k-nearest-neighbor Bayes-risk estimation

Abstract

Nonparametric estimation of the Bayes risk <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^\ast</tex> using a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -nearest-neighbor ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -NN) approach is investigated. Estimates of the conditional Bayes error <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(X)</tex> for use in an unclassified test sample approach to estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^\ast</tex> are derived using maximum-likelihood estimation techniques. By using the volume information as well as the class representations of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -NN's to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> , the mean-squared error of the conditional Bayes error estimate is reduced significantly. Simulations are presented to indicate the performance of the estimates using unclassified testing samples.