9.2.3. Winters’ Additive Seasonal
This adds an additive seasonal component to Holt’s Linear method. For a data series 
forecasts are given by:
where:
·
is the level at
time t.
·
is the trend at
time t.
·
is the seasonal
component at time t.
·
is the level smoothing constant.
·
is the trend smoothing constant.
·
is the seasonal smoothing constant.
· s is the season period.
The initial values are based on the complete data series. Suppose the series includes complete data for L seasons. The initial trend and level values are given by:
where 
is the
average level in season i.
The initial seasonal components 
are
given by the average value of 
, the
observed value minus the expected value if no seasonal component is used. That
is:
where:
· i is the year in which t falls.
· j is season with in t falls.
Example
Open TIMESER and select Statistics 2 → Forecasting → Winters’ Additive Seasonal and select Cola Sales (C2) as [Variable].
Winters’ Additive Seasonal
|
Level Smoothing Constant = |
0.2000 |
|
Trend Smoothing Constant = |
0.1000 |
|
Seasonal Smoothing Constant = |
0.0500 |
|
Seasonal Period = |
12 |
|
Sum of Squares = |
92856.6862 |
Summary Table
|
Row |
Cola Sales |
Forecast |
Lower 95% |
Upper 95% |
Level |
Trend |
Seasonal |
|
1 |
189.0000 |
132.8698 |
* |
* |
411.1948 |
10.6955 |
-264.8537 |
|
2 |
229.0000 |
202.5518 |
65.0353 |
340.0683 |
427.1800 |
11.2245 |
-218.2806 |
|
3 |
249.0000 |
222.1597 |
121.0029 |
323.3164 |
443.7725 |
11.7613 |
-215.1712 |
|
… |
… |
… |
… |
… |
… |
… |
… |
|
34 |
904.0000 |
906.4846 |
814.8228 |
998.1464 |
744.1618 |
13.7218 |
161.7265 |
|
35 |
715.0000 |
755.1910 |
666.0461 |
844.3359 |
749.8454 |
12.9180 |
-4.3002 |
|
36 |
441.0000 |
512.3320 |
422.9208 |
601.7432 |
748.4969 |
11.4913 |
-253.2846 |
|
37 |
|
493.3129 |
401.5309 |
585.0949 |
|
|
|
|
38 |
|
552.6437 |
458.9494 |
646.3381 |
|
|
|
|
39 |
|
567.1610 |
471.4072 |
662.9148 |
|
|
|
|
40 |
|
622.7754 |
524.8243 |
720.7266 |
|
|
|
|
41 |
|
572.9176 |
472.6404 |
673.1948 |
|
|
|
|
42 |
|
808.3282 |
705.6048 |
911.0516 |
|
|
|
|
43 |
|
1092.9881 |
987.7069 |
1198.2693 |
|
|
|
|
44 |
|
1226.4534 |
1118.5106 |
1334.3961 |
|
|
|
|
45 |
|
1416.6969 |
1305.9963 |
1527.3975 |
|
|
|
|
46 |
|
1025.1368 |
911.5891 |
1138.6844 |
|
|
|
|
47 |
|
870.6014 |
754.1240 |
987.0787 |
|
|
|
|
48 |
|
633.1083 |
513.6246 |
752.5920 |
|
|
|
