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**Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR.**
*(English)*
Zbl 1146.62051

Summary: Variable selection can be challenging, particularly in situations with a large number of predictors with possibly high correlations, such as gene expression data. In this article, a new method, called OSCAR (octagonal shrinkage and clustering algorithm for regression), is proposed to simultaneously select variables while grouping them into predictive clusters. In addition to improving prediction accuracy and interpretation, these resulting groups can then be investigated further to discover what contributes to the group having a similar behavior. The technique is based on penalized least squares with a geometrically intuitive penalty function that shrinks some coefficients to exactly zero. Additionally, this penalty yields exact equality of some coefficients, encouraging correlated predictors that have a similar effect on the response to form predictive clusters represented by a single coefficient. The proposed procedure is shown to compare favorably to the existing shrinkage and variable selection techniques in terms of both prediction error and model complexity, while yielding the additional grouping information.

### MSC:

62J07 | Ridge regression; shrinkage estimators (Lasso) |

62P12 | Applications of statistics to environmental and related topics |

65C60 | Computational problems in statistics (MSC2010) |

90C90 | Applications of mathematical programming |

62J05 | Linear regression; mixed models |

### Keywords:

correlation; penalization; predictive group; regression; shrinkage; supervised clustering; variable selection; Appalachian Mountains soil data
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\textit{H. D. Bondell} and \textit{B. J. Reich}, Biometrics 64, No. 1, 115--123 (2008; Zbl 1146.62051)

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