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 [math]\displaystyle{ (X_t,Y_t) }[/math] represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions [math]\displaystyle{ \gamma_{xx} }[/math] and [math]\displaystyle{ \gamma_{yy} }[/math] and cross-covariance function [math]\displaystyle{ \gamma_{xy} }[/math]. Then the cross-spectrum [math]\displaystyle{ \Gamma_{xy} }[/math] is defined as the Fourier transform of [math]\displaystyle{ \gamma_{xy} }[/math]

[math]\displaystyle{ \Gamma_{xy}(f)= \mathcal{F}\{\gamma_{xy}\}(f) = \sum_{\tau=-\infty}^\infty \,\gamma_{xy}(\tau) \,e^{-2\,\pi\,i\,\tau\,f} , }[/math]

where

[math]\displaystyle{ \gamma_{xy}(\tau) = \operatorname{E}[(x_t - \mu_x)(y_{t+\tau} - \mu_y)] }[/math] .

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

[math]\displaystyle{ \Gamma_{xy}(f)= \Lambda_{xy}(f) - i \Psi_{xy}(f) , }[/math]

and (ii) in polar coordinates

[math]\displaystyle{ \Gamma_{xy}(f)= A_{xy}(f) \,e^{i \phi_{xy}(f) } . }[/math]

Here, the amplitude spectrum [math]\displaystyle{ A_{xy} }[/math] is given by

[math]\displaystyle{ A_{xy}(f)= (\Lambda_{xy}(f)^2 + \Psi_{xy}(f)^2)^\frac{1}{2} , }[/math]

and the phase spectrum [math]\displaystyle{ \Phi_{xy} }[/math] is given by

[math]\displaystyle{ \begin{cases} \tan^{-1} ( \Psi_{xy}(f) / \Lambda_{xy}(f) ) & \text{if } \Psi_{xy}(f) \ne 0 \text{ and } \Lambda_{xy}(f) \ne 0 \\ 0 & \text{if } \Psi_{xy}(f) = 0 \text{ and } \Lambda_{xy}(f) \gt 0 \\ \pm \pi & \text{if } \Psi_{xy}(f) = 0 \text{ and } \Lambda_{xy}(f) \lt 0 \\ \pi/2 & \text{if } \Psi_{xy}(f) \gt 0 \text{ and } \Lambda_{xy}(f) = 0 \\ -\pi/2 & \text{if } \Psi_{xy}(f) \lt 0 \text{ and } \Lambda_{xy}(f) = 0 \\ \end{cases} }[/math]

Squared coherency spectrum

The squared coherency spectrum is given by

[math]\displaystyle{ \kappa_{xy}(f)= \frac{A_{xy}^2}{ \Gamma_{xx}(f) \Gamma_{yy}(f)} , }[/math]

which expresses the amplitude spectrum in dimensionless units.

See also

• Cross-correlation
• Power spectrum
• Scaled Correlation