V10/vol2/index/spansizes.tex

\figname{\spansizes}
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\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.}*
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}
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