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Dimension reduction based on constrained canonical correlation and variable filtering. (English) Zbl 1142.62045
Summary: The “curse of dimensionality” has remained a challenge for high-dimensional data analysis in statistics. The sliced inverse regression (SIR) and canonical correlation (CANCOR) methods aim to reduce the dimensionality of data by replacing the explanatory variables with a small number of composite directions without losing much information. However, the estimated composite directions generally involve all of the variables, making their interpretation difficult. To simplify the direction estimates, L. Ni, R. D. Cook and C.-L. Tsai [Biometrika 92, No. 1, 242–247 (2005; Zbl 1068.62080)] proposed the shrinkage sliced inverse regression (SSIR) based on SIR. We propose the constrained canonical correlation $$(C^{3})$$ method based on CANCOR, followed by a simple variable filtering method. As a result, each composite direction consists of a subset of the variables for interpretability as well as predictive power. The proposed method aims to identify simple structures without sacrificing the desirable properties of the unconstrained CANCOR estimates. The simulation studies demonstrate the performance advantage of the proposed $$C^{3}$$ method over the SSIR method. We also use the proposed method in two examples for illustration.

MSC:
 62J07 Ridge regression; shrinkage estimators (Lasso) 62H20 Measures of association (correlation, canonical correlation, etc.) 62G08 Nonparametric regression and quantile regression 65C60 Computational problems in statistics (MSC2010)
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