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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.
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 .
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 ...
If F(r) is the Fisher transformation of r, the sample Spearman rank correlation coefficient, and n is the sample size, then = is a z-score for r, which approximately follows a standard normal distribution under the null hypothesis of statistical independence (ρ = 0). [12] [13]
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.
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
For example, in time series analysis, a plot of the sample autocorrelations versus (the time lags) is an autocorrelogram. If cross-correlation is plotted, the result is called a cross-correlogram . The correlogram is a commonly used tool for checking randomness in a data set .
Since this soft-thresholding procedure applied to a pairwise correlation matrix leads to weighted adjacency matrix, the ensuing analysis is referred to as weighted gene co-expression network analysis. A major step in the module centric analysis is to cluster genes into network modules using a network proximity measure.