A series of 12 seasonal data values are analysed using the Multiplicative and the Additive Holt-Winters method. The season size is nseason
= 4. The objective is to predict one season ahead, nforecasts
= 4 using each method. The forecasts and prediction interval lower and upper bounds are returned. The mean sum of squares of the one-step ahead forecast errors is shown to be smallest using the Multiplicative model.
import com.imsl.stat.*;
import com.imsl.math.PrintMatrix;
public class HoltWintersExponentialSmoothingEx1 {
public static void main(String args[]) {
double[] y = {23, 25, 36, 31, 26, 28, 48, 36, 31, 42, 53, 43};
double confidence = 95.0;
int nobs = y.length, nseason = 4, nforecasts = nseason;
// Compute the time series and forecasts
// using the Multiplicative model.
HoltWintersExponentialSmoothing hw
= new HoltWintersExponentialSmoothing(nseason, y);
hw.setConfidence(confidence);
hw.setNumberForecasts(nforecasts);
double[][] ser = hw.compute();
double[] ysm = hw.getSmoothedSeries();
double[][] forecasts = hw.getForecasts();
new PrintMatrix("Input time series").print(y);
new PrintMatrix("Smoothed Multiplicative series").print(ysm);
new PrintMatrix("Parameters and internal sequence").print(ser);
new PrintMatrix(" Multiplicative forecasts\nwith "
+ confidence + "% prediction interval").print(forecasts);
System.out.println("MSS - Multiplicative "
+ (hw.getSumOfSquares() / (double) (nobs - nseason)));
// Compute the time series and forecasts
// using the Additive model.
hw = new HoltWintersExponentialSmoothing(nseason, y);
hw.setConfidence(confidence);
hw.setNumberForecasts(nforecasts);
hw.setAdditive();
ser = hw.compute();
ysm = hw.getSmoothedSeries();
forecasts = hw.getForecasts();
new PrintMatrix("\n\nSmoothed Additive series").print(ysm);
new PrintMatrix("Parameters and internal sequence").print(ser);
new PrintMatrix(" Additive forecasts\nwith "
+ confidence + "% prediction interval").print(forecasts);
System.out.println("MSS - Additive "
+ (hw.getSumOfSquares() / (double) (nobs - nseason)));
}
}
Input time series
0
0 23
1 25
2 36
3 31
4 26
5 28
6 48
7 36
8 31
9 42
10 53
11 43
Smoothed Multiplicative series
0
0 23
1 25
2 36
3 31
4 24.15
5 27.652
6 41.767
7 38.034
8 30.435
9 33.715
10 54.504
11 45.243
Parameters and internal sequence
0 1 2
0 0.038 1 0.437
1 0 0 0.8
2 0 0 0.87
3 0 0 1.252
4 28.75 1.438 1.078
5 30.275 1.525 0.826
6 31.815 1.54 0.874
7 33.544 1.729 1.33
8 35.202 1.657 1.054
9 36.885 1.683 0.832
10 38.928 2.043 0.964
11 40.927 2 1.315
12 42.846 1.919 1.032
Multiplicative forecasts
with 95.0% prediction interval
0 1 2
0 37.252 27.539 46.964
1 44.99 35.24 54.739
2 63.908 53.989 73.826
3 52.136 42.166 62.105
MSS - Multiplicative 15.347754266704447
Smoothed Additive series
0
0 23
1 25
2 36
3 31
4 23
5 26.323
6 38.58
7 38.542
8 34.048
9 35.73
10 56.627
11 43.827
Parameters and internal sequence
0 1 2
0 0.268 0.643 1
1 0 0 -5.75
2 0 0 -3.75
3 0 0 7.25
4 28.75 0 2.25
5 29.555 0.518 -3.555
6 30.523 0.807 -2.523
7 33.859 2.432 14.141
8 35.609 1.994 0.391
9 36.785 1.468 -5.785
10 39.936 2.55 2.064
11 41.513 1.924 11.487
12 43.215 1.781 -0.215
Additive forecasts
with 95.0% prediction interval
0 1 2
0 39.211 27.801 50.621
1 48.841 36.371 61.311
2 60.046 45.745 74.347
3 50.125 33.244 67.007
MSS - Additive 21.181134361693125
Link to Java source.