Simple exponential smoothing (SES)

A forecast for series without trend or seasonality: a weighted average of past observations with weights that decay geometrically, controlled by the smoothing parameter α.

\ell_t = \alpha y_t + (1-\alpha)\ell_{t-1} = \ell_{t-1} + \alpha e_t, \qquad e_t = y_t - \ell_{t-1}

Level update (weighted-average and error-correction forms)

$\ell_t$: smoothed level after observing $y_t$
$\alpha$: smoothing parameter, $0 < alpha < 1$ ; larger values follow recent data more closely
$e_t$: one-step-ahead forecast error

\hat y_{T+h \given T} = \ell_T, \qquad \sigma_h^2 = \hat\sigma^2\left(1 + \alpha^2(h-1)\right), \quad \hat\sigma^2 = \frac{\text{SSE}}{T-2}

Flat forecasts and approximate h-step forecast variance

$h$: forecast horizon (steps ahead)
$T$: number of observations used to fit
$\text{SSE}$: sum of squared one-step errors over the fit

When to use

Short-term forecasting of a series that wanders around a level with no clear trend or seasonality.

Assumptions

No systematic trend or seasonal pattern (otherwise use Holt or Holt–Winters).
The engine starts from $\ell_0 = y_1$ (the "simple" initialisation) unless an initial level is supplied, and estimates α by minimising the in-sample SSE when α is not given. Prediction intervals assume normal errors and ignore parameter uncertainty, so they are approximate.

How to interpret

The weight on the observation $j$ periods ago is $\alpha(1-\alpha)^j$ . α near 1 ≈ naive forecast (last value); α near 0 ≈ long-run average.
All horizons get the same point forecast; uncertainty grows with $h$ .

Common mistakes

Applying SES to a trending series: forecasts lag systematically behind.
Judging α by in-sample fit only — compare methods on a holdout (forecast accuracy).

Example

Series 20, 22, 21, 25, 24 with α = 0.3 and $\ell_0 = 20$ : levels 20, 20.6, 20.72, 22.00, 22.60. Forecasts for the next two periods are both 22.60; SSE = 26.46.