\figname{\spansizes} \pageinsert \vbox to \vsize{ \centerline{\psfig{file=figures/spansizes.p,width=4in}} \figurecaption{\spansizes}{Bias-variance tradeoff. A sample of size 80 was generated from $Y=sin(X)+\varepsilon$ with $X\sim N(0,1)$ and $\varepsilon\sim N(0,1/9)$. The top two panels represent one realization of the model with a smooth included; panel (a) uses a small neighbourhood and thus does little smoothing, while (b) uses a large neighbourhood and does a lot of smoothing. In terms of approximate degrees of freedom (a concept introduced in Chapter~3), the smoother in (a) uses $\df =15$, whereas for (b) $\df=4$. The $\sin$ function is included in both panels. The lower two panels show the bias-variance behaviour of this smoother. The light shaded regions show twice the pointwise standard errors of the fitted smooth values, and thus reflect the variance of the smooths under repeated sampling of 80 pairs from the model. The darker shaded region shows the bias, which is the difference between the generating curve and the average value of the smooth curves under repeated sampling. In panel (c), the bias is virtually nonexistent, while the standard-error bands are quite wide. In panel (d), there is a considerable cost in bias for a moderate decrease in variance.}* \vfill } \endinsert