M-Md-N

247. M-Md-N

Damped mult. trend, mult. errors

ETS (𝑀, Md, 𝑁)

𝑥_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} (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})

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

Example:

ETS (𝑀, Md, 𝑁)

Given

$𝑡$	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}) (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}

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})

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}}

where $ℎ \in {1, 2, 3, \dots}$ is the forecast horizon (how many steps ahead).

Step 2 — apply at $𝑡 = 1$

𝜇_{1} = 12 \cdot 1^{0.8} = 12

𝜀_{1} = (𝑥_{1} - 𝜇_{1}) / 𝜇_{1} = (12 - 12) / 12 = 0

𝑙_{1} = 12 \cdot 1^{0.8} (1 + 0.5 \cdot 0) = 12

𝑏_{1} = 1^{0.8} (1 + 0.4 \cdot 0) = 1

Step 3 — iterate

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