FORECAST_SEASONAL: Using Triple Exponential Smoothing

How to: Calculate a Triple Exponential Smoothing Column

Triple exponential smoothing produces an exponential moving average that takes into account the tendency of data to repeat itself in intervals over time. For example, sales data that is growing and in which 25% of sales always occur during December contains both trend and seasonality. Triple exponential smoothing takes both the trend and seasonality into account by using three equations with three constants.

For triple exponential smoothing you, need to know the number of data points in each time period (designated as L in the following equations). To account for the seasonality, a seasonal index is calculated. The data is divided by the prior season index and then used in calculating the smoothed average.

• The first equation accounts for the current time period, and is a weighted average of the current data value divided by the seasonal factor and the prior average adjusted for the trend for the previous period. The weight constant is k:

  SEASONAL(t) = k * (datavalue(t)/I(t-L)) + (1-k) * (SEASONAL(t-1) +  b(t-1))

• The second equation is the calculated trend value, and is a weighted average of the difference between the current and previous average and the trend for the previous time period. b(t) represents the average trend. The weight constant is g:

  b(t) = g * (SEASONAL(t)-SEASONAL(t-1)) + (1-g) * (b(t-1))

• The third equation is the calculated seasonal index, and is a weighted average of the current data value divided by the current average and the seasonal index for the previous season. I(t) represents the average seasonal coefficient. The weight constant is p:

  I(t) = p * (datavalue(t)/SEASONAL(t)) + (1 - p) * I(t-L)

These equations are solved to derive the triple smoothed average. The first smoothed average is set to the first data value. Initial values for the seasonality factors are calculated based on the maximum number of full periods of data in the data source, while the initial trend is calculated based on two periods of data. These values are calculated with the following steps:

1. The initial trend factor is calculated by the following formula:

   b(0) = (1/L) ((y(L+1)-y(1))/L + (y(L+2)-y(2))/L + ... + (y(2L) - 
   y(L))/L )

2. The calculation of the initial seasonality factor is based on the average of the data values within each period, A(j) (1<=j<=N):

   A(j) = ( y((j-1)L+1) + y((j-1)L+2) + ... + y(jL) ) / L

3. Then, the initial periodicity factor is given by the following formula, where N is the number of full periods available in the data, L is the number of points per period and n is a point within the period (1<= n <= L):

   I(n) = ( y(n)/A(1) + y(L+n)/A(2) + ... + y((N-1)L+n)/A(N) ) / N

The three constants must be chosen carefully. The best results are usually obtained by choosing the constants to minimize the mean-squared error (MSE) between the data values and the calculated averages. Varying the values of npoint1 and npoint2 affect the results, and some values may produce a better approximation. To search for a better approximation, you may want to find values that minimize the MSE.

The equation used to forecast beyond the last data point with triple exponential smoothing is:

forecast(t+m) = (SEASONAL(t) + m * b(t)) / I(t-L+MOD(m/L))

where:

m

Is the number of periods ahead for the forecast.

FORECAST_SEASONAL(display, infield,
interval, npredict, nperiod, npoint1, npoint2, npoint3)

TABLE FILE VIDEOTRK
SUM TRANSTOT
COMPUTE SEASONAL/D10.1 = FORECAST_SEASONAL(MODEL_DATA,TRANSTOT,1,3,3,3,1000,1);
BY TRANSDATE
WHERE TRANSDATE NE '19910617'
ON TABLE SET STYLE *
GRID=OFF,$
ENDSTYLE
END