Let be a random element in a metric space , and let be a random variable with value or . is called the class, or the label, of . Let be an observed i.i.d. sample having the same law as . The problem of classification is to predict the label of a new random element . The -nearest neighbor classifier is the simple following rule: look at the nearest neighbors of in the trial sample and choose or for its label according to the majority vote. When , Stone (1977) proved in 1977 the universal consistency of this classifier: its probability of error converges to the Bayes error, whatever the distribution of . We show in this paper that this result is no longer valid in general metric spaces. However, if is separable and if some regularity condition is assumed, then the -nearest neighbor classifier is weakly consistent.
Classification : 62H30
Mots clés : classification, consistency, non parametric statistics
@article{PS_2006__10__340_0,
author = {C\'erou, Fr\'ed\'eric and Guyader, Arnaud},
title = {Nearest neighbor classification in infinite dimension},
journal = {ESAIM: Probability and Statistics},
pages = {340--355},
publisher = {EDP-Sciences},
volume = {10},
year = {2006},
doi = {10.1051/ps:2006014},
mrnumber = {2247925},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2006014/}
}
TY - JOUR AU - Cérou, Frédéric AU - Guyader, Arnaud TI - Nearest neighbor classification in infinite dimension JO - ESAIM: Probability and Statistics PY - 2006 DA - 2006/// SP - 340 EP - 355 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006014/ UR - https://www.ams.org/mathscinet-getitem?mr=2247925 UR - https://doi.org/10.1051/ps:2006014 DO - 10.1051/ps:2006014 LA - en ID - PS_2006__10__340_0 ER -
Cérou, Frédéric; Guyader, Arnaud. Nearest neighbor classification in infinite dimension. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 340-355. doi : 10.1051/ps:2006014. http://www.numdam.org/articles/10.1051/ps:2006014/
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