The ``curse of dimensionality'' has remained a challenge for high-dimensionaldata analysis in statistics. The sliced inverse regression (SIR) and canonicalcorrelation (CANCOR) methods aim to reduce the dimensionality of data byreplacing the explanatory variables with a small number of composite directionswithout losing much information. However, the estimated composite directionsgenerally 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 ($C^3$) methodbased on CANCOR, followed by a simple variable filtering method. As a result,each composite direction consists of a subset of the variables forinterpretability as well as predictive power. The proposed method aims toidentify simple structures without sacrificing the desirable properties of theunconstrained CANCOR estimates. The simulation studies demonstrate theperformance advantage of the proposed $C^3$ method over the SSIR method. Wealso use the proposed method in two examples for illustration.
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机译：维度维仍然是统计学中高维数据分析的挑战。切片逆回归（SIR）和典范相关（CANCOR）方法旨在通过用少量合成方向替换解释变量而不会丢失太多信息，从而降低数据的维数。但是，估计的合成方向通常包含所有变量，因此难以解释。为简化方向估计，Ni，Cook和Tsai [Biometrika 92（2005）242--247]提出了基于本文提出了一种基于CANCOR的约束正则相关（$ C ^ 3 $）方法，然后提出了一种简单的变量滤波方法。结果，每个合成方向都由变量的子集组成，这些变量具有可解释性和预测能力。所提出的方法旨在识别简单的结构而不牺牲不受约束的CANCOR估计的期望特性。仿真研究证明了所提出的$ C ^ 3 $方法相对于SSIR方法的性能优势。我们还将在两个示例中使用提出的方法进行说明。
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