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Concept# Multivariate statistics

Summary

Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable, i.e., multivariate random variables.
Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied.
In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both
how these can be used to represent the distributions of observed data;
how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis.
Certain types of problems involving multivariate data, for example simple linear regression and multiple regression, are not usually considered to be special cases of multivariate statistics because the analysis is dealt with by considering the (univariate) conditional distribution of a single outcome variable given the other variables.
Univariate analysis
Multivariate analysis (MVA) is based on the principles of multivariate statistics. Typically, MVA is used to address the situations where multiple measurements are made on each experimental unit and the relations among these measurements and their structures are important. A modern, overlapping categorization of MVA includes:
Normal and general multivariate models and distribution theory
The study and measurement of relationships
Probability computations of multidimensional regions
The exploration of data structures and patterns
Multivariate analysis can be complicated by the desire to include physics-based analysis to calculate the effects of variables for a hierarchical "system-of-systems". Often, studies that wish to use multivariate analysis are stalled by the dimensionality of the problem.

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