Cross-spectrum

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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[edit]

Let (X_t,Y_t) represent a pair of stochastic processes that are jointly wide sense stationary with covariance 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} .

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} given by

\begin{cases}
  \tan^{-1} (  \Psi_{xy}(f) / \Lambda_{xy}(f)  )     & \text{if } \Psi_{xy}(f) \ne 0 \wedge \Lambda_{xy}(f) \ne 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}

Squared coherency spectrum[edit]

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.

See also[edit]

References[edit]

  1. ^ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.