The Maximum Separation Subspace in Sufficient Dimension Reduction with Categorical Response
The Maximum Separation Subspace in Sufficient Dimension Reduction with Categorical Response
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发表时间：
2020
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通讯作者：
Xin Zhang;Qing Mai;H. Zou
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作者：
Xin Zhang;Qing Mai;H. Zou
充分降维（SDR）是回归中探索性分析和数据可视化的一个非常有用的概念，特别是当协变量的数量很大时。许多SDR方法已经被提出用于具有连续响应的回归，其中中心子空间（CS）是估计的目标。施加各种条件，例如线性条件和恒定协方差条件，使得这些方法可以估计CS的至少一部分。本文研究了具有分类响应的回归和判别分析的SDR。出于SDR的探索性分析和数据可视化方面，我们提出了一个新的几何框架来重新制定SDR问题的流形优化，并引入了一个新的概念，称为最大分离子空间（MASES）。MASES自然地保留了SDR中的“充分性”，而没有对预测分布施加额外的条件，并直接激发了半参数估计。数值研究表明，MASES表现出上级性能相比，竞争SDR方法在特定的设置。
Sufficient dimension reduction (SDR) is a very useful concept for exploratory analysis and data visualization in regression, especially when the number of covariates is large. Many SDR methods have been proposed for regression with a continuous response, where the central subspace (CS) is the target of estimation. Various conditions, such as the linearity condition and the constant covariance condition, are imposed so that these methods can estimate at least a portion of the CS. In this paper we study SDR for regression and discriminant analysis with categorical response. Motivated by the exploratory analysis and data visualization aspects of SDR, we propose a new geometric framework to reformulate the SDR problem in terms of manifold optimization and introduce a new concept called Maximum Separation Subspace (MASES). The MASES naturally preserves the “sufficiency” in SDR without imposing additional conditions on the predictor distribution, and directly inspires a semi-parametric estimator. Numerical studies show MASES exhibits superior performance as compared with competing SDR methods in specific settings.