# Holt-Winters Double Exponential Smoothing¶

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

Single Steps using the Algorithm

References