M-A-A

239. M-A-A

Additive HW, multiplicative errors

ETS (𝑀, 𝐴, 𝐴)

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

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝛼 𝜇_{𝑡} 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜇_{𝑡} 𝜀_{𝑡}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜇_{𝑡} 𝜀_{𝑡}

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} + ℎ 𝑏_{𝑡} + 𝑠_{𝑡 + ℎ - 𝑚_{ℎ}^{+}}

where

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

Example:

ETS (𝑀, 𝐴, 𝐴)

Given

Smoothing parameters: $𝛼 = 0.5$ , $𝛽 = 0.4$ , $𝛾 = 0.2$
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} + 𝑏_{𝑡 - 1} + 𝑠_{𝑡 - 𝑚}) (1 + 𝜀_{𝑡})

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

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

Innovation:

𝜀_{𝑡} = (𝑥_{𝑡} - 𝜇_{𝑡}) / 𝜇_{𝑡}

State updates:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝛼 𝜇_{𝑡} 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜇_{𝑡} 𝜀_{𝑡}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜇_{𝑡} 𝜀_{𝑡}

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

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} + 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 + 0.5 + 2 = 14.5

𝜀_{1} = (𝑥_{1} - 𝜇_{1}) / 𝜇_{1} = (12 - 14.5) / 14.5 = - 0.1724

𝑙_{1} = 12 + 0.5 + 0.5 \cdot 14.5 \cdot (- 0.1724) = 11.25

𝑏_{1} = 0.5 + 0.4 \cdot 14.5 \cdot (- 0.1724) = - 0.5

𝑠_{1} = 2 + 0.2 \cdot 14.5 \cdot (- 0.1724) = 1.5

Step 3 — iterate

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

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝛼 𝜇_{𝑡} 𝜀_{𝑡}$	$𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜇_{𝑡} 𝜀_{𝑡}$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜇_{𝑡} 𝜀_{𝑡}$
1	12	$14.5$	$- 0.1724$	$11.25$	$- 0.5$	$1.5$
2	10	$10.75$	$- 0.0698$	$10.375$	$- 0.8$	$- 0.15$
3	8	$6.575$	$0.2167$	$10.2875$	$- 0.23$	$- 2.715$
4	11	$11.0575$	$- 0.0052$	$10.0287$	$- 0.253$	$0.9885$
5	14	$11.2757$	$0.2416$	$11.1379$	$0.8367$	$2.0449$
6	12	$11.8246$	$0.0148$	$12.0623$	$0.9069$	$- 0.1149$
7	9	$10.2542$	$- 0.1223$	$12.3421$	$0.4052$	$- 2.9658$
8	13	$13.7358$	$- 0.0536$	$12.3794$	$0.1109$	$0.8413$
9	16	$14.5351$	$0.1008$	$13.2227$	$0.6968$	$2.3378$
10	14	$13.8046$	$0.0142$	$14.0172$	$0.775$	$- 0.0758$
11	11	$11.8264$	$- 0.0699$	$14.379$	$0.4444$	$- 3.1311$
12	15	$15.6648$	$- 0.0424$	$14.4911$	$0.1785$	$0.7084$
13	18	$17.0074$	$0.0584$	$15.1659$	$0.5756$	$2.5363$
14	16	$15.6656$	$0.0213$	$15.9086$	$0.7093$	$- 0.009$
15	13	$13.4868$	$- 0.0361$	$16.3745$	$0.5146$	$- 3.2285$
16	17	$17.5975$	$- 0.034$	$16.5904$	$0.2756$	$0.5889$