Nonlinear Principal and Canonical Directions from Continuous Extensions of Multidimensional Scaling

作     者：Carles M. Cuadras 

作者机构：Department of Statistics University of Barcelona Barcelona Spain 

出 版 物：《Open Journal of Statistics》 (统计学期刊（英文）)

年 卷 期：2014年第4卷第2期

页      面：154-171页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Statistical Distances Orthogonal Expansions Principal Directions of Random Variables Diagonal Expansions Copulas Uncountable Dimensionality 

摘      要：A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.