We propose a novel semiparametric classifier based on Mahalanobis distances of an observation from the competing classes. Our tool is a generalized additive model with the logistic link function that uses these distances as features to estimate the posterior probabilities of different classes. While popular parametric classifiers like linear and quadratic discriminant analyses are mainly motivated by the normality of the underlying distributions, the proposed classifier is more flexible and free from such parametric modeling assumptions. Since the densities of elliptic distributions are functions of Mahalanobis distances, this classifier works well when the competing classes are (nearly) elliptic. In such cases, it often outperforms popular nonparametric classifiers, especially when the sample size is small compared to the dimension of the data. To cope with non-elliptic and possibly multimodal distributions, we propose a local version of the Mahalanobis distance. Subsequently, we propose another classifier based on a generalized additive model that uses the local Mahalanobis distances as features. This nonparametric classifier usually performs like the Mahalanobis distance based semiparametric classifier when the underlying distributions are elliptic, but outperforms it for several non-elliptic and multimodal distributions. We also investigate the behaviour of these two classifiers in high dimension, low sample size situations. A thorough numerical study involving several simulated and real datasets demonstrate the usefulness of the proposed classifiers in comparison to many state-of-the-art methods.

翻译：本文提出了一种新颖的半参数分类器，其基于观测值到各竞争类别的马氏距离。我们采用具有逻辑链接函数的广义可加模型作为工具，将这些距离作为特征来估计不同类别的后验概率。虽然线性判别分析和二次判别分析等常用参数分类器主要基于底层分布的正态性假设，但所提出的分类器更为灵活，不受此类参数建模假设的限制。由于椭圆分布的密度函数是马氏距离的函数，当竞争类别呈（近似）椭圆分布时，该分类器表现良好。在此类情况下，其性能通常优于流行的非参数分类器，特别是在样本量相对于数据维度较小时。为处理非椭圆及可能的多峰分布，我们提出了局部马氏距离的概念。随后，我们构建了另一种基于广义可加模型的分类器，该模型以局部马氏距离作为特征。当底层分布为椭圆分布时，这种非参数分类器的性能与基于马氏距离的半参数分类器相当，但在处理多种非椭圆及多峰分布时表现更优。我们还研究了这两种分类器在高维度、小样本情境下的表现。通过涉及多个模拟数据集和真实数据集的系统数值实验，证明了所提出分类器相较于多种前沿方法的实用价值。