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.