Single Exponential SmoothingΒΆ

Single Exponential Smoothing algorithm uses a parameter \(\alpha\) to establish an exponentially decreasing weighting factor for time series data. When \(\alpha\) is equal to one, there is no smoothing, and when \(\alpha\) is equal to zero, a constant time series is returned. The basic formulas are stated as follows:

\[(1) \quad s_1 = Y_0 \, \text{,}\]
\[(2) \quad s_t = \alpha(Y_{t-1} - s_{t-1}) + s_{t-1}, \quad t > 1 \, \text{,}\]
\[(3) \quad \hat{Y}_t = s_t \, \text{,}\]

where \(Y\) is the data sequence beginning at time \(t = 0\) and \(\hat{Y}_{t}\) is the smoothed forecast for time \(t\). This function is exponential through the nesting of the function at each subsequent data value.

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

  • K.D. Kammeyer and K. Kroschel, Digitale Signalverarbeitung, 5th ed. Stuttgart: Teubner, 2002.