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 $\gamma_$ and $\gamma_$ and cross-covariance function $\gamma_$. Then the cross-spectrum $\Gamma_$ is defined as the Fourier transform of $\gamma_$

: $\Gamma_(f)= \mathcal\(f) = \sum_^\infty \,\gamma_(\tau) \,e^ ,$
where
: \gamma_(\tau) = \operatorname x_t - \mu_x)(y_ - \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)
: $\Gamma_(f)= \Lambda_(f) - i \Psi_(f) ,$

and (ii) in polar coordinates
: $\Gamma_(f)= A_(f) \,e^ .$
Here, the amplitude spectrum $A_$ is given by
: $A_(f)= (\Lambda_(f)^2 + \Psi_(f)^2)^\frac ,$
and the phase spectrum $\Phi_$ is given by
: $\begin \tan^ ( \Psi_(f) / \Lambda_(f) ) & \text \Psi_(f) \ne 0 \text \Lambda_(f) \ne 0 \\ 0 & \text \Psi_(f) = 0 \text \Lambda_(f) > 0 \\ \pm \pi & \text \Psi_(f) = 0 \text \Lambda_(f) < 0 \\ \pi/2 & \text \Psi_(f) > 0 \text \Lambda_(f) = 0 \\ -\pi/2 & \text \Psi_(f) < 0 \text \Lambda_(f) = 0 \\ \end$

Squared coherency spectrum

The squared coherency spectrum is given by
: $\kappa_(f)= \frac{ \Gamma_{xx}(f) \Gamma_{yy}(f)} ,$

which expresses the amplitude spectrum in dimensionless units.

See also

* Cross-correlation
* Power spectrum
* Scaled Correlation

References

Frequency-domain analysis
Multivariate time series