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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.
Definition
Let
represent a pair of stochastic processes 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
[1]
![{\displaystyle \Gamma _{xy}(f)={\mathcal {F}}\{\gamma _{xy}\}(f)=\sum _{\tau =-\infty }^{\infty }\,\gamma _{xy}(\tau )\,e^{-2\,\pi \,i\,\tau \,f}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/844e03b76c6b53d0ff60115e7e16ca8a1d097780)
The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and its imaginary part (quadrature spectrum)
![{\displaystyle \Gamma _{xy}(f)=\Lambda _{xy}(f)+i\Psi _{xy}(f),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d76d32320f4dbcd7912db35efc620251e95614ba)
and (ii) in polar coordinates
![{\displaystyle \Gamma _{xy}(f)=A_{xy}(f)\,e^{i\phi _{xy}(f)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19ef272adfcb6602a60a590ba98dee6280f6381)
Here, the amplitude spectrum
is given by
![{\displaystyle A_{xy}(f)=(\Lambda _{xy}(f)^{2}+\Psi _{xy}(f)^{2})^{\frac {1}{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0c77665e3f58dab8f4f8cb5933321522c3e9f0)
and the phase spectrum
given by
![{\displaystyle {\begin{cases}\tan ^{-1}(\Psi _{xy}(f)/\Lambda _{xy}(f))&{\text{if }}\Psi _{xy}(f)\neq 0\wedge \Lambda _{xy}(f)\neq 0\\0&{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)>0\\\pm \pi &{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)<0\\\pi /2&{\text{if }}\Psi _{xy}(f)>0{\text{ and }}\Lambda _{xy}(f)=0\\-\pi /2&{\text{if }}\Psi _{xy}(f)<0{\text{ and }}\Lambda _{xy}(f)=0\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05d772732e479a0ee07b59b0935f5d55c235f584)
Squared coherency spectrum
The squared coherency spectrum is given by
![{\displaystyle \kappa _{xy}(f)={\frac {A_{xy}^{2}}{\Gamma _{xx}(f)\Gamma _{yy}(f)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc10db4058a4f3a36f4bd80f9ff55e65ec3947bb)
which expresses the amplitude spectrum in dimensionless units.
See also
References
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