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.
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