Multivariate Analysis: Overview

Abstract

Multivariate analysis is the statistical study of data where multiple measurements are made on each experimental unit and where the relationships among multivariate measurements and their structure are important. A modern, overlapping categorization of multivariate analysis is: normal and general multivariate models and distribution theory; the study and measurement of relationships; probability computations of multidimensional regions; and the exploration of data structures and patterns. The multivariate normal distribution plays a central role in multivariate analysis in the same way that the univariate normal distribution plays a central role. The Student's t generalizes to Hotelling's T 2 and the chi-square distribution to the Wishart distribution. Newer multivariate distributions can model data when the multivariate normal distribution is not adequate. For multidimensional data, relationships among the variables are fundamental to explore. Useful techniques to understand and quantify these include multivariate regression analysis and various correlational notions such as partial correlations and canonical correlations. Approaches to compute complicated multidimensional probabilities including obtaining lower bounds for the probabilities and using numerical approximation techniques. The exploration of structure and patterns for complex multivariate data sets is crucial for modern data analysis and data mining. Multivariate tools, useful in this context, include principal components analysis, canonical analysis, factor analysis, path analysis, structural equation methods, clustering, and discriminant analysis.