Holt-Winters Double Exponential SmoothingΒΆ

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 \(N\) None
\(\alpha\) \(\alpha \in \mathbb R\) \(0 \leq \alpha \leq 1\) None None
\(\beta\) \(\beta \in \mathbb R\) \(0 \leq \beta \leq 1\) None None

Output Parameters

Parameter Type Constraint Description Remarks
\(\hat{Y}\) \(\hat{Y} \in \mathbb R^N\) None None None

Single Steps using the Algorithm

References