Next Article in Journal
Coalescence of Kerr Black Holes—Binary Systems from GW150914 to GW170814
Previous Article in Journal
An Entropy-Based Machine Learning Algorithm for Combining Macroeconomic Forecasts
Open AccessArticle

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

1
National Bank of Costa Rica, 10101 San José, Costa Rica
2
School of Mathematics, Research Center in Pure and Applied Mathematics (CIMPA), University of Costa Rica, 10101 San José, Costa Rica
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(10), 1016; https://doi.org/10.3390/e21101016
Received: 24 September 2019 / Revised: 13 October 2019 / Accepted: 17 October 2019 / Published: 19 October 2019
(This article belongs to the Special Issue Symbolic Entropy Analysis and Its Applications II)
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 R m 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. View Full-Text
Keywords: interval-valued variables; principal component analysis; symbolic data analysis; Best Point method interval-valued variables; principal component analysis; symbolic data analysis; Best Point method
Show Figures

Figure 1

MDPI and ACS Style

Arce Garro, J.; Rodríguez Rojas, O. Optimized Dimensionality Reduction Methods for Interval-Valued Variables and Their Application to Facial Recognition. Entropy 2019, 21, 1016.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
  • Externally hosted supplementary file 1
    Link: http://www.oldemarrodriguez.com/publicaciones
    Description: Optimized Dimensionality Reduction Methods for Interval-Valued Variables and Their Application toFacial Recognition - Iris Benchmark Optimized Dimensionality Reduction Methods for Interval-Valued Variables and Their Application toFacial Recognition - Oils Benchmark
Back to TopTop