# Single Exponential Smoothing¶

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

Single Steps using the Algorithm

References

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