9.2.2. Holt’s Linear
This adds a trend component to Brown’s Exponential method. For a data series xt forecasts are given by:
where:
·
is the level at
time t.
·
is the trend at
time t.
·
is the level smoothing constant.
·
is the trend smoothing constant.
The initial values m0 and b0 are calculated by a linear regression on the first half of
the series. Double exponential smoothing is a special case of Holt’s Linear method with 
and
, where 
is
the discount factor.
Example
Open TIMESER and select Statistics 2 → Forecasting → Holt’s Linear and select Cola Sales (C2) as [Variable]. On the following dialogues accept the program’s suggestions:
Holt’s Linear
|
Level Smoothing Constant = |
0.2000 |
|
Trend Smoothing Constant = |
0.1000 |
|
Sum of Squares = |
3224162.8564 |
Summary Table
|
Row |
Cola Sales |
Forecast |
Lower 95% |
Upper 95% |
Level |
Trend |
|
1 |
189.0000 |
347.1053 |
* |
* |
315.4842 |
8.0747 |
|
2 |
229.0000 |
323.5589 |
-63.7919 |
710.9098 |
304.6472 |
6.1836 |
|
3 |
249.0000 |
310.8307 |
1.3227 |
620.3387 |
298.4646 |
4.9469 |
|
4 |
289.0000 |
303.4115 |
46.5787 |
560.2443 |
300.5292 |
4.6587 |
|
5 |
260.0000 |
305.1879 |
103.7364 |
506.6394 |
296.1503 |
3.7550 |
|
6 |
431.0000 |
299.9053 |
116.6024 |
483.2082 |
326.1242 |
6.3768 |
|
7 |
660.0000 |
332.5011 |
126.2193 |
538.7829 |
398.0009 |
12.9268 |
|
8 |
777.0000 |
410.9277 |
119.4922 |
702.3632 |
484.1422 |
20.2483 |
|
9 |
915.0000 |
504.3904 |
137.2768 |
871.5041 |
586.5123 |
28.4605 |
|
10 |
613.0000 |
614.9728 |
176.8746 |
1053.0710 |
614.5782 |
28.4210 |
|
11 |
485.0000 |
642.9993 |
248.2275 |
1037.7710 |
611.3994 |
25.2610 |
|
12 |
277.0000 |
636.6604 |
242.5869 |
1030.7339 |
564.7283 |
18.0678 |
|
13 |
244.0000 |
582.7962 |
148.1328 |
1017.4595 |
515.0369 |
11.2919 |
|
14 |
296.0000 |
526.3288 |
61.2522 |
991.4054 |
480.2631 |
6.6853 |
|
15 |
319.0000 |
486.9484 |
14.7847 |
959.1120 |
453.3587 |
3.3263 |
|
16 |
370.0000 |
456.6850 |
-11.4321 |
924.8022 |
439.3480 |
1.5926 |
|
… |
… |
… |
… |
… |
… |
… |
|
30 |
660.0000 |
485.3266 |
-47.3881 |
1018.0413 |
520.2613 |
-3.0428 |
|
31 |
1004.0000 |
517.2184 |
-12.0038 |
1046.4407 |
614.5747 |
6.6928 |
|
32 |
1153.0000 |
621.2675 |
70.6462 |
1171.8888 |
727.6140 |
17.3274 |
|
33 |
1388.0000 |
744.9415 |
170.8170 |
1319.0659 |
873.5532 |
30.1886 |
|
34 |
904.0000 |
903.7418 |
299.2737 |
1508.2098 |
903.7934 |
30.1938 |
|
35 |
715.0000 |
933.9872 |
347.2790 |
1520.6954 |
890.1897 |
25.8140 |
|
36 |
441.0000 |
916.0038 |
330.7298 |
1501.2777 |
821.0030 |
16.3139 |
|
37 |
|
837.3170 |
235.9746 |
1438.6593 |
|
|
|
38 |
|
853.6309 |
239.7592 |
1467.5026 |
|
|
|
39 |
|
869.9449 |
242.5799 |
1497.3098 |
|
|
|
40 |
|
886.2588 |
244.4973 |
1528.0203 |
|
|
|
41 |
|
902.5728 |
245.5710 |
1559.5745 |
|
|
|
42 |
|
918.8867 |
245.8581 |
1591.9153 |
|
|
|
43 |
|
935.2006 |
245.4136 |
1624.9877 |
|
|
|
44 |
|
951.5146 |
244.2895 |
1658.7397 |
|
|
|
45 |
|
967.8285 |
242.5346 |
1693.1225 |
|
|
|
46 |
|
984.1425 |
240.1951 |
1728.0899 |
|
|
|
47 |
|
1000.4564 |
237.3138 |
1763.5991 |
|
|
|
48 |
|
1016.7704 |
233.9305 |
1799.6103 |
|
|
