Multivariate Regression Algorithm

A Multivariate Regression Algorithm is a regression algorithm that can be implemented into a Multivariate Regression System (to solve a multivariate regression task).

• AKA: Multiple Dependent Variables Regression Algorithm.
• Example(s):
  • Multivariate Linear Regression Algorithm.
  • Multivariate Polynomial Regression Algorithm.
  • …
• Counter-Example(s):
  • a Univariate Regression Algorithm.
  • a Multinomial Regression Algorithm.
  • a Multivariate Classification Algorithm.
• See: Design Matrix, Multivariate Normal Distribution, Generalized Linear Models, ANOVA, ANCOVA, MANOVA, MANCOVA, t-Test, F-Test, Multivariate Hypothesis Testing.

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References

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/General_linear_model Retrieved:2014-11-13.
  • The general linear model is a statistical linear model.

    It may be written as

     : [math]\displaystyle{ \mathbf{Y} = \mathbf{X}\mathbf{B} + \mathbf{U}, }[/math]

    where Y is a matrix with series of multivariate measurements, X is a matrix that might be a design matrix, B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors or noise.

    The errors are usually assumed to be uncorrelated across measurements, and follow a multivariate normal distribution. If the errors do not follow a multivariate normal distribution, generalized linear models may be used to relax assumptions about Y and U.

    The general linear model incorporates a number of different statistical models: ANOVA, ANCOVA, MANOVA, MANCOVA, ordinary linear regression, t-test and F-test. The general linear model is a generalization of multiple linear regression model to the case of more than one dependent variable. If Y, B, and U were column vectors, the matrix equation above would represent multiple linear regression.

    Hypothesis tests with the general linear model can be made in two ways: multivariate or as several independent univariate tests.

    In multivariate tests the columns of Y are tested together, whereas in univariate tests the columns of Y are tested independently, i.e., as multiple univariate tests with the same design matrix.

2013

• (Hidalgo & Goodman, 2013) ⇒ Bertha Hidalgo, and Melody Goodman. (2013). “Multivariate Or Multivariable Regression?.” In: American journal of public health, 103(1).
  • QUOTE: Most regression models are described in terms of the way the outcome variable is modeled: in linear regression the outcome is continuous, logistic regression has a dichotomous outcome, and survival analysis involves a time to event outcome. Statistically speaking, multivariate analysis refers to statistical models that have 2 or more dependent or outcome variables, and multivariable analysis refers to statistical models in which there are multiple independent or response variables.

1982

• (Chamberlain, 1982) ⇒ Gary Chamberlain (1982). “Multivariate regression models for panel data.” In: Journal of Econometrics, 18(1).

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