Predictive error bars

Chapter 2 · The fitted line is a best guess, but the data only pin down w to within some uncertainty. That uncertainty carries into predictions: for a new year x_new, the prediction is ŵᵀx_new with variance σ² x_newᵀ(XᵀX)⁻¹x_new. The band is the prediction ± that spread.

Event Also include the observation noise σ² Show lines drawn from the uncertainty in w

━ fitted prediction ± 1 standard deviation (darker: ± 2) ● data

Polynomial order

Predict for year

Prediction: s

Standard deviation: s

± 2 standard deviations:

The variance of the prediction is σ² x_newᵀ(XᵀX)⁻¹x_new = x_newᵀ cov{ŵ} x_new, with σ² replaced by its fitted value. It is smallest near the middle of the data and grows as you move away, because there the line could tilt a little and move the prediction a lot. Raising the polynomial order lets the fit follow the data more closely but leaves the parameters less well determined, so the band widens, especially beyond the last Olympic year.