A-N-N

220. A-N-N

Simple exponential smoothing

ETS (𝐴, 𝑁, 𝑁)

𝑥_{𝑡} = 𝑙_{𝑡 - 1} + 𝜀_{𝑡}

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝛼 𝜀_{𝑡}

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡}

Example:

ETS (𝐴, 𝑁, 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} + 𝜀_{𝑡}

Innovation (rearrange):

𝜀_{𝑡} = 𝑥_{𝑡} - 𝑙_{𝑡 - 1}

State update:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝛼 𝜀_{𝑡}

One-step-ahead forecast:

{\hat{𝑥}}_{𝑡 | 𝑡 - 1} = 𝑙_{𝑡 - 1}

Step 2 — apply at $𝑡 = 1$

Use $𝑙_{0} = 12$ and $𝑥_{1} = 12$ :

{\hat{𝑥}}_{1 | 0} = 𝑙_{0} = 12

𝜀_{1} = 𝑥_{1} - 𝑙_{0} = 12 - 12 = 0

𝑙_{1} = 𝑙_{0} + 𝛼 𝜀_{1} = 12 + 0.5 \cdot 0 = 12

Step 3 — iterate

Values rounded to 4 decimal places; arithmetic performed at full precision.

Forecast for any future $𝑡 > 16$ : ${\hat{𝑥}}_{𝑡 | 16} = 𝑙_{16} = 15.7335$ (flat — SES has no trend).