Cross-spectrum

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 $(X_{t},Y_{t})$ represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions $\gamma _{{xx}}$ and $\gamma _{{yy}}$ and cross-covariance function $\gamma _{xy}$ . Then the cross-spectrum $\Gamma _{{xy}}$ is defined as the Fourier transform of $\gamma _{xy}$ ^[1]

\Gamma _{xy}(f)={\mathcal {F}}\{\gamma _{xy}\}(f)=\sum _{\tau =-\infty }^{\infty }\,\gamma _{xy}(\tau )\,e^{-2\,\pi \,i\,\tau \,f},

where

\gamma _{xy}(\tau )=\operatorname {E} [(x_{t}-\mu _{x})(y_{t+\tau }-\mu _{y})]

The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and its imaginary part (quadrature spectrum)

\Gamma _{{xy}}(f)=\Lambda _{{xy}}(f)+i\Psi _{{xy}}(f),

and (ii) in polar coordinates

\Gamma _{{xy}}(f)=A_{{xy}}(f)\,e^{{i\phi _{{xy}}(f)}}.

Here, the amplitude spectrum $A_{{xy}}$ is given by

A_{{xy}}(f)=(\Lambda _{{xy}}(f)^{2}+\Psi _{{xy}}(f)^{2})^{{\frac {1}{2}}},

and the phase spectrum $\Phi _{{xy}}$ is given by

{\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}}

The squared coherency spectrum is given by

\kappa _{{xy}}(f)={\frac {A_{{xy}}^{2}}{\Gamma _{{xx}}(f)\Gamma _{{yy}}(f)}},

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.

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