The statistical properties of the cross spectrum

Abstract

The cross spectrum encodes the correlated variability between two time signals. In recent years, the cross spectrum has been used to study astronomical sources, particularly in the field of X-ray timing. In the literature, it has been common to either simultaneously fit the real and imaginary components of the cross spectrum, or fit the phase and magnitude. Until now, a full discussion of the statistical distribution of the cross spectrum has been missing from the astronomical literature. In this paper, we present a derivation of the full statistical distribution of a cross spectrum between two time series, showing that it follows an asymmetric Laplace distribution. We further provide the probability distribution function for a cross spectrum random variable, along with the marginal distributions for many quantities. We also relate the cross spectrum to the power spectra of the constituent time series. This work will enable the cross spectrum to be used more accurately as a probe of physical processes such as accretion onto black holes and neutron stars.

Figures (8)

• Figure 1: We show simulations of 1,000,000 realisations of a cross spectrum bin, here in an unaveraged ($N=1$) case. The simulations were performed as described in Appendix \ref{['app:simulations']}. The power in the signals is $P_X=P_Y=10$, which includes a noise of $P_{n_x}=P_{n_y}=2$ (consistent with Poisson noise in Leahy normalisation). We show for three values of the intrinsic coherence ($\gamma^2=0,0.25,1$, giving the raw coherences $g^2=0,0.16,0.64$), and use a phase difference of $\phi_G=0.46$ radians.
• Figure 2: Similar simulations as in Fig. \ref{['fig:PxPyGrGi_coh_sims']}, however this time we show the distribution of averaging of $N=50$ realisations of the cross spectrum.
• Figure 3: A simulation of 4,000,000 realisations of just the co and quadrature spectra (as described in Appendix \ref{['app:simulations']}). The 2D histogram has the $1,2,3\,\sigma$ contours from both the simulation (in black) and calculated from the PDF Eq. \ref{['eqn:PDF']} (in red). The marginalised histograms also are also compared to their expected PDFs in Eq. \ref{['eqn:real_imag_PDFs']}.
• Figure 4: The same simulation as in Fig. \ref{['fig:Gr_Gi_comparison']} with 4,000,000 realisations, showing only the joint distribution of the magnitude and phase of the cross spectrum. The 2D histogram has the same contours from the simulation (in black) and from the PDF Eq. \ref{['eqn:polar_PDF']} (in red). The marginalised histograms also are also compared to their expected PDFs in Eqs. \ref{['eqn:mag_PDF']} and \ref{['eqn:ang_PDF']}.
• Figure 5: Left: An example cross spectrum, calculated between the light curves from the FPMA and FPMB detectors of a NuSTAR observation of H 1743-322 which shows a strong QPO. For the frequency bins shown, we use MCMC maximum likelihood estimation to fit the parameters $H_rP_s$, $-H_iP_s$, and $\eta$, which correspond to the mean of the co and quadrature spectra, and the spread. The violin shapes represent the posterior distribution of the parameters sampled at each frequency bin; the 'error-bar' shows the full range of the 100,000 MCMC samples used. Right: We show the MCMC as fit to the $0.25$ Hz frequency bin (corresponding to the peak of the QPO). The $0.5$, $1$, and $2~\sigma$ contours from the data are shown in blue, while the $0.5$, $1$, $2$, and $3~\sigma$ contours of the distribution are over-plotted in red. We show the contours from many parameter draws from the MCMC, transparently, to show the sampled spread. The PDFs of the real and imaginary parts of the cross spectrum are also over-plotted onto their respective histograms, from the same MCMC samples.