TITLE:
Nonlinear Principal and Canonical Directions from Continuous Extensions of Multidimensional Scaling
AUTHORS:
Carles M. Cuadras
KEYWORDS:
Statistical Distances; Orthogonal Expansions; Principal Directions of Random Variables; Diagonal Expansions; Copulas; Uncountable Dimensionality
JOURNAL NAME:
Open Journal of Statistics,
Vol.4 No.2,
February
27,
2014
ABSTRACT:
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.