The Annals of Statistics

Sparse linear discriminant analysis by thresholding for high dimensional data

Jun Shao, Yazhen Wang, Xinwei Deng, and Sijian Wang

Full-text: Open access

Abstract

In many social, economical, biological and medical studies, one objective is to classify a subject into one of several classes based on a set of variables observed from the subject. Because the probability distribution of the variables is usually unknown, the rule of classification is constructed using a training sample. The well-known linear discriminant analysis (LDA) works well for the situation where the number of variables used for classification is much smaller than the training sample size. Because of the advance in technologies, modern statistical studies often face classification problems with the number of variables much larger than the sample size, and the LDA may perform poorly. We explore when and why the LDA has poor performance and propose a sparse LDA that is asymptotically optimal under some sparsity conditions on the unknown parameters. For illustration of application, we discuss an example of classifying human cancer into two classes of leukemia based on a set of 7,129 genes and a training sample of size 72. A simulation is also conducted to check the performance of the proposed method.

Article information

Source
Ann. Statist., Volume 39, Number 2 (2011), 1241-1265.

Dates
First available in Project Euclid: 9 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1304947049

Digital Object Identifier
doi:10.1214/10-AOS870

Mathematical Reviews number (MathSciNet)
MR2816353

Zentralblatt MATH identifier
1215.62062

Subjects
Primary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]
Secondary: 62F12: Asymptotic properties of estimators 62G12

Keywords
Classification high dimensionality misclassification rate normality optimal classification rule sparse estimates

Citation

Shao, Jun; Wang, Yazhen; Deng, Xinwei; Wang, Sijian. Sparse linear discriminant analysis by thresholding for high dimensional data. Ann. Statist. 39 (2011), no. 2, 1241--1265. doi:10.1214/10-AOS870. https://projecteuclid.org/euclid.aos/1304947049


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