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On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification. (English) Zbl 1198.62048
Summary: Let $$X_1, \dots , X_n$$ be identically distributed random vectors in $$\mathbb R^d$$, independently drawn according to some probability density. An observation $$X_i$$ is said to be a layered nearest neighbour (LNN) of a point $$x$$ if the hyperrectangle defined by $$x$$ and $$X_i$$ contains no other data points. We first establish consistency results on $$L_n(x)$$, the number of LNN of $$x$$. Then, given a sample $$(X, Y), (X_1, Y_1), \dots , (X_n, Y_n)$$ of independent identically distributed random vectors from $$\mathbb R^d \times \mathbb R$$, one may estimate the regression function $$r(x) = \mathbb E[Y|X = x]$$ by the LNN estimate $$r_n(x)$$, defined as an average over the $$Y_i$$’s corresponding to those $$X_i$$ which are LNN of $$x$$. Under mild conditions on $$r$$, we establish the consistency of $$\mathbb E|r_n(x) -r(x)|^p$$ towards 0 as $$n \rightarrow \infty$$, for almost all $$x$$ and all $$p\geq 1$$, and discuss the links between $$r_n$$ and the random forest estimates of L. Breiman [Mach. Learn. 45, No. 1, 5–32 (2001; Zbl 1007.68152)]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.

##### MSC:
 62H12 Estimation in multivariate analysis 62G20 Asymptotic properties of nonparametric inference 62G08 Nonparametric regression and quantile regression 62H30 Classification and discrimination; cluster analysis (statistical aspects)
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##### References:
 [1] Amit, Y.; Geman, D., Shape quantization and recognition with randomized trees, Neural computation, 9, 1545-1588, (1997) [2] Bai, Z.-D.; Chao, C.-C.; Hwang, H.-K.; Liang, W.-Q., On the variance of the number of maxima in random vectors and its applications, The annals of applied probability, 8, 886-895, (1998) · Zbl 0941.60021 [3] Bai, Z.-D.; Devroye, L.; Hwang, H.-K.; Tsai, T.-S., Maxima in hypercubes, Random structures and algorithms, 27, 290-309, (2005) · Zbl 1080.60007 [4] Barndorff-Nielsen, O.; Sobel, M., On the distribution of admissible points in a vector random sample, Theory of probability and its applications, 11, 249-269, (1966) · Zbl 0278.60007 [5] Biau, G.; Devroye, L.; Lugosi, G., Consistency of random forests and other averaging classifiers, Journal of machine learning research, 9, 2015-2033, (2008) · Zbl 1225.62081 [6] Breiman, L., Bagging predictors, Machine learning, 24, 123-140, (1996) · Zbl 0858.68080 [7] L. Breiman, Some infinite theory for predictor ensembles, Technical Report 577, Statistics Department, UC Berkeley, 2000. http://www.stat.berkeley.edu/ breiman. [8] Breiman, L., Random forests, Machine learning, 45, 5-32, (2001) · Zbl 1007.68152 [9] L. Breiman, Consistency for a simple model of random forests, Technical Report 670, Statistics Department, UC Berkeley, 2004. http://www.stat.berkeley.edu/ breiman. [10] Cover, T.M.; Hart, P.E., Nearest neighbour pattern classification, IEEE transactions on information theory, 13, 21-27, (1967) · Zbl 0154.44505 [11] Cutler, A.; Zhao, G., Pert—perfect random tree ensembles, Computing science and statistics, 33, 490-497, (2001) [12] Devroye, L., The uniform convergence of nearest neighbour regression function estimators and their application in optimization, IEEE transactions on information theory, 24, 142-151, (1978) · Zbl 0375.62083 [13] Devroye, L.; Györfi, L.; Lugosi, G., A probabilistic theory of pattern recognition, (1996), Springer-Verlag New York · Zbl 0853.68150 [14] Dietterich, T.G., An experimental comparison of three methods for constructing ensembles of decision trees: bagging, boosting, and randomization, Machine learning, 40, 139-157, (2000) [15] Doeblin, W., Exposé de la théorie des chaînes simples constantes de Markov à un nombre fini d’états, Revue mathématique de l’union interbalkanique, 2, 77-105, (1937) [16] E. Fix, J. Hodges, Discriminatory analysis. Nonparametric discrimination: Consistency properties, Technical Report 4, Project Number 21-49-004, USAF School of Aviation Medicine, Randolph Field, Texas, 1951. · Zbl 0715.62080 [17] Györfi, L.; Kohler, M.; Krzyżak, A.; Walk, H., A distribution-free theory of nonparametric regression, (2002), Springer-Verlag New York · Zbl 1021.62024 [18] Lin, Y.; Jeon, Y., Random forests and adaptive nearest neighbors, Journal of the American statistical association, 101, 578-590, (2006) · Zbl 1119.62304 [19] Marcinkiewicz, J.; Zygmund, A., Sur LES fonctions indépendantes, Fundamenta mathematicae, 29, 60-90, (1937) · JFM 63.0946.02 [20] Petrov, V.V., Sums of independent random variables, (1975), Springer-Verlag Berlin · Zbl 0322.60043 [21] Rachev, S.T.; Rüschendorf, L., Mass transportation problems, volume I: theory, (1998), Springer New York · Zbl 0990.60500 [22] Steele, B.M., Exact bootstrap $$k$$-nearest neighbor learners, Machine learning, 74, 235-255, (2009) [23] Stone, C.J., Consistent nonparametric regression, The annals of statistics, 5, 595-645, (1977) · Zbl 0366.62051 [24] Wheeden, R.L.; Zygmund, A., Measure and integral. an introduction to real analysis, (1977), Marcel Dekker New York · Zbl 0362.26004
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