Proyecciones ortogonales y sus aplicaciones

Abstract

The aim of this research was to study and describe from a mathematical point of view, the theoretical basis involved in the reduction of data dimensionality, by means of the Principal Component Analysis or PCA technique; for which the specialized bibliography of the topic was reviewed in detail; in order to understand the elements of mathematics involved in the deduction of this technique. The derivation and mathematical description of the Principal Component Analysis was carried out by means of a detailed description of the topic of orthogonal projections of vectors in subspaces of smaller dimension, to obtain a presentation or approximation of the data that is more manageable and computationally simple to analyze. The aim was achieved by means of the exhaustive and systemic investigation of the existing bibliography on the technique of PCA the obtaining of the principal space. In this research, the principal subspace sought is obtained by minimizing the mean square error of reconstruction, in the Euclidean norm, which leads to the calculation of the principal eigenvalues and eigenvectors of the covariance matrix, which will represent the chordal and optimal vectors of the projection subspace. It is concluded that the elements of mathematics, in this case Linear Algebra, such as vector spaces, inner products, orthogonality, projections, eigenvalues, eigenvectors, are fundamental basis in the description of the PCA. Additionally, the understanding of these notions allowed us to elaborate a basic algorithm for data dimensionality reduction. In the future, this derivation could be performed from another perspective, or by using a norm other than the Euclidean norm.