Seasonal and Non-Seasonal Exponential Smoothing - Quick Tab
Select the Quick tab of the Seasonal and Non-Seasonal Exponential Smoothing dialog box to access the options described here. The general ideas of exponential smoothing and forecasting are explained in exponential smoothing.
| Element Name | Description |
|---|---|
| Model | The exponential smoothing models available in the Model group box are arranged in a two-way classification, as proposed by Gardner (1985): The time series can either contain no seasonal component, an additive component, or a multiplicative seasonal component. In addition, you can select no upward or downward trend in the series, a linear trend, an exponential trend, or a damped trend. The combination of these characteristics results in 12 different exponential smoothing models (which include the classical models developed by Winters, Holt, and Brown). Refer to exponential smoothing models for more detailed descriptions and examples of these models. |
| Seasonal component: lag = | Enter a value in the Seasonal component: lag = box to determine the length of one seasonal cycle. The default value is 12 (e.g., 12 months in each year). When changed, this parameter is retained (remembered) for other time series analyses involving a seasonal component (e.g., in ARIMA models or in Seasonal decomposition (Census I)). Note that the Seasonal component: lag = box is only available when a seasonal model is selected (see below). |
| Alpha | The value entered in the Alpha (α) box is the constant process smoothing parameter. The α (Alpha) parameter is necessary for all models. If α is 0 (zero), then all smoothed values are equal to the initial value (S0). If this parameter is 1, then each smoothed value (forecast) is equal to the respective previous observation. Values greater than 0 (zero) and less than 1 produce a forecast that is a weighted average of the previous observations, with the weights decreasing exponentially the "older" the previous observation. The closer the α parameter is to 0, the more slowly will the weights decrease (that is, the more slowly will the effect of prior observations disappear), the closer it is to 1, the faster will the weights decrease (and the greater will be the effect of immediately preceding or "younger" observations). In plots of the series, small values of α produce smooth fitted lines (forecasts) that only follow major trends or fluctuations spanning many observations; larger values of α (closer to 1) produce more jagged lines that are greatly influenced by even minor disturbances in the series. |
| Delta | Parameter δ (Delta) is the seasonal smoothing parameter, and only needs to be specified for seasonal models. In general the one-step-ahead forecasts are computed as follows (for no trend models, for linear and exponential trend models a trend component is added to the model; see below):
Additive model: Forecastt = St +It-p Multiplicative model: Forecastt = St *It-p In this formula, St stands for the (simple) exponentially smoothed value of the series at time t, and It-p stands for the smoothed seasonal factor at time t minus p (the length of the season). Thus, compared to simple exponential smoothing, the forecast is "enhanced" by adding or multiplying the simple smoothed value by the predicted seasonal component. This seasonal component is derived analogous to the St value from simple exponential smoothing as: Additive model: It = It-p +δ*(1-α)*et Multiplicative model: It = It-p +δ*(1-α)*et/St Put into words, the predicted seasonal component at time t is computed as the respective seasonal component in the last seasonal cycle plus a portion of the error (et; the observed minus the forecast value at time t). Parameter δ can assume values between 0 and 1. If it is zero, then the seasonal component for a particular point in time is predicted to be identical to the predicted seasonal component for the respective time during the previous seasonal cycle, which in turn is predicted to be identical to that from the previous cycle, and so on. Thus, if δ is zero, a constant unchanging seasonal component is used to generate the one-step-ahead forecasts. If the δ parameter is equal to 1, then the seasonal component is modified "maximally" at every step by the respective forecast error. In most cases, when seasonality is present in the time series, the optimum δ parameter will fall somewhere between 0 (zero) and 1(one). |
| Gamma and Phi | Gamma (γ) and Phi (φ) are the trend smoothing parameters. Parameter γ needs to be specified for linear and exponential trend models, and for damped trend models without seasonality. Parameter φ must be specified for damped trend models. Analogous to the seasonal component, when a trend component is included in the exponential smoothing process, an independent trend component is computed for each time, and modified as a function of the forecast error and the respective parameter. If the γ parameter is 0 (zero), then the trend component is constant across all values of the time series (and for all forecasts). If the parameter is 1, then the trend component is modified "maximally" at every step by the respective forecast error. Parameter values that fall in-between represent mixtures of those two extremes. Parameter φ is a trend modification parameter, and affects how strongly changes in the trend will affect estimates of the trend for subsequent forecasts, that is, how quickly the trend will be "damped" or increased. |