M-A-M

240. M-A-M

Multiplicative Holt-Winters

ETS (𝑀, 𝐴, 𝑀)

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

𝑙_{𝑡} = (𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1}) (1 + 𝛼 𝜀_{𝑡})

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 (𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1}) 𝜀_{𝑡}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 𝜀_{𝑡})

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

Example:

ETS (𝑀, 𝐴, 𝑀)

Given

Smoothing parameters: $𝛼 = 0.5$ , $𝛽 = 0.4$ , $𝛾 = 0.2$
Initial states: $𝑙_{0} = 12$ , $𝑏_{0} = 0.5$ , $(𝑠_{- 3}, 𝑠_{- 2}, 𝑠_{- 1}, 𝑠_{0}) = (1.2, 1, 0.8, 1)$ , 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} + 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}) (1 + 𝛼 𝜀_{𝑡})

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 (𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1}) 𝜀_{𝑡}

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 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) \cdot 1.2 = 15

𝜀_{1} = (𝑥_{1} - 𝜇_{1}) / 𝜇_{1} = (12 - 15) / 15 = - 0.2

𝑙_{1} = (12 + 0.5) (1 + 0.5 \cdot (- 0.2)) = 11.25

𝑏_{1} = 0.5 + 0.4 (12 + 0.5) \cdot (- 0.2) = - 0.5

𝑠_{1} = 1.2 (1 + 0.2 \cdot (- 0.2)) = 1.152

Step 3 — iterate

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

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = (𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1}) 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = (𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1}) (1 + 𝛼 𝜀_{𝑡})$	$𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 (𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1}) 𝜀_{𝑡}$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 𝜀_{𝑡})$
1	12	$15$	$- 0.2$	$11.25$	$- 0.5$	$1.152$
2	10	$10.75$	$- 0.0698$	$10.375$	$- 0.8$	$0.986$
3	8	$7.66$	$0.0444$	$9.7875$	$- 0.63$	$0.8071$
4	11	$9.1575$	$0.2012$	$10.0787$	$0.107$	$1.0402$
5	14	$11.734$	$0.1931$	$11.1693$	$0.8938$	$1.1965$
6	12	$11.8948$	$0.0088$	$12.1164$	$0.9365$	$0.9878$
7	9	$10.5351$	$- 0.1457$	$12.102$	$0.1757$	$0.7836$
8	13	$12.7718$	$0.0179$	$12.3874$	$0.2635$	$1.044$
9	16	$15.1367$	$0.057$	$13.0117$	$0.5521$	$1.2101$
10	14	$13.3982$	$0.0449$	$13.8684$	$0.7958$	$0.9967$
11	11	$11.4906$	$- 0.0427$	$14.3512$	$0.5454$	$0.7769$
12	15	$15.5513$	$- 0.0355$	$14.6325$	$0.3341$	$1.0366$
13	18	$18.1117$	$- 0.0062$	$14.9204$	$0.2972$	$1.2086$
14	16	$15.1669$	$0.0549$	$15.6356$	$0.6316$	$1.0076$
15	13	$12.6378$	$0.0287$	$16.5003$	$0.8181$	$0.7813$
16	17	$17.9514$	$- 0.053$	$16.8594$	$0.4509$	$1.0256$