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The correlation matrix is symmetric because the correlation between and is the same as the correlation between and . A correlation matrix appears, for example, in one formula for the coefficient of multiple determination , a measure of goodness of fit in multiple regression .
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
An entity closely related to the covariance matrix is the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector , which can be written as = ( ()) ( ()), where is the matrix of the diagonal elements of (i.e., a diagonal matrix of the variances of for =, …,).
Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. Covariance matrix — a symmetric n×n matrix, formed by the pairwise covariances of several random variables. Sometimes called a dispersion matrix. Dispersion matrix — another name for a covariance matrix.
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a statistical relationship between two variables. [a] The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution.
The coefficient of multiple correlation is known as the square root of the coefficient of determination, but under the particular assumptions that an intercept is included and that the best possible linear predictors are used, whereas the coefficient of determination is defined for more general cases, including those of nonlinear prediction and those in which the predicted values have not been ...
In this definition, it has been assumed that the stochastic variables are scalar-valued. If they are not, then more complicated correlation functions can be defined. For example, if X(s) is a random vector with n elements and Y(t) is a vector with q elements, then an n×q matrix of correlation functions is defined with , element