 Multivariate statistics

Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis. Methods of bivariate statistics, for example simple linear regression and correlation, are special cases of multivariate statistics in which two variables are involved.
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 implementation of multivariate statistics to a particular problem may involve several types of univariate and multivariate analysis in order to understand the relationships between variables and their relevance to the actual 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.
Contents
Types of analysis
There are many different models, each with its own type of analysis:
 Multivariate analysis of variance (MANOVA) extends the analysis of variance to cover cases where there is more than one dependent variable to be analyzed simultaneously: see also MANCOVA.
 Multivariate regression analysis attempts to determine a formula that can describe how elements in a vector of variables respond simultaneously to changes in others. For linear relations, regression analyses here are based on forms of the general linear model.
 Principal components analysis (PCA) creates a new set of orthogonal variables that contain the same information as the original set. It rotates the axes of variation to give a new set of orthogonal axes, ordered so that they summarize decreasing proportions of the variation.
 Factor analysis is similar to PCA but allows the user to extract a specified number of synthetic variables, fewer than the original set, leaving the remaining unexplained variation as error. The extracted variables are known as latent variables or factors; each one may be supposed to account for covariation in a group of observed variables.
 Canonical correlation analysis finds linear relationships among two sets of variables; it is the generalised (i.e. canonical) version of bivariate correlation.
 Redundancy analysis is similar to canonical correlation analysis but allows the user to derive a specified number of synthetic variables from one set of (independent) variables that explain as much variance as possible in another (independent) set. It is a multivariate analogue of regression.
 Correspondence analysis (CA), or reciprocal averaging, finds (like PCA) a set of synthetic variables that summarise the original set. The underlying model assumes chisquared dissimilarities among records (cases). There is also canonical (or "constrained") correspondence analysis (CCA) for summarising the joint variation in two sets of variables (like canonical correlation analysis).
 Multidimensional scaling comprises various algorithms to determine a set of synthetic variables that best represent the pairwise distances between records. The original method is principal coordinates analysis (based on PCA).
 Discriminant analysis, or canonical variate analysis, attempts to establish whether a set of variables can be used to distinguish between two or more groups of cases.
 Linear discriminant analysis (LDA) computes a linear predictor from two sets of normally distributed data to allow for classification of new observations.
 Clustering systems assign objects into groups (called clusters) so that objects (cases) from the same cluster are more similar to each other than objects from different clusters.
 Recursive partitioning creates a decision tree that attempts to correctly classify members of the population based on a dichotomous dependent variable.
 Artificial neural networks extend regression and clustering methods to nonlinear multivariate models.
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 InverseWishart distribution is important in Bayesian inference, for example in Bayesian multivariate linear regression. Additionally, Hotelling's Tsquared distribution is a univariate distribution, generalising Student's tdistribution, that is used in multivariate hypothesis testing.
Software & Tools
There are an enormous number of software packages and other tools for multivariate analysis, including:
 Calc
 MiniTab
 R
 SAS (software)
 sciPy for Python
 SPSS
 Stata
 STATISTICA
 TMVA  Toolkit for Multivariate Data Analysis in ROOT
 The Unscrambler
See also
 Estimation of covariance matrices
 Important publications in multivariate analysis
 Multivariate testing
 Structured data analysis (statistics)
 RV coefficient
References
 Professor Kim H. Esbensen. Multivariate Data Analysis: in Practice : An Introduction to Multivariate Data Analysis and Experimental Design (5th Edition). Camo Process AS. ISBN 8299333032.
 Gerry Quinn and Michael Keough (2002). Experimental Design and Data Analysis for Biologists. Cambridge University Press. ISBN 9780521009768.
External links
 Statnotes: Topics in Multivariate Analysis, by G. David Garson
 Mike Palmer: The Ordination Web Page
 InsightsNow: Makers of ReportsNow, ProfilesNow, and KnowledgeNow
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