Exact representation of the second-order moments for resubstitution and leave-one-out error estimation for linear discriminant analysis in the univariate heteroskedastic Gaussian model

Authors:
Amin Zollanvari;Ulisses Braga-Neto;Edward R. Dougherty
Affiliations:
Children's Hospital Informatics Program at Harvard-MIT Division of Health Science and Technology, Boston, MA 02115, United States and Brigham and Women's Hospital, Boston, MA 02115, United States ...;Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, United States;Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, United States and Translational Genomics Research Institute (TGEN), Phoenix, AZ 85004, United St ...
Venue:
Pattern Recognition
Year:
2012

Citing 12
Cited 5

Estimation of Classifier Performance

IEEE Transactions on Pattern Analysis and Machine Intelligence
Small Sample Size Effects in Statistical Pattern Recognition: Recommendations for Practitioners

IEEE Transactions on Pattern Analysis and Machine Intelligence
Statistical and neural classifiers: an integrated approach to design

Statistical and neural classifiers: an integrated approach to design
Is cross-validation valid for small-sample microarray classification?

Bioinformatics
On the sampling distribution of resubstitution and leave-one-out error estimators for linear classifiers

Pattern Recognition
Exact correlation between actual and estimated errors in discrete classification

Pattern Recognition Letters
Exact performance of error estimators for discrete classifiers

Pattern Recognition
Joint sampling distribution between actual and estimated classification errors for linear discriminant analysis

IEEE Transactions on Information Theory - Special issue on information theory in molecular biology and neuroscience
On Dimensionality, Sample Size, Classification Error, and Complexity of Classification Algorithm in Pattern Recognition

IEEE Transactions on Pattern Analysis and Machine Intelligence
Considerations of sample and feature size

IEEE Transactions on Information Theory
Distribution-free inequalities for the deleted and holdout error estimates

IEEE Transactions on Information Theory
Distribution-free performance bounds for potential function rules

IEEE Transactions on Information Theory

Optimal mean-square-error calibration of classifier error estimators under Bayesian models

Pattern Recognition
The reliability of estimated confidence intervals for classification error rates when only a single sample is available

Pattern Recognition
Optimal classifiers with minimum expected error within a Bayesian framework-Part I: Discrete and Gaussian models

Pattern Recognition
Relationship between the accuracy of classifier error estimation and complexity of decision boundary

Pattern Recognition
Classifier design given an uncertainty class of feature distributions via regularized maximum likelihood and the incorporation of biological pathway knowledge in steady-state phenotype classification

Pattern Recognition

Quantified Score

Hi-index	0.01

Visualization

Abstract

This paper provides exact analytical expressions for the bias, variance, and RMS for the resubstitution and leave-one-out error estimators in the case of linear discriminant analysis (LDA) in the univariate heteroskedastic Gaussian model. Neither the variances nor the sample sizes for the two classes need be the same. The generality of heteroskedasticity (unequal variances) is a fundamental feature of the work presented in this paper, which distinguishes it from past work. The expected resubstitution and leave-one-out errors are represented by probabilities involving bivariate Gaussian distributions. Their second moments and cross-moments with the actual error are represented by 3- and 4-variate Gaussian distributions. From these, the bias, deviation variance, and RMS for resubstitution and leave-one-out as estimators of the actual error can be computed. The RMS expressions are applied to the determination of sample size and illustrated in biomarker classification.