Cross-spectrum
Encyclopedia
In time series analysis, the cross-spectrum is used as part of a frequency domain
analysis of the cross correlation or cross covariance
between two time series.
represent a pair of stochastic process
es that are jointly wide sense stationary with covariance functions
and 
and cross-covariance function 
. Then the cross spectrum 
is defined as the Fourier transform
of
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
analysis of the cross correlation or cross covariance
Cross covariance
In statistics, the term cross-covariance is sometimes used to refer to the covariance cov between two random vectors X and Y, in order to distinguish that concept from the "covariance" of a random vector X, which is understood to be the matrix of covariances between the scalar components of X.In...
between two time series.
Definition
Let represent a pair of stochastic processStochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es that are jointly wide sense stationary with covariance functions
and and cross-covariance function . Then the cross spectrum is defined as the Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of
-
The cross-spectrum has representations as (i) a decomposition into its real part (co-spectrum) and its imaginary part (quadrature spectrum)-
and (ii) in polar coordinates-
Here, the amplitude spectrum
is given by
and the phase spectrum
given by
-
Squared coherency spectrum
The squared coherency spectrum is given by-
which expresses the amplitude spectrum in dimensionless units.
See also
- Cross-correlation
- Power spectrum
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