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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 $A$ 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 up previous
Next: Short observation times Up: OU-processes Previous: Maximum likelihood estimation
Markus Mayer 2009-06-22