Multivariate

Multivariate describes phenomena, models, and analyses that involve more than one variable. In mathematics, a multivariate object is a function, distribution, or equation that depends on two or more variables. In statistics and data analysis, multivariate methods study multiple dependent or outcome variables simultaneously, taking into account their interdependencies.

Multivariate calculus extends calculus to functions of several variables, introducing partial derivatives, gradients, Jacobians, and Hessians, as well as multiple integrals and change-of-variables formulas. It provides tools for optimization, modeling, and geometric analysis in higher dimensions.

In probability and statistics, a multivariate model specifies the joint behavior of several random variables. The joint distribution encodes margins and dependencies; the covariance or correlation matrix summarizes linear relationships. The multivariate normal distribution is a central example with elliptical contours and linear transformations preserving normality.

Common multivariate methods include principal component analysis, factor analysis, discriminant analysis, multivariate regression, canonical correlation analysis, MANOVA, and cluster analysis. These methods address dimensionality reduction, structure discovery, and inference on multiple outcomes.

Applications span finance (risk assessment and portfolio optimization), psychology and social sciences (test batteries and behavioral measures), biology and environmental science (gene expression profiling, spatial data), and image processing (color channels and texture analysis). Distinctions in terminology vary: multivariate typically refers to multiple variables as outcomes or a joint distribution, while multivariable is often used for multiple predictors in regression or functions of several independent variables.