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 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]
where
- .
The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)
and (ii) in polar coordinates
Here, the amplitude spectrum is given by
and the phase spectrum is given by
Squared coherency spectrum[edit]
The squared coherency spectrum is given by
which expresses the amplitude spectrum in dimensionless units.
See also[edit]
References[edit]
- ^ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.