Principal points for an allometric extension model
A set of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> </mrow> </math> </EquationSource> </InlineEquation>-principal points of a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>p</mi> </mrow> </math> </EquationSource> </InlineEquation>-dimensional distribution is an optimal <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> </mrow> </math> </EquationSource> </InlineEquation>-point-approximation of the distribution in terms of a squared error loss. It is in general difficult to derive an explicit expression of principal points. Hence, we may have to search the whole space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$R^p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msup> </mrow> </math> </EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> </mrow> </math> </EquationSource> </InlineEquation>-principal points. Many efforts have been devoted to establish results that specify a linear subspace in which principal points lie. However, the previous studies focused on elliptically symmetric distributions and location mixtures of spherically symmetric distributions, which may not be suitable to many practical situations. In this paper, we deal with a mixture of elliptically symmetric distributions that form an allometric extension model, which has been widely used in the context of principal component analysis. We give conditions under which principal points lie in the linear subspace spanned by the first several principal components. Copyright Springer-Verlag Berlin Heidelberg 2014
