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The definition of the normalized cross-correlation of a stochastic process is (,) = (,) () = [() ¯] () If the function is well-defined, its value must lie in the range [,], with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For jointly wide-sense stationary stochastic processes, the definition is ...
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).
Let (,) represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions and and cross-covariance function . Then the cross-spectrum Γ x y {\displaystyle \Gamma _{xy}} is defined as the Fourier transform of γ x y {\displaystyle \gamma _{xy}} [ 1 ]
A weaker form of stationarity commonly employed in signal processing is known as weak-sense stationarity, wide-sense stationarity (WSS), or covariance stationarity. WSS random processes only require that 1st moment (i.e. the mean) and autocovariance do not vary with respect to time and that the 2nd moment is finite for all times.
In the case of a time series which is stationary in the wide sense, both the means and variances are constant over time (E(X n+m) = E(X n) = μ X and var(X n+m) = var(X n) and likewise for the variable Y). In this case the cross-covariance and cross-correlation are functions of the time difference: cross-covariance
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 .
the constant-correlation model, where the sample variances are preserved, but all pairwise correlation coefficients are assumed to be equal to one another; the two-parameter matrix, where all variances are identical, and all covariances are identical to one another (although not identical to the variances);
Tables of critical values for both statistics are given by Rencher [38] for k = 2, 3, 4. Mardia's tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against symmetric non-normal alternatives.