Robust estimation and its application to a classification problem

Agnieszka Kulawik,Henryk Gacki

doi:10.14708/ma.v47i2.6499

Abstract

In the article, a classification problem with two distributed classes is considered. The problem is solving using empirical discriminant functions for Gaussian classifier and estimators for unknown parameters of multivariate normal distribution. The three etimators, maximum likelihood estimator, Kulawik-Zontek estimator and minimum covariance determinant estimator, are compared in two different empirical examples (small size sample and large size sample).

Highlights

The main aim of classication ([18], [5]) is to make a decision which class should be attributed to a new observation
We will focus on two robust estimators: the Kulawik-Zontek estimator (KZE) and the minimum covariance determinant estimator (MCDE)
The results show that for the small size sample problem it was better to use KZE than maximum likelihood estimator (MLE)

Summary

Introduction

The main aim of classication ([18], [5]) is to make a decision which class should be attributed to a new observation. In the paper a classier based on an estimator for parameters of the multivariate normal model is considered. The classes will be assumed to be multivariate normally distributed and a Gaussian classier will be used. Estimators of the parameters appear in the form of the empirical discriminant functions for Gaussian classiers. Assume that each class is multivariate normally NJ (μi, Σi) distributed with the density function of the form f (z|i). Discriminant functions for the Gaussian classier in the case of considered problem can be dened as gi(z) = lnf (z|i) + lnP (i), for z ∈ RJ , where P (i) denotes "a priori probability" for the i-th class, i = 1, 2. After changing the unknown parameters μi, Σi, P (i) to their estimators μi, Σi, P(i) we get the empirical discriminant functions for the Gaussian classier: Σi.

Estimators for parameters of multivariate normal model

Empirical examples

Conclusions