Gram-Schmidt Orthogonalization of Multinormal Variates: Applications in Genetics

The computation of an N-variate normal density function requires the inversion of an N × N co-variance matrix. Furthermore, if each mean depends on u unobservable factors, a mixture of u^N N-variate normal densities must be computed, making likelihood calculations impractical even for moderate N. The Gram-Schmidt orthogonalization process is used to express a multinormal density as a product of univariate normal densities. When the pattern of the correlation matrix is taken into account the formulas may be considerably simplified. In some cases each of the orthogonal variates can be written as a linear combination of only a few of the original variates. Such results are crucial for applications of multinormal distributions and of mixtures of multinormal distributions. An intraclass correlation model and a genetic variance components model applicable to family data are discussed as examples.