3,616
Views
48
CrossRef citations to date
0
Altmetric
Theory and Methods

Model Selection for High-Dimensional Quadratic Regression via Regularization

, &
Pages 615-625 | Received 01 May 2015, Published online: 08 Feb 2018

References

  • Bien, J., Taylor, J., and Tibshirani, R. (2013), “A Lasso for Hierarchical Interactions,” The Annals of Statistics, 41, 1111–1141.
  • Chen, J., and Chen, Z. (2008), “Extended Bayesian Information Criteria for Model Selection with Large Model Spaces,” Biometrika, 95, 759–771.
  • Cheng, M.-Y., Honda, T., Li, J., and Peng, H. (2014), “Nonparametric Independence Screening and Structure Identification for Ultra-High Dimensional Longitudinal Data,” The Annals of Statistics, 42, 1819–1849.
  • Chipman, H., Hamada, M., and Wu, C. F. J. (1997), “A Bayesian Variable-Selection Approach for Analyzing Designed Experiments with Complex Aliasing,” Technometrics, 39, 372–381.
  • Choi, N. H., Li, W., and Zhu, J. (2010), “Variable Selection with the Strong Heredity Constraint and its Oracle Property,” Journal of the American Statistical Association, 105, 354–364.
  • Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004), “Least Angle Regression,” The Annals of Statistics, 32, 407–451.
  • Fan, J., Feng, Y., and Song, R. (2011), “Nonparametric Independence Screening in Sparse Ultra-High Dimensional Additive Models,” Journal of the American Statistical Association, 106, 544–557.
  • Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties,” Journal of the American Statistical Association, 96, 1348–1360.
  • Fan, J., and Lv, J. (2008), “Sure Independence Screening for Ultrahigh Dimensional Feature Space,” Journal of the Royal Statistical Society, Series B, 70, 849–911.
  • ——— (2011), “Nonconcave Penalized Likelihood with np-Dimensionality,” IEEE Transactions on Information Theory, 57, 5467–5484.
  • Fan, Y., Kong, Y., Li, D., and Lv, J. (2016), “Interaction Pursuit with Feature Screening and Selection,” arXiv:1605.08933.
  • Fan, Y., Kong, Y., Li, D., and Zheng, Z. (2015), “Innovated Interaction Screening for High-Dimensional Nonlinear Classification,” The Annals of Statistics, 43, 1243–1272.
  • Fan, Y., and Lv, J. (2013), “Asymptotic Equivalence of Regularization methods in Thresholded Parameter Space,” Journal of the American Statistical Association, 108, 1044–1061.
  • Fan, Y., and Tang, C. Y. (2013), “Tuning Parameter Selection in High Dimensional Penalized Likelihood,” Journal of the Royal Statistical Society, Series B, 75, 531–552.
  • Friedman, J., Hastie, T., Höfling, H., and Tibshirani, R. (2007), “Pathwise Coordinate Optimization,” Annals of Applied Statistics, 1, 302–332.
  • Friedman, J., Hastie, T., and Tibshirani, R. (2010), “Regularization Paths for Generalized Linear Models via Coordinate Descent,” Journal of Statistical Software, 33, 1–22.
  • Hamada, M., and Wu, C. F. J. (1992), “Analysis of Designed Experiments with Complex Aliasing,” Journal of Quality Technology, 24, 130–137.
  • Hao, N., and Zhang, H. H. (2014), “Interaction Screening for Ultra-High Dimensional Data,” Journal of the American Statistical Association, 109, 1285–1301.
  • ——— (2017), “A Note on High Dimensional Linear Regression with Interactions,” The American Statistician, to appear.
  • Jiang, B., and Liu, J. S. (2014), “Variable Selection for General Index Models via Sliced Inverse Regression,” The Annals of Statistics, 42, 1751–1786.
  • Kong, Y., Li, D., Fan, Y., and Lv, J. (2016), “Interaction Pursuit in High-Dimensional Multi-Response Regression via Distance Correlation,” The Annals of Statistics, 45, 897–922.
  • McCullagh, P., and Nelder, J. (1989), Generalized Linear Models (Monographs on Statistics and Applied Probability), Boca Raton, FL: Chapman and Hall.
  • Nelder, J. A. (1977), “A Reformulation of Linear Models,” Journal of the Royal Statistical Society, Series A, 140, 48–77.
  • Park, M. Y., and Hastie, T. (2007), “L1-Regularization Path Algorithm for Generalized Linear Models,” Journal of the Royal Statistical Society, Series B, 69, 659–677.
  • Peixoto, J. L. (1987), “Hierarchical Variable Selection in Polynomial Regression Models,” The American Statistician, 41, 311–313.
  • Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288.
  • Wainwright, M. J. (2009), “Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using-Constrained Quadratic Programming (lasso),” IEEE Transactions on Information Theory, 55, 2183–2202.
  • Wang, H. (2009), “Forward Regression for Ultra-High Dimensional Variable Screening,” Journal of the American Statistical Association, 104, 1512–1524.
  • Wu, T. T., Chen, Y. F., Hastie, T., Sobel, E., and Lange, K. (2009), “Genome-Wide Association Analysis by Lasso Penalized Logistic Regression,” Bioinformatics, 25, 714–721.
  • Wu, T. T., and Lange, K. (2008), “Coordinate Descent Algorithms for Lasso Penalized Regression,” Annals of Applied Statistics, 2, 224–244.
  • Wu, Y. (2011), “An Ordinary Differential Equation-Based Solution Path Algorithm,” Journal of Nonparametric Statistics, 23, 185–199.
  • Yu, Y., and Feng, Y. (2014), “Apple: Approximate Path for Penalized Likelihood Estimators,” Statistics and Computing, 24, 803–819.
  • Yuan, M., Joseph, V. R., and Zou, H. (2009), “Structured Variable Selection and Estimation,” Annals of Applied Statistics, 3, 1738–1757.
  • Zhang, C.-H. (2010), “Nearly Unbiased Variable Selection under Minimax Concave Penalty,” Annals of Statistics, 38, 894–942.
  • Zhao, P., Rocha, G., and Yu, B. (2009), “The Composite Absolute Penalties Family for Grouped and Hierarchical Variable Selection,” Annals of Statistics, 37, 3468–3497.
  • Zhao, P., and Yu, B. (2006), “On Model Selection Consistency of Lasso,” Journal of Machine Learning Research, 7, 2541–2563.
  • Zhou, H., and Wu, Y. (2014), “A Generic Path Algorithm for Regularized Statistical Estimation,” Journal of the American Statistical Association, 109, 686–699.
  • Zou, H., and Hastie, T. (2005), “Regularization and Variable Selection via the Elastic Net,” Journal of the Royal Statistical Society, Series B, 67, 301–320.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.