Exponential Smoothing

## The method in one recursion Simple exponential smoothing (SES) is the forecasting method for a series with no trend and no seasonal pattern — a quantity that wanders around a slowly-moving level. It carries exactly one piece of state, the **level**, and updates it every period: ``` level_t = alpha * y_t + (1 - alpha) * level_{t-1} ``` Here `y_t` is the observation at time t and `alpha` is the **smoothing parameter**, a number between 0 and 1. The forecast made at time t for any horizon h is flat: ``` forecast_{t+h} = level_t for every h = 1, 2, 3, ... ``` An equivalent and very useful way to write the update is in error-correction form. If `e_t = y_t - level_{t-1}` is the one-step-ahead forecast error, then `level_t = level_{t-1} + alpha * e_t`: alpha is the fraction of each surprise that you fold permanently into your view of the level. ## What alpha actually does: geometric weights Substituting the recursion into itself repeatedly gives ``` level_t = alpha*y_t + alpha(1-alpha)*y_{t-1} + alpha(1-alpha)^2*y_{t-2} + ... + (1-alpha)^t * level_0 ``` The weight on the observation j periods ago is `alpha * (1 - alpha)^j`. Two consequences follow, and both are commonly probed in interviews. First, **nothing is ever discarded**. Unlike a moving average of the last N points, which drops the (N+1)-th point entirely, SES keeps every observation with a small but non-zero weight. That is why the method is sometimes described as an infinite weighted average of the past. Second, **alpha sets the effective memory**. Two summary numbers make this concrete: - Mean age of the data in the weighted average: `(1 - alpha) / alpha`. - Half-life of the weights: `ln(0.5) / ln(1 - alpha)` periods. For alpha = 0.05 the mean age is 19 periods and the weights halve about every 13.5 periods. For alpha = 0.9 the mean age is roughly 0.11 periods and the weights are effectively gone after one or two points. Low alpha means a long memory; high alpha means a short one. The single most common error is to invert this — to say that a small alpha means the model ignores the past, when a small alpha is precisely what makes the past dominate. ## The tradeoff on a real series Take a noisy weekly demand series for one product. Week-to-week variation is a mixture of two things: transient noise (a promotion at a competitor, a delivery landing on Monday instead of Sunday) and genuine, persistent shifts in the level (a price cut, a new distribution channel). With alpha = 0.05 the model treats almost every movement as noise. The fitted line is smooth and the forecast is stable, which is exactly what a downstream planner wants — but when demand genuinely steps up, the model takes many weeks to catch up and forecasts too low the whole time. With alpha = 0.9 the model treats almost every movement as signal. It catches the step change within a week or two, but its forecast series is nearly as jumpy as the raw data, so ordinary one-step errors are large and the plan churns. That is the whole bias-versus-responsiveness tradeoff, controlled by one number. ## How alpha is chosen Alpha (and the initial level `level_0`) are estimated from the data, typically by minimising the sum of squared one-step-ahead errors over the training sample, or equivalently by maximising the likelihood of the corresponding state-space model. Interviewers like to hear that this is a fitted quantity, not a hand-set knob, and that the fitted value is itself a diagnostic: - alpha close to 1 says almost every movement persists — the series behaves like a random walk, and the naive forecast (last observed value) is nearly as good. - alpha close to 0 says almost every movement is noise around a stable level, so a long-run average is nearly as good. - alpha pinned to a boundary, or residuals that still show structure, usually means the series has a trend or a seasonal pattern that a level-only model cannot represent, and you need a richer form. ## Uncertainty The point forecast is flat, but the uncertainty is not. In the additive-error state-space form of SES, the variance of the h-step-ahead forecast is `sigma^2 * (1 + (h - 1) * alpha^2)`, where `sigma^2` is the one-step error variance. Intervals therefore widen with the horizon, and they widen faster when alpha is large — a responsive model is also a more uncertain one far out. A flat point forecast surrounded by widening intervals is the correct picture of SES, and candidates who describe the forecast as a single line with constant uncertainty have missed half the model. ## Common mistakes Calling alpha a significance level, confusing high alpha with more smoothing, believing that only the last few points get weight, or asserting that SES can handle trend. It cannot: with a trending series the flat forecast is biased low (or high) at every horizon, and that is the cue to move to a trend form of the method.