The Annals of Statistics

Choice of neighbor order in nearest-neighbor classification

Peter Hall, Byeong U. Park, and Richard J. Samworth

Full-text: Open access

Abstract

The kth-nearest neighbor rule is arguably the simplest and most intuitively appealing nonparametric classification procedure. However, application of this method is inhibited by lack of knowledge about its properties, in particular, about the manner in which it is influenced by the value of k; and by the absence of techniques for empirical choice of k. In the present paper we detail the way in which the value of k determines the misclassification error. We consider two models, Poisson and Binomial, for the training samples. Under the first model, data are recorded in a Poisson stream and are “assigned” to one or other of the two populations in accordance with the prior probabilities. In particular, the total number of data in both training samples is a Poisson-distributed random variable. Under the Binomial model, however, the total number of data in the training samples is fixed, although again each data value is assigned in a random way. Although the values of risk and regret associated with the Poisson and Binomial models are different, they are asymptotically equivalent to first order, and also to the risks associated with kernel-based classifiers that are tailored to the case of two derivatives. These properties motivate new methods for choosing the value of k.

Article information

Source
Ann. Statist. Volume 36, Number 5 (2008), 2135-2152.

Dates
First available in Project Euclid: 13 October 2008

Permanent link to this document
http://projecteuclid.org/euclid.aos/1223908087

Digital Object Identifier
doi:10.1214/07-AOS537

Mathematical Reviews number (MathSciNet)
MR2458182

Zentralblatt MATH identifier
05368486

Subjects
Primary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]
Secondary: 62G20: Asymptotic properties

Keywords
Bayes classifier bootstrap resampling Edgeworth expansion error probability misclassification error nonparametric classification Poisson distribution

Citation

Hall, Peter; Park, Byeong U.; Samworth, Richard J. Choice of neighbor order in nearest-neighbor classification. Ann. Statist. 36 (2008), no. 5, 2135--2152. doi:10.1214/07-AOS537. http://projecteuclid.org/euclid.aos/1223908087.


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