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Best subset selection via cross-validation criterion

Author

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  • Yuichi Takano

    (University of Tsukuba)

  • Ryuhei Miyashiro

    (Tokyo University of Agriculture and Technology)

Abstract

This paper is concerned with the cross-validation criterion for selecting the best subset of explanatory variables in a linear regression model. In contrast with the use of statistical criteria (e.g., Mallows’ $$C_p$$Cp, the Akaike information criterion, and the Bayesian information criterion), cross-validation requires only mild assumptions, namely, that samples are identically distributed and that training and validation samples are independent. For this reason, the cross-validation criterion is expected to work well in most situations involving predictive methods. The purpose of this paper is to establish a mixed-integer optimization approach to selecting the best subset of explanatory variables via the cross-validation criterion. This subset-selection problem can be formulated as a bilevel MIO problem. We then reduce it to a single-level mixed-integer quadratic optimization problem, which can be solved exactly by using optimization software. The efficacy of our method is evaluated through simulation experiments by comparison with statistical-criterion-based exhaustive search algorithms and $$L_1$$L1-regularized regression. Our simulation results demonstrate that, when the signal-to-noise ratio was low, our method delivered good accuracy for both subset selection and prediction.

Suggested Citation

  • Yuichi Takano & Ryuhei Miyashiro, 2020. "Best subset selection via cross-validation criterion," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 475-488, July.
  • Handle: RePEc:spr:topjnl:v:28:y:2020:i:2:d:10.1007_s11750-020-00538-1
    DOI: 10.1007/s11750-020-00538-1
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    References listed on IDEAS

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    1. Toshiki Sato & Yuichi Takano & Ryuhei Miyashiro & Akiko Yoshise, 2016. "Feature subset selection for logistic regression via mixed integer optimization," Computational Optimization and Applications, Springer, vol. 64(3), pages 865-880, July.
    2. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
    3. Ryuta Tamura & Ken Kobayashi & Yuichi Takano & Ryuhei Miyashiro & Kazuhide Nakata & Tomomi Matsui, 2019. "Mixed integer quadratic optimization formulations for eliminating multicollinearity based on variance inflation factor," Journal of Global Optimization, Springer, vol. 73(2), pages 431-446, February.
    4. Friedman, Jerome H. & Hastie, Trevor & Tibshirani, Rob, 2010. "Regularization Paths for Generalized Linear Models via Coordinate Descent," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 33(i01).
    5. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    6. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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    Cited by:

    1. Serrano, Breno & Minner, Stefan & Schiffer, Maximilian & Vidal, Thibaut, 2024. "Bilevel optimization for feature selection in the data-driven newsvendor problem," European Journal of Operational Research, Elsevier, vol. 315(2), pages 703-714.
    2. Thompson, Ryan, 2022. "Robust subset selection," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    3. Tomokaze Shiratori & Ken Kobayashi & Yuichi Takano, 2020. "Prediction of hierarchical time series using structured regularization and its application to artificial neural networks," PLOS ONE, Public Library of Science, vol. 15(11), pages 1-23, November.
    4. Lorenz Kolley & Nicolas Rückert & Marvin Kastner & Carlos Jahn & Kathrin Fischer, 2023. "Robust berth scheduling using machine learning for vessel arrival time prediction," Flexible Services and Manufacturing Journal, Springer, vol. 35(1), pages 29-69, March.
    5. Andrés Gómez & Oleg A. Prokopyev, 2021. "A Mixed-Integer Fractional Optimization Approach to Best Subset Selection," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 551-565, May.

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