M-Md-M

264. M-Md-M

Fully multiplicative with damped trend

ETS (𝑀, Md, 𝑀)

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

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

𝑏_{𝑡} = 𝑏_{𝑡 - 1}^{𝜑} (1 + 𝛽 𝜀_{𝑡})

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 𝜀_{𝑡})

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

Example:

ETS (𝑀, Md, 𝑀)

Given

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

𝑏_{𝑡} = 𝑏_{𝑡 - 1}^{𝜑} (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}^{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} \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^{0.8} (1 + 0.5 \cdot (- 0.1667)) = 11

𝑏_{1} = 1^{0.8} (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.

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} (1 + 𝛼 𝜀_{𝑡})$	$𝑏_{𝑡} = 𝑏_{𝑡 - 1}^{𝜑} (1 + 𝛽 𝜀_{𝑡})$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 𝜀_{𝑡})$
1	12	$14.4$	$- 0.1667$	$11$	$0.9333$	$1.16$
2	10	$10.4093$	$- 0.0393$	$10.2047$	$0.9314$	$0.9921$
3	8	$7.7127$	$0.0373$	$9.8204$	$0.9588$	$0.806$
4	11	$9.4956$	$0.1584$	$10.2478$	$1.0282$	$1.0317$
5	14	$12.1549$	$0.1518$	$11.2737$	$1.0846$	$1.1952$
6	12	$11.9357$	$0.0054$	$12.0627$	$1.0694$	$0.9932$
7	9	$10.2583$	$- 0.1227$	$11.9474$	$1.0034$	$0.7862$
8	13	$12.3594$	$0.0518$	$12.2902$	$1.0235$	$1.0424$
9	16	$14.965$	$0.0692$	$12.9537$	$1.0469$	$1.2118$
10	14	$13.3466$	$0.049$	$13.7668$	$1.0577$	$1.0029$
11	11	$11.32$	$- 0.0283$	$14.1951$	$1.0341$	$0.7817$
12	15	$15.1986$	$- 0.0131$	$14.4854$	$1.0218$	$1.0397$
13	18	$17.858$	$0.008$	$14.796$	$1.0206$	$1.2137$
14	16	$15.0837$	$0.0607$	$15.4965$	$1.0412$	$1.0151$
15	13	$12.5116$	$0.039$	$16.3171$	$1.0489$	$0.7878$
16	17	$17.6251$	$- 0.0355$	$16.6521$	$1.0242$	$1.0323$