Given

• Smoothing parameters: , , ,
• Initial states: , , , seasonal period
• Data:

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

Step 1 — formula

Observation:

Conditional mean (one-step-ahead forecast ):

Innovation:

State updates:

Forecast steps ahead from time (using current-period states):

where is the forecast horizon (how many steps ahead); picks the right seasonal slot for the period steps ahead (cycles through ).

Step 2 — apply at

Step 3 — iterate

Each column header is the equation that produced its values. Values rounded to 4 decimal places; arithmetic performed at full precision.

1	12
2	10
3	8
4	11
5	14
6	12
7	9
8	13
9	16
10	14
11	11
12	15
13	18
14	16
15	13
16	17