10 - Multivariate Statistics

Summary

This chapter provides a survey of some of the major techniques provided for multivariate random variables. The multivariate Gaussian distribution is characterized, and its key properties are elucidated. The sample mean vector and sample covariance matrix are specified, and shown to have analogous properties to their univariate counterparts. The treatment is extended to the complex multivariate Gaussian distribution. The multivariate counterpart to the Student's t test, Hotelling's T squared test is described and used to obtain simultaneous confidence intervals on multiple parameters. Multivariate analysis of variance is outlined, leading to the multivariate generalization of the F distribution, Wilks' lambda distribution. Three types of hypothesis tests on the covariance matrix are defined. Multivariate linear regression is specified, along with tools for its assessment. Canonical correlation for multiple response and predictor variables is outlined. Empirical orthogonal function (also called principal component) analsysis is defined and illustrated using global temperature data.