Variable selection through CART
ESAIM: Probability and Statistics, Volume 18 (2014), pp. 770-798.

This paper deals with variable selection in regression and binary classification frameworks. It proposes an automatic and exhaustive procedure which relies on the use of the CART algorithm and on model selection via penalization. This work, of theoretical nature, aims at determining adequate penalties, i.e. penalties which allow achievement of oracle type inequalities justifying the performance of the proposed procedure. Since the exhaustive procedure cannot be realized when the number of variables is too large, a more practical procedure is also proposed and still theoretically validated. A simulation study completes the theoretical results.

DOI: 10.1051/ps/2014006
Classification: 62G05,  62G07,  62G20
Keywords: binary classification, CART, model selection, penalization, regression, variable selection
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Sauve, Marie; Tuleau-Malot, Christine. Variable selection through CART. ESAIM: Probability and Statistics, Volume 18 (2014), pp. 770-798. doi : 10.1051/ps/2014006. http://www.numdam.org/articles/10.1051/ps/2014006/

[1] S. Arlot and P. Bartlett, Margin adaptive model selection in statistical learning. Bernoulli 17 (2011) 687-713. | MR | Zbl

[2] L. Birgé and P. Massart, Minimal penalties for gaussian model selection. Probab. Theory Relat. Fields 138 (2007) 33-73. | MR | Zbl

[3] L. Breiman, Random forests. Mach. Learn. 45 (2001) 5-32. | Zbl

[4] L. Breiman and A. Cutler, Random forests. http://www.stat.berkeley.edu/users/breiman/RandomForests/ (2005).

[5] L. Breiman, J. Friedman, R. Olshen and C. Stone, Classification and Regression Trees. Chapman et Hall (1984). | MR | Zbl

[6] R. Díaz-Uriarte and S. Alvarez De Andrés, Gene selection and classification of microarray data using random forest. BMC Bioinform. 7 (2006) 1-13.

[7] B. Efron, T. Hastie, I. Johnstone and R. Tibshirani, Least angle regression. Ann. Stat. 32 (2004) 407-499. | MR | Zbl

[8] J. Fan and J. Lv, A selective overview of variable selection in high dimensional feature space. Stat. Sin. 20 (2010) 101-148. | MR | Zbl

[9] G.M. Furnival and R.W. Wilson, Regression by leaps and bounds. Technometrics 16 (1974) 499-511. | Zbl

[10] R. Genuer, J.M. Poggi and C. Tuleau-Malot, Variable selection using random forests. Pattern Recognit. Lett. 31 (2010) 2225-2236.

[11] S. Gey, Margin adaptive risk bounds for classification trees, hal-00362281.

[12] S. Gey and E. Nédélec, Model Selection for CART Regression Trees. IEEE Trans. Inf. Theory 51 (2005) 658-670. | MR | Zbl

[13] B. Ghattas and A. Ben Ishak, Sélection de variables pour la classification binaire en grande dimension: comparaisons et application aux données de biopuces. Journal de la société française de statistique 149 (2008) 43-66. | EuDML | MR

[14] U. Grömping, Estimators of relative importance in linear regression based on variance decomposition. The American Statistician 61 (2007) 139-147. | MR

[15] I. Guyon and A. Elisseff, An introduction to variable and feature selection. J. Mach. Learn. Res. 3 (2003) 1157-1182. | Zbl

[16] I. Guyon, J. Weston, S. Barnhill and V.N. Vapnik, Gene selection for cancer classification using support vector machines. Mach. Learn. 46 (2002) 389-422. | Zbl

[17] T. Hastié, R. Tibshirani and J. Friedman, The Elements of Statistical Learning. Springer (2001). | MR | Zbl

[18] T. Hesterberg, N.H. Choi, L. Meier and C. Fraley, Least angle regresion and l1 penalized regression: A review. Stat. Surv. 2 (2008) 61-93. | MR | Zbl

[19] R. Kohavi and G.H. John, Wrappers for feature subset selection. Artificial Intelligence 97 (1997) 273-324. | Zbl

[20] V. Koltchinskii, Local rademacher complexities and oracle inequalities in risk minimization. Ann. Stat. 34 (2004) 2593-2656. | MR | Zbl

[21] E. Mammen and A. Tsybakov, Smooth discrimination analysis. Ann. Stat. 27 (1999) 1808-1829. | MR | Zbl

[22] P. Massart, Some applications of concentration inequalities to statistics. Annales de la faculté des sciences de Toulouse 2 (2000) 245-303. | EuDML | Numdam | MR | Zbl

[23] P. Massart, Concentration Inequlaities and Model Selection. Lect. Notes Math. Springer (2003). | Zbl

[24] P. Massart and E. Nédélec, Risk bounds for statistical learning. Ann. Stat. 34 (2006). | MR | Zbl

[25] J.M. Poggi and C. Tuleau, Classification supervisée en grande dimension. Application à l'agrément de conduite automobile. Revue de Statistique Appliquée LIV (2006) 41-60.

[26] E. Rio, Une inégalité de bennett pour les maxima de processus empiriques. Ann. Inst. Henri Poincaré, Probab. Stat. 38 (2002) 1053-1057. | EuDML | Numdam | MR | Zbl

[27] A. Saltelli, K. Chan and M. Scott, Sensitivity Analysis. Wiley (2000). | MR | Zbl

[28] M. Sauvé, Histogram selection in non gaussian regression. ESAIM PS 13 (2009) 70-86. | Numdam | MR | Zbl

[29] M. Sauvé and C. Tuleau-Malot, Variable selection through CART, hal-00551375.

[30] I.M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Mod. Comput. Experiment 1 (1993) 271-280. | MR | Zbl

[31] R. Tibshirani, Regression shrinkage and selection via Lasso. J. R. Stat. Soc. Ser. B 58 (1996) 267-288. | MR | Zbl

[32] A.B. Tsybakov, Optimal aggregation of classifiers in statistical learning. Ann. Stat. 32 (2004) 135-166. | MR | Zbl

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