2.2.4.4.2 Algorithm for Winter's Method

NAG function nag_tsa_exp_smooth (g13amc) is used to smooth with Winter's method[1].

1 Winter's Method Model
2 Initialization Method
3 Forecast
4 Reference

Winter's Method Model

Multiplicative:

$L_t = \alpha (y_t / S_{t-p}) + (1- \alpha)(L_{t-1}+T_{t-1})$

$T_t = \gamma (L_t - L_{t-1}) + (1- \gamma)T_{t-1}$

$S_t = \delta (y_t / L_t) + (1- \delta)S_{t-p}$

$\hat{y_t} = (L_{t-1} + T_{t-1}) S_{t-p}$

$y'_t = L_{t-1} \times S_{t-p}$

Additive:

$L_t = \alpha (y_t - S_{t-p}) + (1- \alpha)(L_{t-1}+T_{t-1})$

$T_t = \gamma (L_t - L_{t-1}) + (1- \gamma)T_{t-1}$

$S_t = \delta (y_t - L_t) + (1- \delta)S_{t-p}$

$\hat{y_t} = L_{t-1} + T_{t-1} + S_{t-p}$

$y'_t = L_{t-1} + S_{t-p}$

where $L_t$ is the level(mean), $T_t$ is the trend and $S_t$ is the seasonal component at time $t$ with $p$ being the seasonal order. The parameters, $\alpha$ , $\delta$ and $\gamma$ control the weight of smoothing. $y_t$ , $\hat{y_t}$ and $y'_t$ are data value, fitted value and smoothed value at time $t$ .

Initialization Method

Intercept Difference between Groups

Linear regression is carried out with the series as the dependent variable and the sequence 1,2,...,k as the independent variable. A separate intercept is used for each of the p seasonal groupings. The shared slope gives an estimate for $T_0$ and the mean of the intercepts is used as the estimate of $L_0$ .

The seasonal parameters $S_{-j}$ , for $j = 0,1,...,p-1$ , are estimated as the $p$ intercepts $- L_0$ .

Averaging First Period[2]

The level is the average of the first period. The slope is set to be the average of the slopes for each period in the first two period.

$L_0 = (y_{1} + ... + y_{p}) / p$

$T_0 = [(y_{p+1} + y_{p+2} + ... + y_{p+p}) - (y_{1} + y_{2} + ... + y_{p})] / p^2$

The initial seasonal values $S_{-p+1},...,S_0$ are calculated with $y_i / L_0$ for multiplicative seasonality, and $y_i - L_0$ for additive seasonality.

Fitting Detrended Data

Multiplicative: Add data with $2 \times (max-min)+2 \times abs(mean)$ . Fit a regression with linear trend to the first period of data (if $p$ is less than 4, at least first 4 points are used). The initial $T_0$ is set to the regression slope. The initial level $L_0$ is set to the intercept subtracted by $2 \times (max-min)+2 \times abs(mean)$ .

Additive: Fit a regression with linear trend to the first period of data (if $p$ is less than 4, at least first 4 points are used). Then the initial level $L_0$ is set to the intercept, and the initial $T_0$ is set to the regression slope.

The initial seasonal values $S_{-p+1},...,S_0$ are computed from the detrended data. Fit a regression with linear trend to the whole time series. The detrended data is calculated by subtracting (additive model) or dividing (multiplicative model) the trend. Perform a multiple linear regresstion to the detrended data with $p$ indicator variables. The coefficients of the regression model are used as the initial values for the seasonal indices.

Forecast

Multiplicative:

$\hat{y}_{t+f} = ( L_t + f T_t ) S_{t-p}$

$var(\hat{y}_{t+f}) = var(\epsilon_t)(1 + \sum_{i=0}^{\infty}\sum_{j=0}^{p-1}(\psi_{j+ip}\frac{S_{t+f}}{S_{t+f-j}})^2)$

$\psi_i=\left\{\begin{array}{ll}0&if\;i \geqslant f\cr\alpha + \alpha \gamma&i\: mod \: p \neq 0\cr\alpha + \alpha \gamma + \delta(1-\alpha)&Otherwise\end{array}\right.$

where $var(\epsilon_t)$ is estimated as the mean deviation.

Additive:

$\hat{y}_{t+f} = L_t + f T_t + S_{t-p}$

$var(\hat{y}_{t+f}) = var(\epsilon_t)(1 + \sum_{i=1}^{f-1}\psi_i^2)$