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 ${\displaystyle (X_{t},Y_{t})}$ represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions ${\displaystyle \gamma _{xx}}$ and ${\displaystyle \gamma _{yy}}$ and cross-covariance function ${\displaystyle \gamma _{xy}}$. Then the cross-spectrum ${\displaystyle \Gamma _{xy}}$ is defined as the Fourier transform of ${\displaystyle \gamma _{xy}}$ ^[1]

${\displaystyle \Gamma _{xy}(f)={\mathcal {F}}\{\gamma _{xy}\}(f)=\sum _{\tau =-\infty }^{\infty }\,\gamma _{xy}(\tau )\,e^{-2\,\pi \,i\,\tau \,f},}$

where

${\displaystyle \gamma _{xy}(\tau )=\operatorname {E} [(x_{t}-\mu _{x})(y_{t+\tau }-\mu _{y})]}$ .

The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)

${\displaystyle \Gamma _{xy}(f)=\Lambda _{xy}(f)-i\Psi _{xy}(f),}$

and (ii) in polar coordinates

${\displaystyle \Gamma _{xy}(f)=A_{xy}(f)\,e^{i\phi _{xy}(f)}.}$

Here, the amplitude spectrum ${\displaystyle A_{xy}}$ is given by

${\displaystyle A_{xy}(f)=(\Lambda _{xy}(f)^{2}+\Psi _{xy}(f)^{2})^{\frac {1}{2}},}$

and the phase spectrum ${\displaystyle \Phi _{xy}}$ is given by

${\displaystyle {\begin{cases}\tan ^{-1}(\Psi _{xy}(f)/\Lambda _{xy}(f))&{\text{if }}\Psi _{xy}(f)\neq 0{\text{ and }}\Lambda _{xy}(f)\neq 0\\0&{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)>0\\\pm \pi &{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)<0\\\pi /2&{\text{if }}\Psi _{xy}(f)>0{\text{ and }}\Lambda _{xy}(f)=0\\-\pi /2&{\text{if }}\Psi _{xy}(f)<0{\text{ and }}\Lambda _{xy}(f)=0\\\end{cases}}}$

Squared coherency spectrum

The squared coherency spectrum is given by

${\displaystyle \kappa _{xy}(f)={\frac {A_{xy}^{2}}{\Gamma _{xx}(f)\Gamma _{yy}(f)}},}$

which expresses the amplitude spectrum in dimensionless units.

See also

• Cross-correlation
• Power spectrum
• Scaled Correlation