A-Ad-M

228. A-Ad-M

Damped Holt-Winters, multiplicative seasonality

ETS (𝐴, Ad, 𝑀)

𝑥_{𝑡} = (𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1}) 𝑠_{𝑡 - 𝑚} + 𝜀_{𝑡}

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}

𝑏_{𝑡} = 𝜑 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡} / (𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1})

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = (𝑙_{𝑡} + (𝜑 + 𝜑^{2} + \dots + 𝜑^{ℎ}) 𝑏_{𝑡}) 𝑠_{𝑡 + ℎ - 𝑚_{ℎ}^{+}}

Example:

ETS (𝐴, Ad, 𝑀)

Given

Smoothing parameters: $𝛼 = 0.5$ , $𝛽 = 0.4$ , $𝜑 = 0.8$ , $𝛾 = 0.2$
Initial states: $𝑙_{0} = 12$ , $𝑏_{0} = 0.5$ , $(𝑠_{- 3}, 𝑠_{- 2}, 𝑠_{- 1}, 𝑠_{0}) = (1.2, 1, 0.8, 1)$ , seasonal period $𝑚 = 4$
Data:

$𝑡$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$x_{𝑡}$	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

Step 1 — formula

Observation:

𝑥_{𝑡} = (𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1}) 𝑠_{𝑡 - 𝑚} + 𝜀_{𝑡}

Conditional mean (one-step-ahead forecast ${\hat{𝑥}}_{𝑡 | 𝑡 - 1} = 𝜇_{𝑡}$ ):

𝜇_{𝑡} = (𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1}) 𝑠_{𝑡 - 𝑚}

Innovation:

𝜀_{𝑡} = 𝑥_{𝑡} - 𝜇_{𝑡}

State updates:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}

𝑏_{𝑡} = 𝜑 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡} / (𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1})

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

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = (𝑙_{𝑡} + (𝜑 + 𝜑^{2} + \dots + 𝜑^{ℎ}) 𝑏_{𝑡}) 𝑠_{𝑡 + ℎ - 𝑚_{ℎ}^{+}}

where $ℎ \in {1, 2, 3, \dots}$ is the forecast horizon (how many steps ahead); $𝑚_{ℎ}^{+} = ((ℎ - 1) mod 𝑚) + 1$ picks the right seasonal slot for the period $ℎ$ steps ahead (cycles through $1, 2, \dots, 𝑚$ ).

Step 2 — apply at $𝑡 = 1$

𝜇_{1} = (12 + 0.8 \cdot 0.5) \cdot 1.2 = 14.88

𝜀_{1} = 𝑥_{1} - 𝜇_{1} = 12 - 14.88 = - 2.88

𝑙_{1} = 12 + 0.8 \cdot 0.5 + 0.5 \cdot (- 2.88) / 1.2 = 11.2

𝑏_{1} = 0.8 \cdot 0.5 + 0.4 \cdot (- 2.88) / 1.2 = - 0.56

𝑠_{1} = 1.2 + 0.2 \cdot (- 2.88) / (12 + 0.8 \cdot 0.5) = 1.1535

Step 3 — iterate

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

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = (𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1}) 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}$	$𝑏_{𝑡} = 𝜑 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡} / (𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1})$
1	12	$14.88$	$- 2.88$	$11.2$	$- 0.56$	$1.1535$
2	10	$10.752$	$- 0.752$	$10.376$	$- 0.7488$	$0.986$
3	8	$7.8216$	$0.1784$	$9.8885$	$- 0.5098$	$0.8037$
4	11	$9.4806$	$1.5194$	$10.2403$	$0.1999$	$1.0321$
5	14	$11.9972$	$2.0028$	$11.2683$	$0.8544$	$1.1921$
6	12	$11.7847$	$0.2153$	$12.0611$	$0.7709$	$0.9896$
7	9	$10.1885$	$- 1.1885$	$11.9383$	$0.0252$	$0.7849$
8	13	$12.3418$	$0.6582$	$12.2774$	$0.2752$	$1.0431$
9	16	$14.8979$	$1.1021$	$12.9598$	$0.59$	$1.2097$
10	14	$13.2924$	$0.7076$	$13.7894$	$0.758$	$1.0002$
11	11	$11.2993$	$- 0.2993$	$14.2052$	$0.4539$	$0.7807$
12	15	$15.1956$	$- 0.1956$	$14.4745$	$0.2881$	$1.0404$
13	18	$17.7887$	$0.2113$	$14.7924$	$0.3004$	$1.2126$
14	16	$15.0349$	$0.9651$	$15.5151$	$0.6263$	$1.013$
15	13	$12.5045$	$0.4955$	$16.3335$	$0.7549$	$0.7869$
16	17	$17.6212$	$- 0.6212$	$16.6388$	$0.3651$	$1.033$