Английская Википедия:Cross-spectrum

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

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

<math>

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

<math>\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>

\Gamma_{xy}(f)= \Lambda_{xy}(f) - i \Psi_{xy}(f) , </math>

and (ii) in polar coordinates

<math>

\Gamma_{xy}(f)= A_{xy}(f) \,e^{i \phi_{xy}(f) } . </math> Here, the amplitude spectrum <math>A_{xy}</math> is given by

<math>A_{xy}(f)= (\Lambda_{xy}(f)^2 + \Psi_{xy}(f)^2)^\frac{1}{2} ,</math>

and the phase spectrum <math>\Phi_{xy}</math> is given by

<math>\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}</math>

Squared coherency spectrum

The squared coherency spectrum is given by

<math>

\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