Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Chaudhuri, Probal (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

Discriminant analysis (see e.g., Devijver and Kittler, 1982; Duda, Hart and Stork, 2000; Hastle, Tibahirani and Friedman, 2001) deals with the separation of different groups of obaervationa and allocation of a new oboervation to one of the previously delined grouga. In a J-class discriminant analysis problem, we usually hae a training sample of the form {(xk, ck) : k = 1,2,...,N}, where xk = (Ik1,Ik2,...J) is a d-dimensional measarement vector, and ca € {1,2,...,J} is its class label. On the basis of thia training sample, one aims to form a decision rule d(x) : Rd + (1,2,...,J} for clasifying the fature oboervstions into one of the J classes with the maximum possible accuracy. The optimsl Bayes rule (see eg, Rao, 1973; Anderson, 1984) assigns an observation to the class which has the largest posteriar probability. It can be described as da(x) = arg maxpli |*)= arg max nds(x). where the n,'s are the prior probsbilities, and the dB(x);s sre the probability density func- tions of the respective clasen (=1,2,..J). These denaity functions fj(x);s are usually unknown in practics, and can be esti- mated from the training sample either parametrically or nonparametrically. Parametrik ap- proaches (see eg, Rao, 1973; Mardin, Kent and Bibby, 1979, Anderaon, 1984; James, 1985; Fukunaga, 1990; Melachlan, 1962) are motivated by some specific distributional assump- tions about the underlying populations, where forma of the density functions are assumed to be knowa except for some unknown real parameters (e.g., means, variances, correlations). For instance, Fisher's linear and quadratic discriminant analysis (Fisher, 1936) are mainly motivated by the normality of the population distributions. Consequently, the performance of these parametric discrimination rules largely dependa on the valldity of those parametrie models. Such model assumptions are usually difficult to verity in practice, and inappropri- ate models may lead to a rather poor clasification. This is why there is a need to develop nonparametric and distribution free methoda for discriminant analysla. These nonparanet-ric classification techniques are more flexible in nature and free from all such parametric model assumptions. Notably, methods like classification trees (see e.g., Breiman et. al., 1984; Loh and Vanichsetakul, 1988; Loh and Shih, 1997; Kim and Loh, 2001, 2003), nearest neighbors (see e.g., Fix and Hodges, 1951; Cover and Hart, 1968; Dasarathy, 1991), flexible discriminant analysis (see e.g., Hastie, Tibshirani and Buja, 1994), splines (see e.g., Bose, 1996; Kooperberg, Bose and Stone, 1997), neural nets (see e.g., Lippman, 1987; Cheng and Titterington, 1994; Ripley, 1994, 1996) and support vector machines (see e.g., Vapnik, 1995, 1998; Burges, 1998) are known to outperform the parametric approaches in a wide variety of problems. A comparative study of the performance of several parametric and nonparametric classification algorithms can be found in the recent paper by Lim, Loh and Shih (2000).


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