$$\begin{align*}
\hat{y}_{t+h|t} &= (l_{t} \circ_{b} (b_{t} \circ_{d} \phi )) \circ_{s} s_{t+h-m(k+1)} \\
l_{t} &= \alpha (y_{t} \ominus_{s} s_{t-m}) + (1-\alpha)(l_{t-1} \circ_{b} (b_{t-1} \circ_{d} \phi ))\\
b_{t} &= \frac{\beta}{\alpha}(l_{t} \ominus_{b} l_{t-1}) + (1-\frac{\beta}{\alpha})b_{t-1} \\
s_{t} &= \gamma (y_{t} \ominus_{s}(l_{t-1} \circ_{b}(b_{t-1} \circ_{d} \phi))) +(1-\gamma)s_{t-m}\\
\end{align*}$$
- $\alpha, \beta, \gamma, \phi$ : smoothing parameters. In particular, $\ \phi_{h} = \phi + \phi^2 + \cdots + \phi^{h}$ means the damping parameter.
- $k$ : the quotient of $\frac{h-1}{m}$
Exponential Smoothing
N,N : Simple exponential smoothing
$$\begin{align*}
y_{t+h|t} &= l_{t} \\
l_{t} &= \alpha y_{t} + (1-\alpha) l_{t-1} \\
\end{align*}$$
N,A : Simple exponential smoothing with additive seasonal
$$\begin{align*}
y_{t+h|t} &= l_{t} + s_{t+h-m(k+1)}\\
l_{t} &= \alpha (y_{t} - s_{t-m}) +(1-\alpha)l_{t-1} \\
s_{t} &= \gamma(y_{t} - l_{t-1}) +(1-\gamma)s_{t-m}\\
\end{align*}$$
N,M : Simple exponential smoothing with multiplicative seasonal
$$\begin{align*}
y_{t+h|t} &= l_{t} s_{t+h-m(k+1)}\\
l_{t} &= \alpha (y_{t} / s_{t-m}) +(1-\alpha)l_{t-1} \\
s_{t} &= \gamma(y_{t} / l_{t-1}) +(1-\gamma)s_{t-m}\\
\end{align*}$$
A,N : Holt’s linear method
$$\begin{align*}
y_{t+h|t} &= l_{t} + hb_{t}\\
l_{t} &= \alpha y_{t} +(1-\alpha)(l_{t-1} + b_{t-1})\\
b_{t} &= \beta^{*}(l_{t} - l_{t-1}) +(1-\beta^{*})b_{t-1}\\
\end{align*}$$
A,A : Additive Holt-Winters’ method
$$\begin{align*}
y_{t+h|t} &= l_{t} + hb_{t} + s_{t+h-m(k+1)}\\
l_{t} &= \alpha (y_{t} - s_{t-m}) +(1-\alpha)(l_{t-1} + b_{t-1})\\
b_{t} &= \beta^{*}(l_{t} - l_{t-1}) +(1-\beta^{*})b_{t-1}\\
s_{t} &= \gamma(y_{t} - l_{t-1} - b_{t-1}) +(1-\gamma)s_{t-m}\\
\end{align*}$$
A,M : Multiplicative Holt-Winters’ method
$$\begin{align*}
y_{t+h|t} &= ( l_{t} + hb_{t} ) s_{t+h-m(k+1)}\\
l_{t} &= \alpha (y_{t} / s_{t-m}) +(1-\alpha)(l_{t-1} + b_{t-1})\\
b_{t} &= \beta^{*}(l_{t} - l_{t-1}) +(1-\beta^{*})b_{t-1}\\
s_{t} &= \gamma(y_{t} / (l_{t-1} - b_{t-1})) +(1-\gamma)s_{t-m}\\
\end{align*}$$
Ad,N : Additive damped trend method
$$\begin{align*}
y_{t+h|t} &= l_{t} + \phi_{h}b_{t}\\
l_{t} &= \alpha y_{t} +(1-\alpha)(l_{t-1} + \phi b_{t-1})\\
b_{t} &= \beta^{*}(l_{t} - l_{t-1}) +(1-\beta^{*})\phi b_{t-1}\\
\end{align*}$$
Ad,A : Additive damped trend method with additive seasonal
$$\begin{align*}
y_{t+h|t} &= l_{t} + \phi_{h}b_{t} + s_{t+h-m(k+1)}\\
