Order statistics-based parametric classification for multi-dimensional distributions

Traditionally, in the field of Pattern Recognition (PR), the moments of the class-conditional densities of the respective classes have been used to perform classification. However, the use of phenomena that utilized the properties of the Order Statistics (OS) were not reported. Recently, in [10,8], we proposed a new paradigm named CMOS, Classification by the Moments of Order Statistics, which specifically used these quantifiers. It is fascinating that CMOS is essentially “anti”-Bayesian in its nature because the classification is performed in a counter-intuitive manner, i.e., by comparing the testing sample to a few samples distant from the mean, as opposed to the Bayesian approach in which the task is based on the central points of the distributions. In our initial works, we proposed the foundational theory of CMOS for the uni-dimensional Uniform and some other distributions. These results were extended for various symmetric and asymmetric uni-dimensional distributions within the exponential family in [8]. In this paper, we generalize these results for multi-dimensional distributions. The multi-dimensional generalization is particularly non-trivial because there is no well-established method for achieving the ordering of multi-dimensional data specified in terms of its uni-dimensional components. The strategy is analogous to a Naïve-Bayesʹ approach, although it really is of an anti-Naïve-Bayesʹ paradigm. We provide here the analytical and experimental results for the 2-dimensional Uniform, Doubly Exponential and Gaussian distributions, and also clearly specify the way by which one should extend the results for higher dimensions. The analogous results for the other distributions in the exponential family, which were discussed in [10,8] are alluded to, but omitted to avoid repetition.