Multivariate analysis

This article is about statistics. For multivariate analysis in mathematics, see Several complex variables.

Multivariate analysis (MVA) is based on the statistical principle of multivariate statistics, which involves observation and analysis of more than one statistical outcome variable at a time. In design and analysis, the technique is used to perform trade studies across multiple dimensions while taking into account the effects of all variables on the responses of interest.

Uses for multivariate analysis include:

• Design for capability (also known as capability-based design)
• Inverse design, where any variable can be treated as an independent variable
• Analysis of Alternatives (AoA), the selection of concepts to fulfill a customer need
• Analysis of concepts with respect to changing scenarios
• Identification of critical design drivers and correlations across hierarchical levels.

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.

Factor analysis

Main article: Factor analysis

Overview: Factor analysis is used to uncover the latent structure (dimensions) of a set of variables. It reduces attribute space from a larger number of variables to a smaller number of factors. Factor analysis originated a century ago with Charles Spearman's attempts to show that a wide variety of mental tests could be explained by a single underlying intelligence factor.

Applications:

• To reduce a large number of variables to a smaller number of factors for data modeling

• To validate a scale or index by demonstrating that its constituent items load on the same factor, and to drop proposed scale items which cross-load on more than one factor.

• To select a subset of variables from a larger set, based on which original variables have the highest correlations with some other factors.

• To create a set of factors to be treated as uncorrelated variables as one approach to handling multi-collinearity in such procedures as multiple regression

Factor analysis is part of the general linear model (GLM) family of procedures and makes many of the same assumptions as multiple regression, but it uses multiple outcomes.

History

Anderson's 1958 textbook, An Introduction to Multivariate 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.

See also

• Bivariate analysis
• Design of experiments (DoE)
• Exploratory data analysis
• Multilinear PCA
• Multilinear subspace learning
• OLS
• Partial least squares regression
• Pattern recognition
• Principal component analysis (PCA)
• Regression analysis
• Soft independent modelling of class analogies (SIMCA)
• Univariate analysis

Further reading

• T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley, New York, 1958.

• KV Mardia, JT Kent, and JM Bibby (1979). Multivariate Analysis. Academic Press,.  (M.A. level "likelihood" approach)

• Feinstein, A. R. (1996) Multivariable Analysis. New Haven, CT: Yale University Press.

• Hair, J. F. Jr. (1995) Multivariate Data Analysis with Readings, 4th ed. Prentice-Hall.

• Johnson, Richard A.; Wichern, Dean W. (2007). Applied Multivariate Statistical Analysis (Sixth ed.). Prentice Hall. ISBN 0-13-187715-1, ISBN 978-0-13-187715-3 . 

• Schafer, J. L. (1997) Analysis of Incomplete Multivariate Data. CRC Press. (Advanced)

• Sharma, S. (1996) Applied Multivariate Techniques. Wiley. (Informal, applied)