Gram-Schmidt Orthogonalization of Multinormal Variates: Applications in Genetics

✍ Scribed by Prof. G. E. Bonney; G. E. Kissling

• Book ID: 101712749
• Publisher: John Wiley and Sons
• Year: 1986
• Tongue: English
• Weight: 345 KB
• Volume: 28
• Category: Article
• ISSN: 0323-3847
• DOI: 10.1002/bimj.4710280407

✦ Synopsis

The computation of an N-variata normal density function requires the inversion of a n N x N covariance matrix. Furthermore, if each mean depends o n u unobservable factors, a mixture of u N N-variata normal densities must be computed,making likelihood calculet~oneimpractical even for moderate N. The Gram-Schmidt orthogona~ization process is used to express a mu~tinormal density as a product of univariate normal densities. When the patternof the Correlation matrix is taken into account the formulas may be considerably simplified. I n some cases each of the orthogonal variates can be written as a linear combinationof only a few of the original variates. Such r e s u b are crucial for applicationsof multinormal distributions and of mixtures of multinormal distributions. An intraclacrs correlation model and a genetic variance components model apphabh3 to family data are discussed a8 examples.