M-Md-A

248. M-Md-A

Damped mult. trend + add. season., mult. errors

ETS (𝑀, Md, 𝐴)

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

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

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

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜇_{𝑡} 𝜀_{𝑡}

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

where

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

Example:

ETS (𝑀, Md, 𝐴)

Given

Smoothing parameters: $𝛼 = 0.5$ , $𝛽 = 0.4$ , $𝜑 = 0.8$ , $𝛾 = 0.2$
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}^{𝜑} + 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}^{𝜑} + 𝑠_{𝑡 - 𝑚}

Innovation:

𝜀_{𝑡} = (𝑥_{𝑡} - 𝜇_{𝑡}) / 𝜇_{𝑡}

State updates:

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

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

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} 𝑏_{𝑡}^{𝜑 + 𝜑^{2} + \dots + 𝜑^{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^{0.8} + 2 = 14

𝜀_{1} = (𝑥_{1} - 𝜇_{1}) / 𝜇_{1} = (12 - 14) / 14 = - 0.1429

𝑙_{1} = 12 \cdot 1^{0.8} + 0.5 \cdot 14 \cdot (- 0.1429) = 11

𝑏_{1} = 1^{0.8} + 0.4 \cdot 14 \cdot (- 0.1429) / 12 = 0.9333

𝑠_{1} = 2 + 0.2 \cdot 14 \cdot (- 0.1429) = 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.

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} + 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} + 𝛼 𝜇_{𝑡} 𝜀_{𝑡}$	$𝑏_{𝑡} = 𝑏_{𝑡 - 1}^{𝜑} + 𝛽 𝜇_{𝑡} 𝜀_{𝑡} / 𝑙_{𝑡 - 1}$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜇_{𝑡} 𝜀_{𝑡}$
1	12	$14$	$- 0.1429$	$11$	$0.9333$	$1.6$
2	10	$10.4093$	$- 0.0393$	$10.2047$	$0.9314$	$- 0.0819$
3	8	$6.6408$	$0.2047$	$10.3204$	$0.998$	$- 2.7282$
4	11	$11.3041$	$- 0.0269$	$10.152$	$0.9866$	$0.9392$
5	14	$11.6433$	$0.2024$	$11.2217$	$1.0821$	$2.0713$
6	12	$11.8714$	$0.0108$	$12.0176$	$1.0698$	$- 0.0561$
7	9	$9.9557$	$- 0.096$	$12.206$	$1.0236$	$- 2.9193$
8	13	$13.3754$	$- 0.0281$	$12.2485$	$1.0066$	$0.8641$
9	16	$14.3841$	$0.1123$	$13.1207$	$1.058$	$2.3945$
10	14	$13.6701$	$0.0241$	$13.8912$	$1.0562$	$0.0098$
11	11	$11.5931$	$- 0.0512$	$14.2158$	$1.0276$	$- 3.0379$
12	15	$15.3934$	$- 0.0256$	$14.3326$	$1.011$	$0.7854$
13	18	$16.8529$	$0.0681$	$15.0319$	$1.0408$	$2.6239$
14	16	$15.5303$	$0.0302$	$15.7553$	$1.045$	$0.1038$
15	13	$13.2821$	$- 0.0212$	$16.179$	$1.0287$	$- 3.0943$
16	17	$17.3345$	$- 0.0193$	$16.3818$	$1.0146$	$0.7185$