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].

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:

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

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

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

Reference

1. nag_tsa_exp_smooth (g13amc)
2. Wongoutong, Chantha. (2021). Improvement of the Holt-Winters Multiplicative Method with a New Initial Value Settings Method.