M-N-M

237. M-N-M

SES + mult. seasonality and errors

ETS (𝑀, 𝑁, 𝑀)

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

𝑙_{𝑡} = 𝑙_{𝑡 - 1} (1 + 𝛼 𝜀_{𝑡})

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} (1 + 𝛾 𝜀_{𝑡})

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

Example:

ETS (𝑀, 𝑁, 𝑀)

Given

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

Conditional mean (one-step-ahead forecast ${\hat{𝑥}}_{𝑡 | 𝑡 - 1} = 𝜇_{𝑡}$ ):

𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑠_{𝑡 - 𝑚}

Innovation:

𝜀_{𝑡} = (𝑥_{𝑡} - 𝜇_{𝑡}) / 𝜇_{𝑡}

State updates:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} (1 + 𝛼 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} 𝑠_{𝑡 + ℎ - 𝑚_{ℎ}^{+}}

where $𝑚_{ℎ}^{+} = ((ℎ - 1) mod 𝑚) + 1$ 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 = 14.4

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

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

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.