# 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 $$N$$
$$\alpha$$ $$\alpha \in \mathbb R$$ $$0 \leq \alpha \leq 1$$

Output Parameters

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

Tool Support

Single Steps using the Algorithm

References