Autocorrelation function and power spectrum

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Autocorrelation function and power spectrum

Define $\tilde C_t=\mathsf{E}\,\xi_t\xi_0^\prime$ which satisfies the ODE $\mathrm{d}\tilde C_t/\mathrm{d}t = A\tilde C_t+a\mu_\infty^\prime$ , i.e.

$\begin{displaymath} \mathsf{E}\xi_t\xi_0^\prime = e^{A\, t}\,V_\infty + \mu_\infty\mu_\infty^\prime \quad . \end{displaymath}$

(76)

The auto(cross-)covariance and auto(cross-)correlation functions are therefore

$\displaystyle \mathsf{cov}_{t-t^\prime}$	$\textstyle =$	$\displaystyle e^{\vert t-t^\prime\vert\,A}\,V_\infty$	(77)
$\displaystyle C_{t-t^\prime}$	$\textstyle =$	$\displaystyle \mathsf{cov}_{t-t^\prime}\,\mathsf{cov}_0^{-1} = e^{\vert t-t^\prime\vert\,A} \quad .$	(78)

The Fourier transform of the autocorrelation function is the power spectrum:

$\begin{displaymath} S(\omega) = \int_{-\infty}^\infty e^{-i\,\omega\,\tau} C_\t... ...ega\,\tau + \vert\tau\vert\,A} = \frac{2 A}{\omega^2 + A^2} \end{displaymath}$

(79)

(use $\mathcal{F}(e^{-\gamma \vert s\vert})(\omega) = \frac{2\gamma}{\gamma^2+\omega^2}$ .) If the eigenvalues of

are $(\nu_i)_{i=1,\dots,n}$ the eigenvalues of $S(\omega)$ are $( 2\nu_i/(\omega^2+\nu_i^2) )_{i=1,\dots,n}$ . If the observed process is an affine function of the state process $\xi_t$ , i.e. $y_t = U\xi_t+v$ , the observed autocovariance and autocorrelation function are

$\displaystyle \mathsf{E}\,y_t y_0^\prime - \mathsf{E}y_t\,\mathsf{E}y_0$	$\textstyle =$	$\displaystyle U\,C_t\,V_\infty\,U^\prime$	(80)
$\displaystyle C^y_t$	$\textstyle =$	$\displaystyle U\,C_t\,V_\infty\,U^\prime\,\left(U\,V_\infty\,U^\prime\right)^{-1} \quad .$	(81)

and

$\begin{displaymath} S^y(\omega) = \int_{-\infty}^\infty e^{-i\,\omega\,\tau} C_... ...\,S(\omega)\,V_\infty\,U^\prime\,(U\,V_\infty\,U^\prime)^{-1} \end{displaymath}$

(82)

Next: Short observation times Up: OU-processes Previous: Maximum likelihood estimation

Markus Mayer 2009-06-22