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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.
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
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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7ccbb3f85c312fe0b5a5ee61ca7341712274afaa)
where
.
The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)
![{\displaystyle \Gamma _{xy}(f)=\Lambda _{xy}(f)-i\Psi _{xy}(f),}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8b9dbee9ac43728dd40b056cf01755820fe665fe)
and (ii) in polar coordinates
![{\displaystyle \Gamma _{xy}(f)=A_{xy}(f)\,e^{i\phi _{xy}(f)}.}](https://backend.710302.xyz/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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0d0c77665e3f58dab8f4f8cb5933321522c3e9f0)
and the phase spectrum
is given by
![{\displaystyle {\begin{cases}\tan ^{-1}(\Psi _{xy}(f)/\Lambda _{xy}(f))&{\text{if }}\Psi _{xy}(f)\neq 0{\text{ and }}\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/bf38701af3e23b585bdb6545cf2c033d950e83e0)
Squared coherency spectrum
[edit]
The squared coherency spectrum is given by
![{\displaystyle \kappa _{xy}(f)={\frac {A_{xy}^{2}}{\Gamma _{xx}(f)\Gamma _{yy}(f)}},}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/cc10db4058a4f3a36f4bd80f9ff55e65ec3947bb)
which expresses the amplitude spectrum in dimensionless units.
- ^ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.