On data depth and distribution-free discriminant analysis using separating surfaces
On data depth and distribution-free discriminant analysis using separating surfaces
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DOI：
10.3150/bj/1110228239
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发表时间：
2005
期刊：
影响因子：
1.5
通讯作者：
A. Ghosh;P. Chaudhuri
中科院分区：
文献类型：
--
作者：
A. Ghosh;P. Chaudhuri
判别分析中一个非常著名的传统方法是使用一些测量变量的线性（或非线性）组合来提高类的可分性。例如，线性（或二次）分类器找到测量变量的线性投影（或二次函数），使类之间的分离最大化。这些技术对于获得类可分离性的良好的低维视图非常有用。Fisher的判别分析主要由多元正态分布驱动，使用训练样本的一阶和二阶矩来构建这样的分类器。然而，这些估计对异常值高度敏感，并且它们对重尾分布不可靠。本文研究了两种基于统计深度函数概念的无分布线性分类方法。其中一个分类器与Tukey的半空间深度密切相关，而另一个分类器则基于回归深度的概念。这两种方法都可以推广到构造非线性曲面来区分竞争类。这些基于深度的方法假设识别曲面的有限维参数形式，并利用数据云的分布几何来构建分类器。我们使用一些模拟和真实的数据集来检验这些判别分析工具的性能，并研究它们在适当的正则性条件下的渐近性质。
A very well-known traditional approach in discriminant analysis is to use some linear (or nonlinear) combination of measurement variables which can enhance class separability. For instance, a linear (or a quadratic) classifier finds the linear projection (or the quadratic function) of the measurement variables that will maximize the separation between the classes. These techniques are very useful in obtaining good lower-dimensional views of class separability. Fisher's discriminant analysis, which is primarily motivated by the multivariate normal distribution, uses the first-and second-order moments of the training sample to build such classifiers. These estimates, however, are highly sensitive to outliers, and they are not reliable for heavy-tailed distributions. This paper investigates two distribution-free methods for linear classification, which are based on the notions of statistical depth functions. One of these classifiers is closely related to Tukey's half-space depth, while the other is based on the concept of regression depth. Both these methods can be generalized for constructing nonlinear surfaces to discriminate among competing classes. These depth-based methods assume some finite-dimensional parametric form of the discriminating surface and use the distributional geometry of the data cloud to build the classifier. We use a few simulated and real data sets to examine the performance of these discriminant analysis tools and study their asymptotic properties under appropriate regularity conditions.