Model Initialization 

The forecast model is initialized automatically. To do this the system needs a particular minimum number of historic values for the particular model.  This is presented in the following table:

Minimum number of historic values per model

Model	Number of historic values
Average	1
Moving average	Order of moving average
Weighted moving average	Order of moving average
Linear regression	2
Constant	1
Trend	3
Seasonal	1 seasonal length
Seasonal trend (mult./add.)	2 * seasonal length

 

For the exponential smoothing algorithm, starting values are required for the base value (level), the trend value (growth rate), and the seasonal indices.   The calculation of these starting values is called model initialization. This initialization is performed heuristically and depends on the specific model:

·        Constant model(simple exponential smoothing)

The starting value for the base value is calculated as the medial average of the first five values (that is, the average of the values after removing the minimum and maximum value).

·        Linear model(Holt’s exponential smoothing)

The starting values for the base value and trend value are determined by the regression line for the first eight values with the least squares method (Ordinary Least Squares [OLS]).

·        Trend-seasonal model (Holt-Winter’s exponential smoothing)

The trend component is estimated by calculating the centered moving average of order s  for the first four seasons. s denotes the seasonal length.

If, for example, s = 12 for monthly observation values, the trend component is estimated for the first four seasons, the first 48 months.

The system can also estimate a trend component if less data is available. However, at least two seasons must be provided.

The centered moving average is a time series T and represents the estimation of the trend component. Starting values for the base value and trend value are calculated using the regression line (OLS) by time series T.

The seasonal component, that is the seasonal indices, is estimated in the following ways:

¡        For the multiplicative seasonal component the average of the local indices is calculated: S(t) = Y (t) / T (t).

¡        For the additive seasonal component the difference between the local indices is calculated: S(t) = Y (t) / T (t).

Y(t) denotes the observed value and T(t) denotes the value of the trend component for that time.

·        Season model (Winter’s exponential smoothing)

This model corresponds to the trend-seasonal model but without the estimation of the trend component.