A comparison of bootstrap methods and an adjusted bootstrap ...

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ecomes e , this results in systematic downward bias when there are no classdifferences (Breiman et al. [13], Efron and Tibshirani [3]). The .632+ bootstrap aims tocircumvent this problem by increasing the weight w with respect to the growing level ofoverfitting. It often performs well in classification problems with n>p. For microarray0.632 LOOBSndata with n
Let cl ( ) be the chances that an observation appears in a bootstrap sample of size ln . Asimple probabilistic argument indicates that cl ( ) ≈1−e −l . A bootstrap sample of sizeln contains approximately cl ( ) ⋅ndistinct observations from the original sample. Forexample, for l = 1,2,3 , the number of distinct observations is about 0.632n , 0.865n ,0.95n respectively. With l = 1, the repeated leave-one-out bootstrap closely resemblesthe leave-one-out bootstrap procedure. Asl increases, a bootstrap learning set for a leftoutitem contains more distinct observations. On one hand, the method acquiresadditional accuracy and brings a reduction on the upward bias. On the other hand, thebootstrap learning sets obtained from the same leave-one-out set become more similar instructure and this raises the variability of the estimation.The learning behavior of the repeated leave-one-out bootstrap can be modeled as afunction of the number of distinct observations included in the bootstrap learning sets.The trend of a learning process as a function of sample size is often modeled in themachine learning literature by a flexible curve following an inverse power law. Letthe expected number of distinct observations to appear in a bootstrap sample of sizeln .Letem ( )be the expected error rate given the observed sample using the repeated leaveone-outbootstrap method with bootstrap learning sets of sizeln . Ideally,follow the inverse power lawwhere a , α and b are the parameters.−αem ( ) = am + bem ( )shouldm beThe Adjusted Bootstrap method estimates the prediction error as follows. PickJ bootstrap learning set sizes ln,j = 1,..., J . Compute the repeated leave-one-outjRLOOBbootstrap estimate e RLOOBn ( lj) with bootstrap learning sets of size lnj.Denote en ( lj)by e where m = c( l ) ⋅nis the expected number of distinct original observations in am jjjbootstrap learning set of size ln. Fit an empirical learning curve of thej9
Page 4 and 5: n> p situation, the .632+ bootstrap
Page 6 and 7: Ordinary Bootstrap method [4] has t
Page 10 and 11: form e = am −α+b with j=1, …,
Page 12 and 13: In Table 1, we report the simulatio
Page 14 and 15: signals ( µ = 1.5 i.e. µ1= µ2in
Page 16 and 17: As CART is not a satisfactory class
Page 18 and 19: ootstrap method and demonstrated th
Page 20 and 21: Table 1. Simulation study with samp
Page 22 and 23: Table 3. Simulation study with samp
Page 24 and 25: BIASSTDMSECase 10.150.10.050-0.05-0
Page 26: BIASSTDMSECase 10.10.050-0.05-0.1-0

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