A-M-A

245. A-M-A

Multiplicative trend, additive seasonality

ETS (𝐴, 𝑀, 𝐴)

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

𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / 𝑙_{𝑡 - 1}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡}

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

Example:

ETS (𝐴, 𝑀, 𝐴)

Given

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

State updates:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / 𝑙_{𝑡 - 1}

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} 𝑏_{𝑡}^{ℎ} + 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 \cdot 1 + 2 = 14

𝜀_{1} = 𝑥_{1} - 𝜇_{1} = 12 - 14 = - 2

𝑙_{1} = 12 \cdot 1 + 0.5 \cdot (- 2) = 11

𝑏_{1} = 1 + 0.4 \cdot (- 2) / 12 = 0.9333

𝑠_{1} = 2 + 0.2 \cdot (- 2) = 1.6

Step 3 — iterate

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

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} + 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡}$	$𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / 𝑙_{𝑡 - 1}$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡}$
1	12	$14$	$- 2$	$11$	$0.9333$	$1.6$
2	10	$10.2667$	$- 0.2667$	$10.1333$	$0.9236$	$- 0.0533$
3	8	$6.3595$	$1.6405$	$10.1798$	$0.9884$	$- 2.6719$
4	11	$11.0616$	$- 0.0616$	$10.0308$	$0.986$	$0.9877$
5	14	$11.4901$	$2.5099$	$11.145$	$1.0861$	$2.102$
6	12	$12.0509$	$- 0.0509$	$12.0788$	$1.0842$	$- 0.0635$
7	9	$10.4243$	$- 1.4243$	$12.3841$	$1.0371$	$- 2.9568$
8	13	$13.8308$	$- 0.8308$	$12.4277$	$1.0102$	$0.8215$
9	16	$14.6569$	$1.3431$	$13.2265$	$1.0535$	$2.3706$
10	14	$13.8701$	$0.1299$	$13.9985$	$1.0574$	$- 0.0375$
11	11	$11.8452$	$- 0.8452$	$14.3794$	$1.0332$	$- 3.1258$
12	15	$15.6789$	$- 0.6789$	$14.5179$	$1.0144$	$0.6857$
13	18	$17.097$	$0.903$	$15.1779$	$1.0392$	$2.5512$
14	16	$15.7359$	$0.2641$	$15.9055$	$1.0462$	$0.0153$
15	13	$13.5145$	$- 0.5145$	$16.383$	$1.0333$	$- 3.2287$
16	17	$17.6137$	$- 0.6137$	$16.6211$	$1.0183$	$0.563$