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The autocorrelation matrix is a positive semidefinite matrix, [3]: p.190 i.e. for a real random vector, and respectively in case of a complex random vector. All eigenvalues of the autocorrelation matrix are real and non-negative.
Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
Autocovariance can be used to calculate turbulent diffusivity. [4] Turbulence in a flow can cause the fluctuation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations [citation needed].
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 fact, the autocorrelation method is the most common [2] and it is used, for example, for speech coding in the GSM standard. Solution of the matrix equation R A = r {\displaystyle \mathbf {RA} =\mathbf {r} } is computationally a relatively expensive process.
Visual comparison of convolution, cross-correlation and autocorrelation. A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. [1]
The autocorrelation function of an AR(p) process is a sum of decaying exponentials. ... as well as the eigenvalues of the temporal update matrix: ...
The value of can depend quite a bit on the assumptions built into the spatial weights matrix .The matrix is required because, in order to address spatial autocorrelation and also model spatial interaction, we need to impose a structure to constrain the number of neighbors to be considered.