Exponential Smoothing

P7.forecasting.02 · Audience: guest, it-ml, language-pro · Prerequisites: Baselines You Must Beat

The baselines set the bar; now comes the first real forecasting method. The moving average from the last module treated its window bluntly — the value from five days ago counted exactly as much as yesterday's, and everything older counted for nothing. Exponential smoothing replaces that cliff with a slope: every past value gets a vote, but the votes fade the further back you look. One dial controls how fast they fade — and two upgrades of the same idea carry it all the way to trending, seasonal series.

Step 1 / 5 — An average that forgets

Two extremes bracket the problem. Averaging all history gives a stiff forecast that barely moves when the series shifts — old data outvotes the news. The naive forecast is the other extreme: it listens only to yesterday and inherits every wiggle of the noise. What you want is in between — recent values should count most, older values should fade smoothly rather than fall off a cliff.

ⓘ Concept: Simple exponential smoothing (SES)

level(t) = α · y(t) + (1 − α) · level(t−1)

Why it matters — SES tracks a single quantity — the current level of the series — and updates it with each new observation. The smoothing parameter α (between 0 and 1) is the forgetting dial: α near 1 trusts the latest value and reacts fast like naive; α near 0 trusts the accumulated level and moves slowly like a long average. Weights on past values decay geometrically — exponentially — which gives the method its name.

🗣️ From a linguist's perspective: A conversation partner who forgets gracefully

An interpreter tracking a speaker's pace works exactly this way: the estimate of "how fast are they talking" leans mostly on the last few sentences, a little on the previous minute, and almost nothing on the start of the speech. Nobody recomputes an average over the whole talk, and nobody resets it on every sentence either. Exponential smoothing is that instinct written as one line of arithmetic.

✅ Key takeaways

Simple exponential smoothing tracks one quantity — the level — and updates it as level = α·y + (1−α)·level: an average whose weights fade exponentially into the past.
α is the forgetting dial: near 1 behaves like naive, near 0 like a long stiff average. Between them the noise is smoothed without going deaf to change.
SES forecasts a flat line. On the seeded demo series it says 24.969 while the last value is 25.917 — a level-only model always trails a trending series.
Holt adds a trend dial (forecast becomes a straight line); Holt-Winters adds a seasonal dial and period on top — level, trend and season, each smoothed with its own parameter.
Fitted values describe the past one step at a time and flatter every model; only forecasts on unseen data — scored against the baselines with MASE — count.

Next: P7.forecasting.03 — ARIMA, Gently

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