Brown’s Double Exponential Smoothing

The basic formulas of the Brown’s Double Exponential Smoothing algorithm are stated as follows:

\[(1) \quad s^{'}_0 = Y_0 \, \text{,}\]
\[(2) \quad s^{''}_0 = Y_0 \, \text{,}\]
\[(3) \quad s^{'}_t = \alpha Y_t + (1-\alpha) s^{'}_{t-1} \, \text{,}\]
\[(4) \quad s^{''}_t = \alpha s^{'}_t + (1-\alpha) s^{''}_{t-1} \, \text{,}\]
\[(5) \quad a_t = 2 s^{'}_t - s^{''}_t \, \text{,}\]
\[(6) \quad b_t = \frac{\alpha}{1-\alpha} (s^{'}_t - s^{''}_t) \, \text{,}\]
\[(7) \quad \hat{Y}_{t+m} = a_t + m b_t \, \text{,}\]

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

Output Parameters

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

Single Steps using the Algorithm

References