Functional equation helps in finding affine-covariant probability distributions

To select variables which provide the most relevant clustering, researchers use a probability density f that depends on the means E and the covariance matrix C of the data. Under the assumption that the distributions are normal, this density becomes proportional to f(C) = |det(C)|α for some real value α, where det(C) is the determinant of the matrix C. This function is affine-covariant in the sense that the ratios of the two density values do not change under an arbitrary affine transformation. In this paper, we show that only the functions f(C) = |det(C)|α satisfy the functional equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{f(E,C)}{f(E',C')}= \frac{f(U^TE+S,U^TCU)}{f(U^TE'+S,U^TC'U)}$$\end{document}that describes affine covariance. This result justifies the use of the functions f(C) = |det(C)|α in non-Gaussian situations as well.