Single Exponential Smoothing

Local Algorithm - One-Dimensional Algorithm

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 \(N\)
\(\alpha\)	\(\alpha \in \mathbb R\)	\(0 \leq \alpha \leq 1\)

Output Parameters

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

Tool Support

• Matlab

  For details refer to the online documentation of the function ‘smoothts’.

Single Steps using the Algorithm

• Data Denoising with Single Exponential Smoothing

References

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