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Correlation matrices

The correlation matrix of n {\displaystyle n} random variables X 1, …, X n {\displaystyle X_{1},\ldots,X_{n}} is the n × n {\displaystyle n\times n} matrix C {\displaystyle C} whose (i, j) {\displaystyle (i,j)} entry is

Thus the diagonal entries are all identically one. If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X i / σ (X i) {\displaystyle X_{i}/\sigma (X_{i})} for i = 1, …, n {\displaystyle i=1,\dots,n}. This applies both to the matrix of population correlations (in which case σ {\displaystyle \sigma } is the population standard deviation), and to the matrix of sample correlations (in which case σ {\displaystyle \sigma } denotes the sample standard deviation). Consequently, each is necessarily a positive-semidefinite matrix. Moreover, the correlation matrix is strictly positive definite if no variable can have all its values exactly generated as a linear function of the values of the others.

The correlation matrix is symmetric because the correlation between X i {\displaystyle X_{i}} and X j {\displaystyle X_{j}} is the same as the correlation between X j {\displaystyle X_{j}} and X i {\displaystyle X_{i}}.

A correlation matrix appears, for example, in one formula for the coefficient of multiple determination, a measure of goodness of fit in multiple regression.

In statistical modelling, correlation matrices representing the relationships between variables are categorized into different correlation structures, which are distinguished by factors such as the number of parameters required to estimate them. For example, in an exchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. On the other hand, an autoregressive matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Other examples include independent, unstructured, M-dependent, and Toeplitz.

In exploratory data analysis, the iconography of correlations consists in replacing a correlation matrix by a diagram where the "remarkable" correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).