Lectures on the Nearest Neighbor Method (Springer Series in by Gérard Biau, Luc Devroye

By Gérard Biau, Luc Devroye

This textual content offers a wide-ranging and rigorous evaluate of nearest neighbor tools, probably the most very important paradigms in desktop studying. Now in a single self-contained quantity, this booklet systematically covers key statistical, probabilistic, combinatorial and geometric principles for knowing, reading and constructing nearest neighbor methods.

Gérard Biau is a professor at Université Pierre et Marie Curie (Paris). Luc Devroye is a professor on the university of computing device technology at McGill collage (Montreal).   

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X/ x…A? x…A? Ä max ! sup x…A? x/; sup x…A? x/ xk: Therefore, ! x/ ; nVd x…A? x/ x…A? x/ is the k-nearest neighbor estimate for Y1 ; : : : ; Yn (since, for x … A? x/ < 1). x/ x…A? x// Ä max k ; 2dC1 kgk1 nVd Ã Â Ä max à k ;" : nVd Since k=n ! x/ x…A? x/j Ä ": Next, partition A? D Œ a 1; a C 1d into nd equal squares of volume . 2aC2 /d n d each. Denote these squares by Ci , 1 Ä i Ä n , and let G D fx1 ; : : : ; xnd g be the collection of their centers. xi /j Ä " for all n large enough. x/ x2A? 2a C 2/=n !

0/ k D 1 p C k  Ã2 ! 1. W for some random variable W that is not identically 0 with probability one. 0/ n ˛=2 ! 0/ n2˛ 2 ! 0/ n 2=5 ! 0/ D D if 0 < ˛ < C if 2 4 5 4 5 <˛<1 if ˛ D 45 . 0/ ¤ 0, the best possible rate for the k-nearest neighbor estimate is n4=5 , > 0. n 2=5 / is achievable. However, without further conditions on g, one cannot precisely determine the best possible rate. This leaves us with the choice of . 0/ : cg for constant c, or minimize EjWjˇ for suitable One can opt to minimize PfjWj ˇ.

X; K// > " C 1n . x; K// " 1, and thus, for all n large enough, it is possible to choose such a K. We conclude by the Borel-Cantelli lemma. 1). We leave the proof of the last part to the interested reader. x/ < 1: n 1 1ÄjÄn For the first part, let x be a Lebesgue point of f . x; ı// H. x; ı// 0<ıÄ and recall that H. / ! 0 as # 0. 1/. x/ <1 with probability one: Therefore, II ! 0 almost surely. x// < 1 with probability one. 1 where K" is any constant strictly larger than H. x; // D 2": If x belongs to the support of , then, clearly, K" is as small as desired by choice of ".

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