A-Ad-A

227. A-Ad-A

Damped additive Holt-Winters

ETS (𝐴, Ad, 𝐴)

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}

𝑏_{𝑡} = 𝜑 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡}

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

Example:

ETS (𝐴, Ad, 𝐴)

Given

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

𝑏_{𝑡} = 𝜑 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡}

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 + 2 = 14.4

𝜀_{1} = 𝑥_{1} - 𝜇_{1} = 12 - 14.4 = - 2.4

𝑙_{1} = 12 + 0.8 \cdot 0.5 + 0.5 \cdot (- 2.4) = 11.2

𝑏_{1} = 0.8 \cdot 0.5 + 0.4 \cdot (- 2.4) = - 0.56

𝑠_{1} = 2 + 0.2 \cdot (- 2.4) = 1.52

Step 3 — iterate

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

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = 𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1} + 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝜑 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡}$	$𝑏_{𝑡} = 𝜑 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡}$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡}$
1	12	$14.4$	$- 2.4$	$11.2$	$- 0.56$	$1.52$
2	10	$10.752$	$- 0.752$	$10.376$	$- 0.7488$	$- 0.1504$
3	8	$6.777$	$1.223$	$10.3885$	$- 0.1098$	$- 2.7554$
4	11	$11.3006$	$- 0.3006$	$10.1503$	$- 0.2081$	$0.9399$
5	14	$11.5038$	$2.4962$	$11.2319$	$0.832$	$2.0192$
6	12	$11.7471$	$0.2529$	$12.0239$	$0.7667$	$- 0.0998$
7	9	$9.882$	$- 0.882$	$12.1964$	$0.2606$	$- 2.9318$
8	13	$13.3447$	$- 0.3447$	$12.2325$	$0.0706$	$0.8709$
9	16	$14.3082$	$1.6918$	$13.1349$	$0.7332$	$2.3576$
10	14	$13.6216$	$0.3784$	$13.9106$	$0.7379$	$- 0.0241$
11	11	$11.5692$	$- 0.5692$	$14.2164$	$0.3627$	$- 3.0456$
12	15	$15.3774$	$- 0.3774$	$14.3178$	$0.1392$	$0.7954$
13	18	$16.7867$	$1.2133$	$15.0358$	$0.5966$	$2.6003$
14	16	$15.4889$	$0.5111$	$15.7686$	$0.6817$	$0.0781$
15	13	$13.2684$	$- 0.2684$	$16.1798$	$0.438$	$- 3.0993$
16	17	$17.3257$	$- 0.3257$	$16.3674$	$0.2202$	$0.7303$