Multivariate analysis

Multivariate analysis (MVA) is based on the statistical principle of multivariate statistics, which involves observation and analysis of more than one statistical 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 capabilitybased 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 physicsbased analysis to calculate the effects of variables for a hierarchical "systemofsystems." 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 physicsbased code. Since surrogate models take the form of an equation, they can be evaluated very quickly. This becomes an enabler for largescale MVA studies: while a Monte Carlo simulation across the design space is difficult with physicsbased codes, it becomes trivial when evaluating surrogate models, which often take the form of response surface equations.
Contents
Factor analysis
Main article: Factor analysisOverview: 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 crossload 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 the principal component factors.
• To create a set of factors to be treated as uncorrelated variables as one approach to handling multicollinearity 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
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.^{[1]}^{[2]}
See also
 Univariate analysis
 Bivariate analysis
 Pattern recognition
 Exploratory data analysis
 Principal component analysis (PCA)
 Design of experiments (DoE)
 Soft independent modelling of class analogies (SIMCA)
 Regression analysis
 OLS
 Partial least squares regression
Software and tools
 TMVA  Toolkit for Multivariate Data Analysis in ROOT
 XLSTAT Addin for Excel for statistics and multivariate analysis
 The Unscrambler (freetotry commercial MVA software for Windows)
 ControlMV, PharmaMV and WaterMV from Perceptive Engineering
 SIMCAP+ (Professional MVA Software, free demo)
 R (See Task View: Multivariate for relevant packages)
 OCCAM is a webbased discrete multivariate modeling tool based on the methodology of reconstructability analysis, from Portland State University
Notes
 ^ Sen, Pranab Kumar; Anderson, T. W.; Arnold, S. F.; Eaton, M. L.; Giri, N. C.; Gnanadesikan, R.; Kendall, M. G.; Kshirsagar, A. M. et al. (June 1986). "Review: Contemporary Textbooks on Multivariate Statistical Analysis: A Panoramic Appraisal and Critique". Journal of the American Statistical Association 81 (394): 560–564. doi:10.2307/2289251. ISSN 01621459. JSTOR 2289251.(Pages 560–561)
 ^ Schervish, Mark J. (November 1987). "A Review of Multivariate Analysis". Statistical Science 2 (4): 396–413. doi:10.1214/ss/1177013111. ISSN 08834237. JSTOR 2245530.
Further reading
 KV Mardia, JT Kent, and JM Bibby (1979). Multivariate Analysis. Academic Press,. http://www.amazon.com/dp/0124712525/. (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. PrenticeHall.
 Johnson, Richard A.; Wichern, Dean W. (2007). Applied Multivariate Statistical Analysis (Sixth ed.). Prentice Hall. ISBN 0131877151, ISBN 9780131877153.
 Schafer, J. L. (1997) Analysis of Incomplete Multivariate Data. CRC Press. (Advanced)
 Sharma, S. (1996) Applied Multivariate Techniques. Wiley. (Informal, applied)
Categories: Multivariate statistics
 Statistical methods
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