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.

Contents

DefinitionEdit

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 spectrumEdit

The squared coherency spectrum is given by

 

which expresses the amplitude spectrum in dimensionless units.

See alsoEdit

ReferencesEdit

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