Optimized Dimensionality Reduction Methods for Interval-Valued Variables and Their Application to Facial Recognition

Abstract

The center method, which was first proposed in Rev. Stat. Appl. 1997 by Cazes et al. and Stat. Anal. Data Mining 2011 by Douzal-Chouakria et al., extends the well-known Principal Component Analysis (PCA) method to particular types of symbolic objects that are characterized by multivalued interval-type variables. In contrast to classical data, symbolic data have internal variation. The authors who originally proposed the center method used the center of a hyper-rectangle in as a base point to carry out PCA, followed by the projection of all vertices of the hyper-rectangles as supplementary elements. Since these publications, the center point of the hyper-rectangle has typically been assumed to be the best point for the initial PCA. However, in this paper, we show that this is not always the case, if the aim is to maximize the variance of projections or minimize the squared distance between the vertices and their respective projections. Instead, we propose the use of an optimization algorithm that maximizes the variance of the projections (or that minimizes the distances between the squares of the vertices and their respective projections) and finds the optimal point for the initial PCA. The vertices of the hyper-rectangles are, then, projected as supplementary variables to this optimal point, which we call the “Best Point” for projection. For this purpose, we propose four new algorithms and two new theorems. The proposed methods and algorithms are illustrated using a data set comprised of measurements of facial characteristics from a study on facial recognition patterns for use in surveillance. The performance of our approach is compared with that of another procedure in the literature, and the results show that our symbolic analyses provide more accurate information. Our approach can be regarded as an optimization method, as it maximizes the explained variance or minimizes the squared distance between projections and the original points. In addition, the symbolic analyses generate more informative conclusions, compared with the classical analysis in which classical surrogates replace intervals. All the methods proposed in this paper can be executed in the RSDA package developed in R.