l_{t} &= \alpha (y_{t} - s_{t-m}) +(1-\alpha)(l_{t-1} + \phi b_{t-1})\\
b_{t} &= \beta^{*}(l_{t} - l_{t-1}) +(1-\beta^{*})\phi b_{t-1}\\
s_{t} &= \gamma(y_{t} - l_{t-1} - \phi b_{t-1}) +(1-\gamma)s_{t-m}\\
\end{align*}$$
Ad,M : Holt-Winters’ damped method
$$\begin{align*}
y_{t+h|t} &= ( l_{t} + \phi_{h}b_{t} ) s_{t+h-m(k+1)}\\
l_{t} &= \alpha (y_{t} / s_{t-m}) +(1-\alpha)(l_{t-1} + \phi b_{t-1})\\
b_{t} &= \beta^{*}(l_{t} - l_{t-1}) +(1-\beta^{*})\phi b_{t-1}\\
s_{t} &= \gamma(y_{t} / (l_{t-1} - \phi b_{t-1})) +(1-\gamma)s_{t-m}\\
\end{align*}$$
ETS Models
Additive models
$$e_{t} = y_{t} - \hat{y}_{t|t-1}$$
ETS(A,N,N)
$$\begin{align*}
y_{t} &= l_{t-1} + \epsilon_{t} \\
l_{t} &= l_{t-1} + \alpha \epsilon_{t} \\
\end{align*}$$
ETS(A,N,A)
$$\begin{align*}
y_{t} &= l_{t-1} + s_{t-m} + \epsilon_{t} \\
l_{t} &= l_{t-1} + \alpha \epsilon_{t} \\
s_{t} &= s_{t-m} + \gamma \epsilon_{t} \\
\end{align*}$$
ETS(A,N,M)
$$\begin{align*}
y_{t} &= l_{t-1} s_{t-m} + \epsilon_{t} \\
l_{t} &= l_{t-1} + \alpha \epsilon_{t}/s_{t-m} \\
s_{t} &= s_{t-m} + \gamma \epsilon_{t}/l_{t-1} \\
\end{align*}$$
ETS(A,A,N)
$$\begin{align*}
y_{t} &= l_{t-1} + b_{t-1} + \epsilon_{t} \\
l_{t} &= l_{t-1} + b_{t-1} + \alpha \epsilon_{t} \\
b_{t} &= b_{t-1} + \beta \epsilon_{t} \\
\end{align*}$$
ETS(A,A,A)
$$\begin{align*}
y_{t} &= l_{t-1} + b_{t-1} + s_{t-m} + \epsilon_{t} \\
l_{t} &= l_{t-1} + b_{t-1} + \alpha \epsilon_{t} \\
b_{t} &= b_{t-1} + \beta \epsilon_{t} \\
s_{t} &= s_{t-m} + \gamma \epsilon_{t} \\
\end{align*}$$
ETS(A,A,M)
$$\begin{align*}
y_{t} &= (l_{t-1} + b_{t-1}) s_{t-m} + \epsilon_{t} \\
l_{t} &= l_{t-1} + b_{t-1} + \alpha \epsilon_{t}/s_{t-m} \\
b_{t} &= b_{t-1} + \beta \epsilon_{t}/s_{t-m} \\
s_{t} &= s_{t-m} + \gamma \epsilon_{t}/(l_{t-1} + b_{t-1}) \\
\end{align*}$$
ETS(A,Ad,N)
$$\begin{align*}
y_{t} &= l_{t-1} + \phi b_{t-1} + \epsilon_{t} \\
l_{t} &= l_{t-1} + \phi b_{t-1} + \alpha \epsilon_{t} \\
b_{t} &= \phi b_{t-1} + \beta \epsilon_{t} \\
\end{align*}$$
ETS(A,Ad,A)
$$\begin{align*}
y_{t} &= l_{t-1} + \phi b_{t-1} + s_{t-m} + \epsilon_{t} \\
l_{t} &= l_{t-1} + \phi b_{t-1} + \alpha \epsilon_{t} \\
b_{t} &= \phi b_{t-1} + \beta \epsilon_{t} \\
s_{t} &= s_{t-m} + \gamma \epsilon_{t} \\
\end{align*}$$
ETS(A,Ad,M)
$$\begin{align*}
y_{t} &= (l_{t-1} + \phi b_{t-1}) s_{t-m} + \epsilon_{t} \\
l_{t} &= l_{t-1} + \phi b_{t-1} + \alpha \epsilon_{t}/s_{t-m} \\
