Principal Curves and Surfaces to Interval Valued Variables

Abstract

In this paper we propose a generalization to symbolic interval valued variables, of the Principal Curves and Surfaces method proposed by Hastie in [6]. Given a data set X with n observations and m continuous variables, the main idea of Principal Curves and Surfaces method is to generalize the principal component line, providing a smooth one-dimensional curved approximation to a set of data points in $\mathbb {R}^m$. A principal surface is more general, providing a curved manifold approximation of dimension 2 or more. In our case we are interested in finding the main principal curve that approximates better symbolic interval data variables. In [3, 4], authors proposed the Centers Method and the Vertices Method to extend the well-known principal components analysis method to a particular kind of symbolic objects characterized by multi-valued variables of interval type. In this paper we generalize both, the Centers Method and the Vertices Method, finding a smooth curve that passes through the middle of the data X in an orthogonal sense. Some comparisons of the proposed method regarding the Centers and the Vertices Methods are made, this was done with the RSDA package using Ichino data set, see [1, 10]. To make these comparisons we have used the correlation index.