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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 =, …,).
where = [] and = [] are vectors containing the expected values of and .The vectors and need not have the same dimension, and either might be a scalar value.. The cross-covariance matrix is the matrix whose (,) entry is the covariance
The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms.
In the analysis of data, a correlogram is a chart of correlation statistics. 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.
CCA can be computed using singular value decomposition on a correlation matrix. [8] It is available as a function in [9] MATLAB as canoncorr (also in Octave) R as the standard function cancor and several other packages, including CCA and vegan. CCP for statistical hypothesis testing in canonical correlation analysis. SAS as proc cancorr
The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R p×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. [1]
The model order may be determined by means of criteria developed in the framework of information theory and the coefficients of the model are found by means of the minimalization of the residual noise. In the procedure correlation matrix between signals is calculated. By the transformation to the frequency domain we get:
The cross-covariance is also relevant in signal processing where the cross-covariance between two wide-sense stationary random processes can be estimated by averaging the product of samples measured from one process and samples measured from the other (and its time shifts).