Aplicação da álgebra linear aos métodos de análise discriminante

Abstract

This monograph has the purpose of exposing the functions of Discriminant Analysis with emphasis on Linear Algebra techniques. In fact, here we restrict ourselves to the case of discrimination methods based on quantitative variables. Thus, we find ourselves in the presence of a set of individuals or elements, divided into a certain number of classes defined "a priori"; In addition, it is assumed that for each individual or element the results of p measurements are made (or equivalently, of observations or values ​​assumed by p variables). In discriminating for descriptive purposes it is desired to show the possible discriminatory power of the variables in question; that is to say whether the measures or observations made justify the separation according to the different classes considered a priori. For example, in a typical medical application, it is important to determine if the results of certain clinical or laboratory tests (expressed quantitatively) justify or not the separation of a group of patients into two classes according to the most indicated therapeutic measures: i) the class of patients for whom a surgical procedure is reserved; ii) the class of patients for whom the best course of treatment involves drug treatment. This "descriptive step", in turn, may be followed by a "decisional stage", or discretion for decision or identification purposes, which is intended to carry out the assignment of each new individual, to one of the classes, under the lower risk of incorrect assignment. Regarding the previous example, which relates to the indication of the best therapy, this choice is required since a new patient is presented. It is noted that Discriminant Analysis opposes so-called "classification methods" according to which there are no classes determined "a priori". Discriminant Analysis methods have been used in several domains of applied research: Anthropology; Policy; Psychiatry; Clinical Medicine; Agrarian Geography. To conclude this presentation, we caution that we have not entirely exhausted the problem of the study of methods usable in discrimination with quantitative variables, since in fact this is a very rich and complex field of study.