Computing a Simple Exponential Smoothing Forecast
Weekly signups are y = 20, 24, 22, 28. You fit simple exponential
smoothing with \alpha = 0.5 and initial level \ell_1 = y_1 = 20.
- Compute \ell_2, \ell_3, \ell_4 and the forecast for week 5.
- Compute the in-sample one-step MAE and compare it with the naive forecast's MAE. Which does better, and by how much?
- What happens to the forecast for week 20 if no new data arrives between week 5 and week 20? Why is this a limitation for a series with trend?
1. Recursion \ell_t = 0.5\, y_t + 0.5\, \ell_{t-1}:
\ell_2 = 0.5(24) + 0.5(20) = 22.0 \ell_3 = 0.5(22) + 0.5(22.0) = 22.0 \ell_4 = 0.5(28) + 0.5(22.0) = 25.0
Forecast for week 5: \hat y_5 = \ell_4 = 25.0.
2. In-sample one-step errors (\hat y_t = \ell_{t-1}):
- t=2: forecast 20, actual 24, error 4
- t=3: forecast 22, actual 22, error 0
- t=4: forecast 22, actual 28, error 6
SES MAE = (4 + 0 + 6)/3 = 10/3 \approx 3.33.
Naive forecast (\hat y_t = y_{t-1}): errors |24-20|=4, |22-24|=2, |28-22|=6. Naive MAE = 12/3 = 4.0.
SES beats naive here (3.33 vs 4.0, about 17 % lower MAE), because \alpha = 0.5 lets it react quickly to the jump at week 4 while slightly smoothing the dip at week 3. On only three errors this is not strong evidence — the honest test is a rolling-origin backtest over much more history, and MASE = 3.33/4.0 \approx 0.83 is the way to report the comparison in a scale-free form.
3. Forecast for week 20: SES forecasts are flat: \hat y_h = \ell_4 = 25.0 for every future h, including week 20, because SES has no trend term — the level simply stops updating once data stops arriving. If the underlying series has an upward trend (signups are growing), a flat forecast will systematically under-forecast further out. This is exactly the gap Holt's linear method fills: it adds a trend state b_t so the h-step forecast is \ell_t + h\, b_t instead of a constant, and a damped trend variant prevents that extrapolated trend from running away at long horizons.
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