b_{t} &= \phi b_{t-1} + \beta \epsilon_{t}/s_{t-m} \\
s_{t} &= s_{t-m} + \gamma \epsilon_{t}/(l_{t-1} + \phi b_{t-1}) \\
\end{align*}$$
Multiplicative models
$$e_{t} = \frac{y_{t}}{\hat{y}_{t|t-1}} - 1$$
ETS(M,N,N)
$$\begin{align*}
y_{t} &= l_{t-1}(1 + \epsilon_{t}) \\
l_{t} &= l_{t-1}(1 + \alpha \epsilon_{t}) \\
\end{align*}$$
ETS(M,N,A)
$$\begin{align*}
y_{t} &= (l_{t-1} + s_{t-m})(1 + \epsilon_{t}) \\
l_{t} &= l_{t-1} + \alpha (l_{t-1} + s_{t-m}) \epsilon_{t} \\
s_{t} &= s_{t-m} + \gamma (l_{t-1} + s_{t-m}) \epsilon_{t} \\
\end{align*}$$
ETS(M,N,M)
$$\begin{align*}
y_{t} &= l_{t-1} s_{t-m}(1 + \epsilon_{t}) \\
l_{t} &= l_{t-1} (1 + \alpha \epsilon_{t}) \\
s_{t} &= s_{t-m} (1 + \gamma \epsilon_{t}) \\
\end{align*}$$
ETS(M,A,N)
$$\begin{align*}
y_{t} &= (l_{t-1} + b_{t-1})(1 + \epsilon_{t}) \\
l_{t} &= (l_{t-1} + b_{t-1})(1 + \alpha \epsilon_{t}) \\
b_{t} &= b_{t-1} + \beta (l_{t-1} + b_{t-1}) \epsilon_{t} \\
\end{align*}$$
ETS(M,A,A)
$$\begin{align*}
y_{t} &= (l_{t-1} + b_{t-1} + s_{t-m})(1 + \epsilon_{t}) \\
l_{t} &= l_{t-1} + b_{t-1} + \alpha (l_{t-1} + b_{t-1} + s_{t-m}) \epsilon_{t} \\
b_{t} &= b_{t-1} + \beta (l_{t-1} + b_{t-1} + s_{t-m}) \epsilon_{t} \\
s_{t} &= s_{t-m} + \gamma (l_{t-1} + b_{t-1} + s_{t-m}) \epsilon_{t} \\
\end{align*}$$
ETS(M,A,M)
$$\begin{align*}
y_{t} &= (l_{t-1} + b_{t-1}) s_{t-m} (1 + \epsilon_{t}) \\
l_{t} &= (l_{t-1} + b_{t-1})(1 + \alpha \epsilon_{t}) \\
b_{t} &= b_{t-1} + \beta (l_{t-1} + b_{t-1}) \epsilon_{t} \\
s_{t} &= s_{t-m}( 1 + \gamma \epsilon_{t}) \\
\end{align*}$$
ETS(M,Ad,N)
$$\begin{align*}
y_{t} &= (l_{t-1} + \phi b_{t-1})(1 + \epsilon_{t}) \\
l_{t} &= (l_{t-1} + \phi b_{t-1})(1 + \alpha \epsilon_{t}) \\
b_{t} &= \phi b_{t-1} + \beta (l_{t-1} + \phi b_{t-1}) \epsilon_{t} \\
\end{align*}$$
ETS(M,Ad,A)
$$\begin{align*}
y_{t} &= (l_{t-1} + \phi b_{t-1} + s_{t-m})(1 + \epsilon_{t}) \\
l_{t} &= l_{t-1} + \phi b_{t-1} + \alpha (l_{t-1} + \phi b_{t-1} + s_{t-m}) \epsilon_{t} \\
b_{t} &= \phi b_{t-1} + \beta (l_{t-1} + \phi b_{t-1} + s_{t-m}) \epsilon_{t} \\
s_{t} &= s_{t-m} + \gamma (l_{t-1} + \phi b_{t-1} + s_{t-m}) \epsilon_{t} \\
\end{align*}$$
ETS(M,Ad,M)
$$\begin{align*}
y_{t} &= (l_{t-1} + \phi b_{t-1}) s_{t-m} (1 + \epsilon_{t}) \\
l_{t} &= (l_{t-1} + \phi b_{t-1})(1 + \alpha \epsilon_{t}) \\
b_{t} &= \phi b_{t-1} + \beta (l_{t-1} + \phi b_{t-1}) \epsilon_{t} \\
s_{t} &= s_{t-m}( 1 + \gamma \epsilon_{t}) \\
\end{align*}$$