trend_derivations.qmd

---
title: Deriving models for ETS trend models
author: Rob J Hyndman
branding: false
fontfamily: mathpazo
format: memo-pdf
---

## ETS(A,A,N) {-}

Component form:\vspace*{-0.8cm}
\begin{align*}
\hat{y}_{t+h|t} &= \ell_{t} + hb_{t} \\
\hat{y}_{t|t-1} &= \ell_{t-1} + b_{t-1} \\
\ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\
b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*) b_{t-1}.
\end{align*}
Model form:\vspace*{-0.8cm}
\begin{align*}
y_t &= \hat{y}_{t|t-1} + \varepsilon_t \\
   &= \ell_{t-1} + b_{t-1} + \varepsilon_t \\
\ell_t &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\
  &= \alpha (\ell_{t-1} + b_{t-1} + \varepsilon_t ) + (1 - \alpha)(\ell_{t-1} +  b_{t-1})\\
  &= \ell_{t-1} + b_{t-1} + \alpha \varepsilon_t \\
b_t &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*) b_{t-1}\\
   &= \beta^*(b_{t-1} + \alpha \varepsilon_t ) + (1 -\beta^*) b_{t-1}\\
   &=  b_{t-1} + \alpha\beta^* \varepsilon_t \\
   &= b_{t-1} + \beta \varepsilon_t \qquad\text{where $\beta = \alpha\beta^*$.}
\end{align*}

## ETS(A,Ad,N) {-}

Component form:\vspace*{-0.8cm}
\begin{align*}
\hat{y}_{t+h|t} &= \ell_{t} + (\phi+\phi^2 + \dots + \phi^{h})b_{t} \\
\hat{y}_{t|t-1} &= \ell_{t-1} + \phi b_{t-1} \\
\ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + \phi b_{t-1})\\
b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*)\phi b_{t-1}.
\end{align*}
Model form:\vspace*{-0.8cm}
\begin{align*}
y_t &= \hat{y}_{t|t-1} + \varepsilon_t \\
   &= \ell_{t-1} + \phi b_{t-1} + \varepsilon_t \\
\ell_t &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + \phi b_{t-1})\\
  &= \alpha (\ell_{t-1} + \phi b_{t-1} + \varepsilon_t ) + (1 - \alpha)(\ell_{t-1} + \phi b_{t-1})\\
  &= \ell_{t-1} + \phi b_{t-1} + \alpha \varepsilon_t \\
b_t &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*)\phi b_{t-1}\\
   &= \beta^*(\phi b_{t-1} + \alpha \varepsilon_t ) + (1 -\beta^*)\phi b_{t-1}\\
   &= \phi b_{t-1} + \alpha\beta^* \varepsilon_t \\
   &= \phi b_{t-1} + \beta \varepsilon_t\qquad\text{where $\beta = \alpha\beta^*$.}
\end{align*}