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 $(X_t,Y_t)$ 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 ${\mathrm{\Gamma }}_{xy}$ is defined as the Fourier transform of ${\gamma }_{xy}$

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

where

${\gamma }_{xy}(\tau )=\mathrm{E}[(x_t-{\mu }_x)(y_{t+\tau }-{\mu }_y)]$ .

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

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

and (ii) in polar coordinates

${\mathrm{\Gamma }}_{xy}(f)=A_{xy}(f)e^{i{\varphi }_{xy}(f)}.$

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

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

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

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

Squared coherency spectrum

The squared coherency spectrum is given by

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

which expresses the amplitude spectrum in dimensionless units.

See also

• Cross-correlation
• Power spectrum
• Scaled Correlation