A weighted k-nearest neighbor density estimate for geometric inference

By Gérard Biau, Frédéric Chazal, David Cohen-Steiner, Luc Devroye, Carlos W. Rodríguez. 2011

79 citations recorded by the source corpus.

Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the k-nearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong approximation result is obtained and the choice of the optimal set of weights is discussed. In particular, the classical k-nearest neighbor estimate is not optimal in a sense described in the manuscript. The proposed method has been implemented to recover level sets in both simulated and real-life data.

Subfields: Computational Theory and Mathematics, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6270ddc7272fd0642a34d30b46bc1bcf8bbc8c2f

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Citations in the corpus

Automated Mapping of Phenotype Space with Single-Cell Data (2016, methodology): “Density estimation was computed as described in 5 , setting nu=const and, p=1 Where n is the size of the dataset, V d is the volume of a unit sphere in d dimensions, X j (x) represents the j-th nearest neighbor of x, d is the number of dimensions and |x - y| is the distance between x and y.”
Fast k-Nearest Neighbour Search via Prioritized DCI (2017, methodology): “The method of k -nearest neighbours is a fundamental building block of many machine learning algorithms and also has broad applications beyond artiﬁcial intelligence, including in statistics, bioinformatics and database systems, e.g. (Biau et al., 2011; Behnam et al., 2013; El-dawy & Mokbel, 2015).”
Demystifying fixed k-nearest neighbor information estimators (2016, background): “[6] Gérard Biau, Frédéric Chazal, David Cohen-Steiner, Luc Devroye, Carlos Rodriguez, et al.”
A Robust-Equitable Measure for Feature Ranking and Selection (2017, methodology): “Using Taylor expansion, we have (from last line page 228 in Biau et al. (2011))∫ B(Z;r) c(z)dz = c(Z)vdr d + Q̃(Z)vdr d+2 + o(rd+2).”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “…has motivated further works and applications in various directions such as topological data analysis (Buchet et al., 2015a), GPS traces analysis (Chazal et al., 2011a), density estimation (Biau et al., 2011), hypothesis testing Brécheteau (2017), clustering (Chazal et al., 2013) just to name a few.”