Cross-covariance function

Cross-covariance function#

\[\rho_{xy}(\tau) = \frac{\gamma_{xy}(\tau)}{\sigma_X\sigma_y},\]

where \(\gamma_{xy}(\tau)\) is the sample cross-covariance function for \(\tau \geq 0\):

\[\gamma_{xy}(\tau) = \frac{1}{T} \sum_{t=1}^{T-\tau}\mathbf{X'_t}\mathbf{Y'_{t+\tau}},\]

and for \(\tau < 0\):

\[\gamma_{xy}(\tau) = \frac{1}{T} \sum_{t=1-\tau}^{T}\mathbf{X'_t}\mathbf{Y'_{t+\tau}}\]

the sample cross-covariance function is set to be zero for \(|\tau| \geq T\)
\(\tau > 0 \Rightarrow\) the time evolution of \(\mathbf{X_t}\) leads those of \(\mathbf{Y_t}\) and vice versa for \(|\tau| \leq T\)
\(\gamma_{xy}\) can be asymmetric
\(\gamma_{xy}(\tau)\) can be larger than \(\gamma_{xy}(0)\)
\(|\rho_{xy}(\tau)| \leq 1\)

Examples#

\(\mathbf{Y_t} = \alpha \mathbf{X_t} ~~\Rightarrow ~~\gamma_{yy}=\varepsilon(\mathbf{Y_tY_t}) = \alpha^2 \gamma_{xx}(\tau)\); cross-covariance \(\gamma_{xy} =\varepsilon(\mathbf{X_tY_t}) = \alpha\gamma_{xx}(\tau)\)
cross-correlation function \(\rho_{xy}(\tau) = \frac{\gamma_{xy}(\tau)}{\sigma_x\sigma_y} = \frac{\alpha \gamma_{xx}(\tau)}{\sigma_x \alpha \sigma_x} \rho_{xx}(\tau)\)

Cross spectrum#

Let \(\mathbf{X_t}\) and \(\mathbf{Y_t}\) be two weakly stationary stochastic processes with covariance functions \(\gamma_{xx}\) and \(\gamma_{yy}\), and a cross-covariance function \(\gamma_{xy}\). Then the cross-spectrum \(\Gamma_{xy}\) is defined as the Fourier transform of \(\gamma_{xy}\):

\[\begin{split}\begin{align*} \Gamma_{xy} &= \cal{F}\{\gamma_{xy}\}(\omega)\\ &= \sum_{t=-\infty}^{\infty} \gamma_{xy}(\tau)e^{-2\pi i\tau\omega} \end{align*}\end{split}\]

for all \(\omega \in [-0.5, 0.5]\). The cross-spectrum is generally a complex-valued function since the cross-covariance function is, in general, neither strictly symmetric nor anti-symmetric.

Cross-covariance function

Contents

Cross-covariance function#

Examples#

Cross spectrum#