-1}) \varepsilon_t \\
&&s_t &= s_{t-m}(1 + \gamma\varepsilon_t)
\end{align*}
\end{block}
* Forecast errors: $\varepsilon_{t} = (y_t - \hat{y}_{t|t-1})/\hat{y}_{t|t-1}$
* $k$ is integer part of $(h-1)/m$.
## ETS model specification
```{r ann-spec, echo = TRUE, eval=FALSE}
ETS(y ~ error("A") + trend("N") + season("N"))
```
By default, optimal values for $\alpha$, $\beta$, $\gamma$, and the states at time 0 are used.
The values for $\alpha$, $\beta$ and $\gamma$ can be specified:
```{r alpha-spec, echo = TRUE, eval = FALSE}
trend("A", alpha = 0.5, beta = 0.2)
trend("A", alpha_range = c(0.2, 0.8), beta_range = c(0.1, 0.4))
season("M", gamma = 0.04)
season("M", gamma_range = c(0, 0.3))
```
## Exponential smoothing methods
\fontsize{12}{13}\sf
\begin{block}{}
\begin{tabular}{ll|ccc}
& &\multicolumn{3}{c}{\bf Seasonal Component} \\
\multicolumn{2}{c|}{\bf Trend}& N & A & M\\
\multicolumn{2}{c|}{\bf Component} & (None) & (Additive) & (Multiplicative)\\
\cline{3-5} &&&&\\[-0.4cm]
N & (None) & (N,N) & (N,A) & (N,M)\\
&&&&\\[-0.4cm]
A & (Additive) & (A,N) & (A,A) & (A,M)\\
&&&&\\[-0.4cm]
A\damped & (Additive damped) & (A\damped,N) & (A\damped,A) & (A\damped,M)
\end{tabular}
\end{block}\fontsize{12}{13}\sf
\begin{tabular}{lp{9.7cm}}
\alert{(N,N)}: & Simple exponential smoothing\\
\alert{(A,N)}: & Holt's linear method\\
\alert{(A\damped,N)}: & Additive damped trend method\\
\alert{(A,A)}:~~ & Additive Holt-Winters' method\\
\alert{(A,M)}: & Multiplicative Holt-Winters' method\\
\alert{(A\damped,M)}: & Damped multiplicative Holt-Winters' method
\end{tabular}
\only<2>{\begin{textblock}{5}(10,6)
\begin{alertblock}{}\fontsize{12}{14}\sf
There are also multiplicative trend methods (not recommended).
\end{alertblock}
\end{textblock}}
## ETS models
\fontsize{11}{12}\sf
\begin{block}{}
\begin{tabular}{ll|ccc}
\multicolumn{2}{l}{\alert{\bf Additive Error}} & \multicolumn{3}{c}{\bf Seasonal Component} \\
\multicolumn{2}{c|}{\bf Trend} & N & A & M \\
\multicolumn{2}{c|}{\bf Component} & ~(None)~ & (Additive) & (Multiplicative) \\ \cline{3-5}
& & & & \\[-0.4cm]
N & (None) & A,N,N & A,N,A & A,N,M \\
& & & & \\[-0.4cm]
A & (Additive) & A,A,N & A,A,A & A,A,M \\
& & & & \\[-0.4cm]
A\damped & (Additive damped) & A,A\damped,N & A,A\damped,A & A,A\damped,M
\end{tabular}
\end{block}
\begin{block}{}
\begin{tabular}{ll|ccc}
\multicolumn{2}{l}{\alert{\bf Multiplicative Error}} & \multicolumn{3}{c}{\bf Seasonal Component} \\
\multicolumn{2}{c|}{\bf Trend} & N & A & M \\
\multicolumn{2}{c|}{\bf Component} & ~(None)~ & (Additive) & (Multiplicative) \\ \cline{3-5}
& & & & \\[-0.4cm]
N & (None) & M,N,N & M,N,A & M,N,M \\
& & & & \\[-0.4cm]
A & (Additive) & M,A,N & M,A,A & M,A,M \\
& & & & \\[-0.4cm]
A\damped & (Additive damped) & M,A\damped,N & M,A\damped,A & M,A\damped,M
\end{tabular}
\end{block}
## ETS models
\fontsize{11}{12}\sf
\begin{block}{}
\begin{tabular}{ll|ccc}
\multicolumn{2}{l}{\alert{\bf Additive Error}} & \multicolumn{3}{c}{\bf Seasonal Component} \\
\multicolumn{2}{c|}{\bf Trend} & N & A & M \\
\multicolumn{2}{c|}{\bf Component} & ~(None)~ & (Additive) & (Multiplicative) \\ \cline{3-5}
& & & & \\[-0.4cm]
N & (None) & A,N,N & A,N,A & \str{A,N,M} \\
& & & & \\[-0.4cm]
A & (Additive) & A,A,N & A,A,A & \str{A,A,M} \\
& & & & \\[-0.4cm]
A\damped & (Additive damped) & A,A\damped,N & A,A\damped,A & \str{A,A\damped,M}
\end{tabular}
\end{block}
\begin{block}{}
\begin{tabular}{ll|ccc}
\multicolumn{2}{l}{\alert{\bf Multiplicative Error}} & \multicolumn{3}{c}{\bf Seasonal Component} \\
\multicolumn{2}{c|}{\bf Trend} & N & A & M \\
\multicolumn{2}{c|}{\bf Component} & ~(None)~ & (Additive) & (Multiplicative) \\ \cline{3-5}
& & & & \\[-0.4cm]
N & (None) & M,N,N & M,N,A & M,N,M \\
& & & & \\[-0.4cm]
A & (Additive) & M,A,N & M,A,A & M,A,M \\
& & & & \\[-0.4cm]
A\damped & (Additive damped) & M,A\damped,N & M,A\damped,A & M,A\damped,M
\end{tabular}
\end{block}
## AIC and cross-validation
\Large
\begin{alertblock}{}
Minimizing the AIC assuming Gaussian residuals is asymptotically equivalent to minimizing one-step time series cross validation MSE.
\end{alertblock}
## Automatic forecasting
**From Hyndman et al.\ (IJF, 2002):**
* Apply each model that is appropriate to the data.
Optimize parameters and initial values using MLE (or some other
criterion).
* Select best method using AICc:
* Produce forecasts using best method.
* Obtain forecast intervals using underlying state space model.
Method performed very well in M3 competition.
## Residuals
\vspace*{0.2cm}
### Response residuals
\centerline{$\hat{e}_t = y_t - \hat{y}_{t|t-1}$}
### Innovation residuals
Additive error model:
\centerline{$\hat\varepsilon_t = y_t - \hat{y}_{t|t-1}$}
Multiplicative error model:
\centerline{$\hat\varepsilon_t = \frac{y_t - \hat{y}_{t|t-1}}{\hat{y}_{t|t-1}}$}