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Unbiased bootstrap error estimation for linear discriminant analysis

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 unbia...

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Detalles Bibliográficos
Autores principales: Vu, Thang, Sima, Chao, Braga-Neto, Ulisses M, Dougherty, Edward R
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2014
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5270504/
https://www.ncbi.nlm.nih.gov/pubmed/28194165
http://dx.doi.org/10.1186/s13637-014-0015-0
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author Vu, Thang
Sima, Chao
Braga-Neto, Ulisses M
Dougherty, Edward R
author_facet Vu, Thang
Sima, Chao
Braga-Neto, Ulisses M
Dougherty, Edward R
author_sort Vu, Thang
collection PubMed
description 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. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s13637-014-0015-0) contains supplementary material, which is available to authorized users.
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spelling pubmed-52705042017-02-13 Unbiased bootstrap error estimation for linear discriminant analysis Vu, Thang Sima, Chao Braga-Neto, Ulisses M Dougherty, Edward R EURASIP J Bioinform Syst Biol Research 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. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s13637-014-0015-0) contains supplementary material, which is available to authorized users. Springer International Publishing 2014-10-03 /pmc/articles/PMC5270504/ /pubmed/28194165 http://dx.doi.org/10.1186/s13637-014-0015-0 Text en © Vu et al.; licensee Springer. 2014 This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
spellingShingle Research
Vu, Thang
Sima, Chao
Braga-Neto, Ulisses M
Dougherty, Edward R
Unbiased bootstrap error estimation for linear discriminant analysis
title Unbiased bootstrap error estimation for linear discriminant analysis
title_full Unbiased bootstrap error estimation for linear discriminant analysis
title_fullStr Unbiased bootstrap error estimation for linear discriminant analysis
title_full_unstemmed Unbiased bootstrap error estimation for linear discriminant analysis
title_short Unbiased bootstrap error estimation for linear discriminant analysis
title_sort unbiased bootstrap error estimation for linear discriminant analysis
topic Research
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5270504/
https://www.ncbi.nlm.nih.gov/pubmed/28194165
http://dx.doi.org/10.1186/s13637-014-0015-0
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