Asymptotic Expansions of the k Nearest Neighbor Risk
The finite-sample risk of the k nearest neighbor classifier that uses a weighted Lp-metric as a measure of class similarity is examined. For a family of classification problems with smooth distributions in Rn, an asymptotic expansion for the risk is obtained in decreasing fractional powers of the re...
| Veröffentlicht in: | The Annals of Statistics. - Institute of Mathematical Statistics. - 26(1998), 3, Seite 850-878 |
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| Format: | Online-Aufsatz |
| Sprache: | English |
| Veröffentlicht: |
1998
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| Zugriff auf das übergeordnete Werk: | The Annals of Statistics |
| Schlagworte: | Primary 62G20, 62H30, 41A60 k nearest neighbor classifier Finite-sample risk Asymptotic expansions Laplace's method Mathematics Social sciences Philosophy Physical sciences Behavioral sciences |
| Zusammenfassung: | The finite-sample risk of the k nearest neighbor classifier that uses a weighted Lp-metric as a measure of class similarity is examined. For a family of classification problems with smooth distributions in Rn, an asymptotic expansion for the risk is obtained in decreasing fractional powers of the reference sample size. An analysis of the leading expansion coefficients reveals that the optimal weighted Lp-metric, that is, the metric that minimizes the finite-sample risk, tends to a weighted Euclidean (i.e., L2) metric as the sample size is increased. Numerical simulations corroborate this finding for a pattern recognition problem with normal class-conditional densities. |
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| ISSN: | 00905364 |