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Multivariate analysis (MVA) is based on the principles of multivariate statistics, which involves observation and analysis of more than one statistical outcome variable at a time. 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.[1] A modern, overlapping categorization of MVA includes[1]:

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. 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.

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.[2][3] 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.
Applications

Multivariate hypothesis testing
Dimensionality reduction
Latent structure discovery
Clustering
Multivariate regression analysis
Classification and discrimination analysis
Variable selection
Multidimensional Scaling
Data mining

Tools

NCSS (statistical software) includes multivariate analysis.
The Unscrambler® X is a multivariate analysis tool.
SIMCA
Python
R
MATLAB

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

References

Olkin, I.; Sampson, A. R. (2001-01-01), "Multivariate Analysis: Overview", in Smelser, Neil J.; Baltes, Paul B. (eds.), International Encyclopedia of the Social & Behavioral Sciences, Pergamon, pp. 10240–10247, ISBN 9780080430768, retrieved 2019-09-02
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 0162-1459. 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 0883-4237. JSTOR 2245530.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley, New York, 1958.
KV Mardia; JT Kent & JM Bibby (1979). Multivariate Analysis. Academic Press. ISBN 978-0124712522. (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 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)
Izenman, Alan J. (2008). Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning. Springer Texts in Statistics. New York: Springer-Verlag. ISBN 9780387781884.
"Handbook of Applied Multivariate Statistics and Mathematical Modeling | ScienceDirect". Retrieved 2019-09-03.

Discriminant Correlation Analysis (DCA)
M. Haghighat, M. Abdel-Mottaleb, & W. Alhalabi (2016). Discriminant Correlation Analysis: Real-Time Feature Level Fusion for Multimodal Biometric Recognition. IEEE Transactions on Information Forensics and Security, 11(9), 1984–1996.