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Variance reduction in purely random forests. (English) Zbl 1254.62050
Summary: Random forests (RFs), introduced by L. Breiman [Mach. Learn. 45, No. 1, 5–32 (2001; Zbl 1007.68152)], are a very effective statistical method. The complex mechanism of the method makes the theoretical analysis difficult. Therefore, simplified versions of RF, called purely RFs (PRFs), which can be theoretically handled more easily, have been considered. We study the variance of such forests. First, we show a general upper bound which emphasises the fact that a forest reduces the variance. We then introduce a simple variant of PRFs, that we call purely uniformly RFs. For this variant and in the context of regression problems with a one-dimensional predictor space, we show that both random trees and RFs reach the minimax rate of convergence. In addition, we prove that compared with random trees, RFs improve accuracy by reducing the estimator variance by a factor of three-fourths.

62G08 Nonparametric regression and quantile regression
68T05 Learning and adaptive systems in artificial intelligence
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
Full Text: DOI
[1] Arlot, S. (2008), ’V-Fold Cross-Validation Improved: V-Fold Penalization’, preprint, arXiv:0802.0566v2.
[2] Biau, G. (2012), ’Analysis of a random forests model’,Journal of Machine Learning Research, in press. · Zbl 1283.62127
[3] Biau G., Journal of Machine Learning Research 9 pp 2039– (2008)
[4] DOI: 10.1023/A:1010933404324 · Zbl 1007.68152 · doi:10.1023/A:1010933404324
[5] Cutler A., Computing Science and Statistics 33 pp 490– (2001)
[6] DOI: 10.1002/0471722162 · Zbl 1053.62060 · doi:10.1002/0471722162
[7] Díaz-Uriarte, R., and Alvarez de Andrés, S. (2006), ’Gene Selection and Classification of Microarray Data using Random Forest’,BMC Bioinformatics, 7, 3.
[8] DOI: 10.1016/j.patrec.2010.03.014 · doi:10.1016/j.patrec.2010.03.014
[9] DOI: 10.1186/1471-2156-11-49 · doi:10.1186/1471-2156-11-49
[10] Graham R. L., Concrete Mathematics (1989)
[11] DOI: 10.1007/978-0-387-84858-7 · Zbl 1273.62005 · doi:10.1007/978-0-387-84858-7
[12] Ibragimov I. A., Statistical Estimation: Asymptotic Theory (1981) · doi:10.1007/978-1-4899-0027-2
[13] Liaw A., R News 2 (3) pp 18– (2002)
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