Exponential smoothing and ETS

The most cost-effective forecasting method on Earth was invented in the 1950s for inventory planning, runs in microseconds, and still embarrasses neural networks on short business series. Exponential smoothing starts from one psychological truth about time: recent observations matter more. Formalize that with geometric decay, add components the same way the decomposition lesson took them apart, and you get the ETS family — this lesson, and one of the two classical pillars (ARIMA, next lesson, is the other).

Simple exponential smoothing: one number, updated

Forecast with a single adaptive level $\ell_t$:

Unroll the recursion and the name explains itself: $\ell_t = \alpha \sum_{j \ge 0} (1-\alpha)^j y_{t-j}$ — a weighted average with weights decaying geometrically into the past. $\alpha$ is the memory dial: $\alpha \to 1$ reproduces the naive forecast (last value, no memory), $\alpha \to 0$ the long-run average (all memory, no reactivity). Written as $\ell_t = \ell_{t-1} + \alpha (y_t - \ell_{t-1})$ it is error-correction: move the estimate a fraction α toward each surprise — the same update-on-surprise shape as SGD and, exactly, the Kalman filter with a fixed gain (lesson 4 makes that identity precise).

Add trend, add season: Holt–Winters

The decomposition lesson’s components return, but now each is tracked online with its own smoothing constant. Level, trend $b_t$, and seasonal $s_t$ (period $m$), additive form:

Each line is the SES pattern applied to one component: deseasonalized surprise updates the level, level changes update the trend, level-adjusted surprise updates that season’s slot. Three dials, each meaning “how fast does this component change” — and the right values depend on the series, which is why you should now go feel them:

From smoother to model: ETS state space

As presented so far, Holt–Winters is an algorithm with no error bars — the band in the widget was bolted on from one-step errors. The modern fix (Hyndman et al., 2002) rewrites each smoother as a state-space model:

with the recursions above as the state transitions driven by the same $\varepsilon_t$. Now it is a likelihood: the smoothing constants are estimated by MLE instead of hand-tuning, models are compared by AIC, and prediction intervals come from the model rather than a heuristic. The taxonomy ETS(Error, Trend, Seasonal) — each Additive/Multiplicative/None, damped or not — gives ~30 family members, and the ets() automation that searches them by AIC is the single most used forecasting routine in industry. The lesson’s through-line, one more time: heuristic, made probabilistic, gains uncertainty and principled selection — the same promotion k-means got from mixtures.

Two honest boundaries. ETS forecasts one series from its own past — no covariates, no cross-series learning (the neural lesson’s opening). And its seasonal component is periodic-with-drift, so genuinely new patterns (a pandemic, a product launch) are structural breaks it must re-learn through the α/γ dials, slowly.

import numpy as np, pandas as pd
from statsmodels.tsa.holtwinters import ExponentialSmoothing

rng = np.random.default_rng(0)
t = np.arange(96)
y = pd.Series(10 + 0.08*t + 2*np.sin(2*np.pi*t/12) + 0.7*rng.standard_normal(96),
              index=pd.date_range("2018-01-01", periods=96, freq="MS"))

fit = ExponentialSmoothing(y, trend="add", seasonal="add",
                           seasonal_periods=12, damped_trend=True).fit()
print(fit.params["smoothing_level"], fit.params["smoothing_trend"],
      fit.params["smoothing_seasonal"])       # MLE found the dials for you
print(fit.forecast(24).head())                # the widget's blue line

Exercises