基于约束典型相关和变量滤波的维数约简
Dimension reduction based on constrained canonical correlation and variable filtering
摘要：
“维数灾难”仍然是统计学中高维数据分析的一个挑战。切片逆回归（SIR）和典型相关（CANCOR）方法旨在通过用少量复合方向代替解释变量来降低数据的维数，而不会丢失太多信息。然而，估计的复合方向通常涉及所有的变量，使得它们的解释困难。为了简化方向估计，Ni，Cook和Tsai [Biometrika 92（2005）242- 247]提出了基于SIR的收缩切片逆回归（SSIR）。在本文中，我们提出了约束典型相关（$C^3$）方法的基础上CANCOR，其次是一个简单的变量过滤方法。因此，每个复合方向由可解释性和预测能力的变量子集组成。该方法的目的是识别简单的结构，而不牺牲理想的性能的无约束CANCOR估计。仿真研究表明，所提出的$C^3$方法的SSIR方法的性能优势。我们还使用所提出的方法在两个例子中进行说明。
英文摘要：
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, Ni, Cook and Tsai [Biometrika 92 (2005) 242--247] proposed the shrinkage sliced inverse regression (SSIR) based on SIR. In this paper, 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.
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