Robust generalised quadratic discriminant analysis. (September 2021) - Ghosh, Abhik; SahaRay, Rita; Chakrabarty, Sayan; Bhadra, Sayan
Highlights: Investigation of robustness of a Generalized Quadratic Discriminant Analysis (GQDA) under the presence of Noise in data. The GQDA is a novel approach integrating linear & quadratic discriminant analyses, but is extremely sensitive under mild contamination. Development of roust versions of GQDA classier by using robust estimators of the mean vector and the dispersion matrix. Detailed empirical comparison of robust GQDA proposals with 6 robust estimators, 3 classes of model distribution and 4 real data examples. Abstract: Quadratic discriminant analysis (QDA) is a widely used statistical tool to classify observations from different multivariate Normal populations. The generalized quadratic discriminant analysis (GQDA) classification rule/classifier, which generalizes the QDA and the minimum Mahalanobis distance (MMD) classifiers to discriminate between populations with underlying elliptically symmetric distributions competes quite favorably with the QDA classifier when it is optimal and performs much better when QDA fails under non-Normal underlying distributions with heavy tail, e.g. Cauchy distribution. However, the classification rule in GQDA is still based on the sample mean vector and the sample dispersion matrix of a training set, which are extremely non-robust under data contamination. In real world, however, it is quite common to face data which are highly vulnerable to outliers and so the lack of robustness of the classical estimators of the mean vector and the dispersion matrix reduces the efficiency of the GQDA classifier significantly, increasing the misclassification errors. The present paper investigates the performance of the GQDA classifier when the classical estimators of the mean vector and the dispersion matrix used therein are replaced by various robust counterparts. Applications to various real data sets as well as simulation studies reveal far better performance of the proposed robust versions of the GQDA classifier. A comparative study has been made to advocate the appropriate choice of the robust estimators to be used in a specific situation.