Aprimoramento do método de análise de componentes principais para matrizes com dados intervalares

Abstract

Principal Component Analysis (PCA) is a multivariate statistical technique that performs a linear transformation on a data set composed of observations and interrelated variables, which forms a new data set. This new set has uncorrelated or less correlated variables, and its dimension may be smaller than the original set. The main objectives of the PCA technique are: (i) to extract more relevant information from the observed set; (ii) to compress this set; and (iii) to simplify the description of the data set. It is a technique widely used in several contexts, such as the development of psychometric scales, neuroscience, identification of process failures, grouping or classification of information, among others. However, in everyday life, it is common to encounter situations in which the data set has uncertain values. So, the Classical PCA method will have difficulty in providing a satisfactory solution without adequate treatment from uncertain data, which may be dealt with by using interval mathematics. Thus, it is interesting to develop techniques for PCA in its interval context so that uncertainty can be considered in its data analysis, and this needs to be done without increasing the dimension of the data matrix in order to avoid a large increase in computational cost. Therefore, a new modeling of the interval data matrix is proposed to find the appropriate covariance matrix and, from this, apply the methods to find interval eigenvalues and eigenvectors in such a way that a linear transformation can be applied and interval principal components obtained.