- Cross-spectrum
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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.
Contents
Definition
Let (Xt,Yt) represent a pair of stochastic processes that are jointly wide sense stationary with covariance functions γxx and γyy and cross-covariance function γxy. Then the cross spectrum Γxy is defined as the Fourier transform of γxy [1]
The cross-spectrum has representations as (i) a decomposition into its real part (co-spectrum) and its imaginary part (quadrature spectrum)
- Γxy(f) = Λxy(f) + iΨxy(f),
and (ii) in polar coordinates
Here, the amplitude spectrum Axy is given by
and the phase spectrum Φxy given by
Squared coherency spectrum
The squared coherency spectrum is given by
which expresses the amplitude spectrum in dimensionless units.
See also
References
Categories:- Frequency domain analysis
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