Page 264 - Introduction to Statistical Pattern Recognition

246 Introduction to Statistical Pattern Recognition

Experiment 10: Bootstrap errors
Data: 1-1, 1-41, I-A (Normal, n = 8)
Classifier: Quadratic classifier of (5.54)
No. of bootstrap sets: S = 100
Sample size: N I = N2 = 24, 40, 80, 160, 320
No. of trials: z = 10
Results: Table 5-12 [6]
A #. A A
of
In Table 5-12(a), the means and standard deviations of &L and &h (= E~-&~)
the 10 trials are presented for the conventional L and R methods. Table 5-
A A*
12(b) shows the corresponding terms in the bootstrap method: ER + E* { &h IS ]
AX A*
and Ex { &h IS ] respectively. E* { &h IS, } is obtained by taking the average of
..* ,*
&h, I,. . . ,E,,, [see Fig. 5-51. This is the bootstrap estimation of the bias
between the H and R errors given S,, and varies with S,. The random variable
,.
E,{;; IS] with a random S should be compared with ;,, of Table 5-12(a). If
,.
this bias estimate is close to the bias between E~, and E~, of SL,$ the bootstrap
,. A A
bias could be added to cR, to estimate . The term eR + E, { &h I S ] of Table
A A
5-12(b) shows this estimation of E~, and should be cornpared with of Table
5-12(a). The table shows that & of (a) and iR + Ex { 2, IS ] of (b) are close in
A
both mean and standard deviation for a reasonable size of N. The biases, &h of
A-
(a) and { &,, I S] of (b), are also close in the mean, but the standard deviations
A A
of E* { &h I S 1 are consistently smaller than the ones of &h.
Bootstrap Variance

The variance with respect to the bootstrap samples can be evaluated in a
fashion similar to (5.154)

(5.183)