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Flexible variable selection for recovering sparsity in nonadditive nonparametric models

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  • Zaili Fang
  • Inyoung Kim
  • Patrick Schaumont

Abstract

Variable selection for recovering sparsity in nonadditive and nonparametric models with high‐dimensional variables has been challenging. This problem becomes even more difficult due to complications in modeling unknown interaction terms among high‐dimensional variables. There is currently no variable selection method to overcome these limitations. Hence, in this article we propose a variable selection approach that is developed by connecting a kernel machine with the nonparametric regression model. The advantages of our approach are that it can: (i) recover the sparsity; (ii) automatically model unknown and complicated interactions; (iii) connect with several existing approaches including linear nonnegative garrote and multiple kernel learning; and (iv) provide flexibility for both additive and nonadditive nonparametric models. Our approach can be viewed as a nonlinear version of a nonnegative garrote method. We model the smoothing function by a Least Squares Kernel Machine (LSKM) and construct the nonnegative garrote objective function as the function of the sparse scale parameters of kernel machine to recover sparsity of input variables whose relevances to the response are measured by the scale parameters. We also provide the asymptotic properties of our approach. We show that sparsistency is satisfied with consistent initial kernel function coefficients under certain conditions. An efficient coordinate descent/backfitting algorithm is developed. A resampling procedure for our variable selection methodology is also proposed to improve the power.

Suggested Citation

  • Zaili Fang & Inyoung Kim & Patrick Schaumont, 2016. "Flexible variable selection for recovering sparsity in nonadditive nonparametric models," Biometrics, The International Biometric Society, vol. 72(4), pages 1155-1163, December.
  • Handle: RePEc:bla:biomet:v:72:y:2016:i:4:p:1155-1163
    DOI: 10.1111/biom.12518
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    References listed on IDEAS

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    1. Arnab Maity & Xihong Lin, 2011. "Powerful Tests for Detecting a Gene Effect in the Presence of Possible Gene–Gene Interactions Using Garrote Kernel Machines," Biometrics, The International Biometric Society, vol. 67(4), pages 1271-1284, December.
    2. Radchenko, Peter & James, Gareth M., 2010. "Variable Selection Using Adaptive Nonlinear Interaction Structures in High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1541-1553.
    3. Dawei Liu & Xihong Lin & Debashis Ghosh, 2007. "Semiparametric Regression of Multidimensional Genetic Pathway Data: Least-Squares Kernel Machines and Linear Mixed Models," Biometrics, The International Biometric Society, vol. 63(4), pages 1079-1088, December.
    4. Pradeep Ravikumar & John Lafferty & Han Liu & Larry Wasserman, 2009. "Sparse additive models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 1009-1030, November.
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    Cited by:

    1. Lulu Cheng & Inyoung Kim & Herbert Pang, 2016. "Bayesian Semiparametric Model for Pathway-Based Analysis with Zero-Inflated Clinical Outcomes," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(4), pages 641-662, December.
    2. Zaili Fang & Inyoung Kim & Jeesun Jung, 2018. "Semiparametric Kernel-Based Regression for Evaluating Interaction Between Pathway Effect and Covariate," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 23(1), pages 129-152, March.

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