cross-spectrum

Let $(X\_t,Y\_t)$ represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions $\backslash gamma\_\{xx\}$ and $\backslash gamma\_\{yy\}$ and cross-covariance function $\backslash gamma\_\{xy\}$. Then the cross-spectrum $\backslash Gamma\_\{xy\}$ is defined as the Fourier transform of $\backslash gamma\_\{xy\}$ {{Cite book

| publisher = Cambridge Univ Pr

| isbn = 0-521-01230-9

| last = von Storch

| first = H.

|author2=F. W Zwiers

| title = Statistical analysis in climate research

| year = 2001

}}

: $$

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

where

: $\backslash gamma\_\{xy\}(\backslash tau)\; =\; \backslash operatorname\{E\}[(x\_t\; -\; \backslash mu\_x)(y\_\{t+\backslash tau\}\; -\; \backslash mu\_y)]$ .

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

: $$

\Gamma_{xy}(f)= \Lambda_{xy}(f) - i \Psi_{xy}(f) ,

and (ii) in polar coordinates

: $$

\Gamma_{xy}(f)= A_{xy}(f) \,e^{i \phi_{xy}(f) } .

Here, the amplitude spectrum $A\_\{xy\}$ is given by

: $A\_\{xy\}(f)=\; (\backslash Lambda\_\{xy\}(f)^2\; +\; \backslash Psi\_\{xy\}(f)^2)^\backslash frac\{1\}\{2\}\; ,$

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

: $\backslash begin\{cases\}$

\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) > 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}

The squared coherency spectrum is given by

: $$

\kappa_{xy}(f)= \frac{A_{xy}^2}{ \Gamma_{xx}(f) \Gamma_{yy}(f)} ,

which expresses the amplitude spectrum in dimensionless units.

• Cross-correlation
• Power spectrum
• Scaled Correlation

Category:Frequency-domain analysis

Category:Multivariate time series