On the eigenvalue distribution of genetic and phenotypic dispersion matrices: Evidence for a nonrandom organization of quantitative character variation
The idea is to develop a quantitative genetic model of ''random pleiotropy'' and to study the eigenvalue distribution of its (Gaussian) genetic dispersion matrix. This will serve as a reference model. By comparing the eigenvalues of empirical correlation matrices of several morphological data sets in mammalian and insect species with those of the reference model, the author draws some inferences concerning morphological integration and ''the kind and degree of organization found in the correlation matrices of quantitative characters''. The issue is of biological interest and the idea of using the eigenvalues seems to be worth exploring, for they may reflect ''the distribution of variation within the phenotype space.'' Looking at the mathematical side of the paper, a random vector \(x=(x_ 1,...,x_ m)\) of allelic values at m loci is constructed assuming the \(x_ i's\) to be i.i.d. Gaussian (0,1). A random developmental \(n\times m\) matrix B of i.i.d. Gaussian zero mean elements is also constructed. Phenotypical characters \(z=(z_ 1,....,z_ n)\) are assumed to be given by \(z=Bx\) and its dispersion matrix is, therefore, \(G=BB^ T.\) Considering the ensemble of all such matrices G, the distribution of the ordered sequence of their eigenvalues is Wishart. The marginal and the conditional distributions of the eigenvalues and their moments are obtained based on two (somewhat disputable) simplifying assumptions. The author then makes the comparisons referred at the beginning and draws tentative conclusions based on deviations of the empirical data from the reference model, although admiting that no statistical test is so far available for such comparisons.
eigenvalues of empirical correlation matrices
scientific article; zbMATH DE number 3882250