| The basis for statistical pattern classification is that the individual classes are characterized by their distributions . These distributions have numerous indicators such as their means, variances etc., and these have, traditionally, played a prominent role in achieving pattern classification. The gold standard for a classifier is the condition of optimality attained by the Bayesian classifier. Within a Bayesian paradigm, if we are allowed to compare the testing sample with only a single point in the feature space from each class, the optimal Bayesian strategy would be to achieve this based on the (Mahalanobis) distance from the corresponding means. Apart from the indicators mentioned above, a distribution has many other characterizing indicators, for example, those related to its Order Statistics (OS). The interesting point about these indicators is that some of them are quite unrelated to the traditional moments themselves, and in spite of this, have not been used in achieving PR. The main question that we shall consider in this thesis is whether these indicators/indices possess any potential in PR. The amazing answer to this question is that OS can be used in PR, and that such classifiers operate in a completely "anti-Bayesian" manner.;In this thesis, we introduce the theory of optimal PR using the OS of the features rather than the distributions of the features themselves. Our novel methodology, is referred to as Classification by Moments of Order Statistics (CMOS). This claim has been proven for many uni-dimensional and multi-dimensional distributions within the exponential family namely the Uniform, Doubly-exponential, Gaussian, and the theoretical results have been verified by rigorous experimental testing. We have also extended these results significantly by considering asymmetric distributions within the exponential family like the Rayleigh, Gamma, and Beta distributions, for which a near-optimal accuracy has been achieved. The results have also been extended for the corresponding multi-dimensional distributions, and to yield Prototype Reduction Schemes (PRS) which contain only a single element for each class. Apart from the fact that these results are quite fascinating and pioneering in their own right, they also give a theoretical foundation for the families of Border Identification (BI) algorithms. |
