title: Unbiased bootstrap error estimation for linear discriminant analysis.
abstract: Convex bootstrap error estimation is a popular tool for classifier error estimation in gene expression studies. A basic question is how to determine the weight for the convex combination between the basic bootstrap estimator and the resubstitution estimator such that the resulting estimator is unbiased at finite sample sizes. The well-known 0.632 bootstrap error estimator uses asymptotic arguments to propose a fixed 0.632 weight, whereas the more recent 0.632+ bootstrap error estimator attempts to set the weight adaptively. In this paper, we study the finite sample problem in the case of linear discriminant analysis under Gaussian populations. We derive exact expressions for the weight that guarantee unbiasedness of the convex bootstrap error estimator in the univariate and multivariate cases, without making asymptotic simplifications. Using exact computation in the univariate case and an accurate approximation in the multivariate case, we obtain the required weight and show that it can deviate significantly from the constant 0.632 weight, depending on the sample size and Bayes error for the problem. The methodology is illustrated by application on data from a well-known cancer classification study.
authors.affiliation: Department of Electrical and Computer Engineering, Texas A&M University, 3128 TAMU, College Station, 77843 TX USA.
authors.forename: Thang
authors.initials: T
authors.lastname: Vu
authors.affiliation: Center for Bioinformatics and Genomic Systems Engineering, Texas A&M University, 101 Gateway, Suite A, College Station, 77845 TX USA.
authors.forename: Chao
authors.initials: C
authors.lastname: Sima
authors.affiliation: Department of Electrical and Computer Engineering, Texas A&M University, 3128 TAMU, College Station, 77843 TX USA.|Center for Bioinformatics and Genomic Systems Engineering, Texas A&M University, 101 Gateway, Suite A, College Station, 77845 TX USA.
authors.forename: Ulisses M
authors.initials: UM
authors.lastname: Braga-Neto
authors.affiliation: Department of Electrical and Computer Engineering, Texas A&M University, 3128 TAMU, College Station, 77843 TX USA.|Center for Bioinformatics and Genomic Systems Engineering, Texas A&M University, 101 Gateway, Suite A, College Station, 77845 TX USA.
authors.forename: Edward R
authors.initials: ER
authors.lastname: Dougherty
keywords: Bias; Bootstrap; Error estimation; Gene expression classification; Linear discriminant analysis