Holt-Winters Double Exponential Smoothing

Local Algorithm - One-Dimensional Algorithm

Holt-Winters Double Exponential Smoothing algorithm is an improved version of the Single Exponential Smoothing algorithm. It works well when there is a a trend in the input data. The basic formulas are stated as follows:

$$(1) \quad s_1 = Y_0 \, \text{,}$$

,

$$(2) \quad b_1 = Y_1-Y_0 \, \text{,}$$

$$(3) \quad s_t = \alpha Y_t + (1 - \alpha)(s_{t-1} + b_{t-1}), \quad t > 1 \, \text{,}$$

$$(4) \quad b_t = \beta (s_t - s_{t-1}) + (1 - \beta) b_{t-1}, \quad t > 1\, \text{,}$$

$$(5) \quad \hat{Y}_{t+m} = s_t + m b_t \, \text{,}$$

where $Y$ is the data sequence beginning at time $t = 0$ and $\hat{Y}_{t+m}$ is the smoothed forecast for time $t + m$.

Input Parameters

Parameter	Type	Constraint	Description	Remarks
$Y$	$Y \in \mathbb R^N$	$N \in \mathbb{N}$	Input data sequence of length $N$
$\alpha$	$\alpha \in \mathbb R$	$0 \leq \alpha \leq 1$
$\beta$	$\beta \in \mathbb R$	$0 \leq \beta \leq 1$

Output Parameters

Parameter	Type	Constraint	Description	Remarks
$\hat{Y}$	$\hat{Y} \in \mathbb R^N$

Tool Support

• Excel

Single Steps using the Algorithm

• Data Denoising with Holt-Winters Double Exponential Smoothing

References

• NIST/SEMATECH e-Handbook of Statistical Methods

  http://www.itl.nist.gov/div898/handbook