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 (ii) 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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