M-M-M

261. M-M-M

Fully multiplicative

ETS (𝑀, 𝑀, 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} (1 + 𝛽 𝜀_{𝑡})

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 𝜀_{𝑡})

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

Example:

ETS (𝑀, 𝑀, 𝑀)

Given

Smoothing parameters: $𝛼 = 0.5$ , $𝛽 = 0.4$ , $𝛾 = 0.2$
Initial states: $𝑙_{0} = 12$ , $𝑏_{0} = 1$ , $(𝑠_{- 3}, 𝑠_{- 2}, 𝑠_{- 1}, 𝑠_{0}) = (1.2, 1, 0.8, 1)$ , 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} (1 + 𝛽 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} 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 \cdot 1.2 = 14.4

𝜀_{1} = (𝑥_{1} - 𝜇_{1}) / 𝜇_{1} = (12 - 14.4) / 14.4 = - 0.1667

𝑙_{1} = 12 \cdot 1 (1 + 0.5 \cdot (- 0.1667)) = 11

𝑏_{1} = 1 (1 + 0.4 \cdot (- 0.1667)) = 0.9333

𝑠_{1} = 1.2 (1 + 0.2 \cdot (- 0.1667)) = 1.16

Step 3 — iterate

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

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} (1 + 𝛼 𝜀_{𝑡})$	$𝑏_{𝑡} = 𝑏_{𝑡 - 1} (1 + 𝛽 𝜀_{𝑡})$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 𝜀_{𝑡})$
1	12	$14.4$	$- 0.1667$	$11$	$0.9333$	$1.16$
2	10	$10.2667$	$- 0.026$	$10.1333$	$0.9236$	$0.9948$
3	8	$7.4876$	$0.0684$	$9.6798$	$0.9489$	$0.8109$
4	11	$9.1853$	$0.1976$	$10.0927$	$1.0239$	$1.0395$
5	14	$11.9874$	$0.1679$	$11.2015$	$1.0927$	$1.199$
6	12	$12.1759$	$- 0.0144$	$12.1511$	$1.0864$	$0.9919$
7	9	$10.7049$	$- 0.1593$	$12.1493$	$1.0172$	$0.7851$
8	13	$12.8459$	$0.012$	$12.4317$	$1.022$	$1.042$
9	16	$15.2334$	$0.0503$	$13.0253$	$1.0426$	$1.211$
10	14	$13.4706$	$0.0393$	$13.8471$	$1.059$	$0.9997$
11	11	$11.5129$	$- 0.0446$	$14.3373$	$1.0401$	$0.7781$
12	15	$15.5389$	$- 0.0347$	$14.6539$	$1.0257$	$1.0348$
13	18	$18.202$	$- 0.0111$	$14.9469$	$1.0211$	$1.2083$
14	16	$15.2587$	$0.0486$	$15.6336$	$1.041$	$1.0094$
15	13	$12.6634$	$0.0266$	$16.4906$	$1.052$	$0.7823$
16	17	$17.9523$	$- 0.053$	$16.8888$	$1.0297$	$1.0238$