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On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

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  • Biau, Gérard
  • Devroye, Luc

Abstract

Let be identically distributed random vectors in , independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from , one may estimate the regression function by the LNN estimate , defined as an average over the Yi's corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n-->[infinity], for almost all and all p>=1, and discuss the links between rn and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.

Suggested Citation

  • Biau, Gérard & Devroye, Luc, 2010. "On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2499-2518, November.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:10:p:2499-2518
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    References listed on IDEAS

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    1. Lin, Yi & Jeon, Yongho, 2006. "Random Forests and Adaptive Nearest Neighbors," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 578-590, June.
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    2. Uguccioni, James, 2022. "The long-run effects of parental unemployment in childhood," CLEF Working Paper Series 45, Canadian Labour Economics Forum (CLEF), University of Waterloo.
    3. Jincheng Shen & Lu Wang & Jeremy M. G. Taylor, 2017. "Estimation of the optimal regime in treatment of prostate cancer recurrence from observational data using flexible weighting models," Biometrics, The International Biometric Society, vol. 73(2), pages 635-645, June.
    4. Kudraszow, Nadia L. & Vieu, Philippe, 2013. "Uniform consistency of kNN regressors for functional variables," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1863-1870.
    5. Tung Duy Luu & Jalal Fadili & Christophe Chesneau, 2021. "Sampling from Non-smooth Distributions Through Langevin Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1173-1201, December.
    6. Zhexiao Lin & Fang Han, 2022. "On regression-adjusted imputation estimators of the average treatment effect," Papers 2212.05424, arXiv.org, revised Jan 2023.
    7. Timothy I. Cannings & Richard J. Samworth, 2017. "Random-projection ensemble classification," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(4), pages 959-1035, September.
    8. Biau, Gérard & Devroye, Luc & Dujmović, Vida & Krzyżak, Adam, 2012. "An affine invariant k-nearest neighbor regression estimate," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 24-34.
    9. Susan Athey & Julie Tibshirani & Stefan Wager, 2016. "Generalized Random Forests," Papers 1610.01271, arXiv.org, revised Apr 2018.
    10. Ramosaj, Burim & Pauly, Markus, 2019. "Consistent estimation of residual variance with random forest Out-Of-Bag errors," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 49-57.
    11. Marie-Hélène Roy & Denis Larocque, 2012. "Robustness of random forests for regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 993-1006, December.
    12. Gérard Biau & Erwan Scornet, 2016. "A random forest guided tour," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 197-227, June.
    13. Guoyi Zhang & Yan Lu, 2012. "Bias-corrected random forests in regression," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(1), pages 151-160, March.
    14. Paola Zuccolotto & Marco Sandri & Marica Manisera, 2023. "Spatial performance analysis in basketball with CART, random forest and extremely randomized trees," Annals of Operations Research, Springer, vol. 325(1), pages 495-519, June.
    15. Luu, Tung Duy & Fadili, Jalal & Chesneau, Christophe, 2019. "PAC-Bayesian risk bounds for group-analysis sparse regression by exponential weighting," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 209-233.
    16. Mendez, Guillermo & Lohr, Sharon, 2011. "Estimating residual variance in random forest regression," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 2937-2950, November.
    17. Mathlouthi, Walid & Fredette, Marc & Larocque, Denis, 2015. "Regression trees and forests for non-homogeneous Poisson processes," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 204-211.

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