Multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. 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 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. Multivariate 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: 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. These concerns are often eased through the use of surrogate models, highly accurate approximations of the physics-based code. Since surrogate models take the form of an equation, they can be evaluated very quickly. This becomes an enabler for large-scale MVA studies: while a Monte Carlo simulation across the design space is difficult with physics-based codes, it becomes trivial when evaluating surrogate models, which often take the  form of response-surface equations. Types of analysis. There are many different models, each with its own type of analysis: Important probability distributions. There is a set of probability distributions used in multivariate analyses that play a similar role to the corresponding set of distributions that are used in univariate analysis when the normal distribution is appropriate to a dataset. These multivariate distributions are: The Inverse-Wishart distribution is important in Bayesian inference, for example in Bayesian multivariate linear regression. Additionally, Hotelling's T-squared distribution is a multivariate distribution, generalising Student's t-distribution, that is used in multivariate  hypothesis testing. History. Anderson's 1958 textbook," An Introduction to Multivariate Statistical Analysis", educated a generation of theorists and applied statisticians; Anderson's book emphasizes hypothesis testing via likelihood ratio tests and the properties of power functions: admissibility, unbiasedness and monotonicity. MVA once solely stood in the statistical theory realms due to the size, complexity of underlying data set and high computational consumption. With the dramatic growth of computational power, MVA now plays an increasingly important role in data analysis and has wide application in OMICS fields. Software and tools. There are an enormous number of software packages and other  tools for multivariate analysis, including: