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For jointly wide-sense stationary stochastic processes, the definition is = = [() (+) ¯] The normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of statistical dependence, and because the normalization has an effect on the statistical ...
Cross-covariance may also refer to a "deterministic" cross-covariance between two signals. This consists of summing over all time indices. For example, for discrete-time signals f [ k ] {\displaystyle f[k]} and g [ k ] {\displaystyle g[k]} the cross-covariance is defined as
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
Consequently, the correlation is a dimensionless quantity that can be used to compare the linear relationships between pairs of variables in different units. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρ XY is near +1 (or −1).
The parameter belongs to the set of positive-definite matrices, which is a Riemannian manifold, not a vector space, hence the usual vector-space notions of expectation, i.e. "[^]", and estimator bias must be generalized to manifolds to make sense of the problem of covariance matrix estimation.
Several generalizations of mutual information to more than two random variables have been proposed, such as total correlation (or multi-information) and dual total correlation. The expression and study of multivariate higher-degree mutual information was achieved in two seemingly independent works: McGill (1954) [ 12 ] who called these ...