Cross-spectrum explained

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) = l{F}\{\gamma _{xy} \}(f) = \sum ^{inf ty} _{\tau =-inf ty} \gamma _{xy} (\tau) e ^{-2\pii\tauf} , 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) = Λ _{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 byA _{xy} (f) = ( Λ _{xy} (f) ^{2} + \Psi _{xy} (f) ^{2)} ^{(1)/(2)} , and the phase spectrum \Phi _{xy} is given by\begin{cases} \tan ^{-1} ( \Psi _{xy} (f) / Λ _{xy} (f) ) & if \Psi _{xy} (f) \ne 0 and Λ _{xy} (f) \ne 0 \\ 0 & if \Psi _{xy} (f) = 0 and Λ _{xy} (f) > 0 \\ \pm \pi & if \Psi _{xy} (f) = 0 and Λ _{xy} (f) < 0 \\ \pi/ 2 & if \Psi _{xy} (f) > 0 and Λ _{xy} (f) = 0 \\ - \pi/ 2 & if \Psi _{xy} (f) < 0 and Λ _{xy} (f) = 0 \\ \end{cases}

Squared coherency spectrum

The squared coherency spectrum is given by

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

which expresses the amplitude spectrum in dimensionless units.

See also

• Cross-correlation
• Power spectrum
• Scaled Correlation

References

1. Book: von Storch , H. . Cambridge Univ Pr. 0-521-01230-9. F. W Zwiers. Statistical analysis in climate research. 2001.