> [!tldr] > Simple exponential smoothing (SES) is a type of exponential smoothing for forecasting time series. > Denoting $\hat{x}_{t}$ as the forecasted value of $x_{t}$, it is an AR process with exponentially decaying weights: $\hat{x}_{N+1}=\sum_{i=0}^{N}\alpha(1-\alpha)^{i}x_{t-i}.$ Equivalently, SES can be defined with the **updating equation** $\hat{x}_{t+1}=\alpha x_{t}+(1-\alpha)\hat{x}_{t}$ Rearranging gives the **error-correction form**: $\hat{x}_{t+1}=\alpha \cdot e_{t}+\hat{x}_{t}$where $e_{t}=x_{t} - \hat{x}_{t}$ is the forecasting error at time $t$. This form is useful for computing the training error when modeling the observed time series with different $\alphas.