Nonlinear Principal and Canonical Directions from Continuous Extensions of Multidimensional Scaling

• 作者: Carles M. Cuadras
• 来源期刊: 统计学期刊(英文)
• 学科: 数学
• 关键词: Statistical DISTANCES Orthogonal EXPANSIONS Principal DIRECTIONS of Random Variables DIAGONAL EXPANSIONS COPULAS Uncountable Dimensionality
• 年，卷（期）: 2014,（2）
• 页码范围: 154-171
• 页数: 18页
• 分类号: O1

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.