Bayesian modeling of source confusion in LISA data

One of the greatest data analysis challenges for the Laser Interferometer Space Antenna (LISA) is the need to account for a large number of gravitational wave signals from compact binary systems expected to be present in the data. We introduce the basis of a Bayesian method that we believe can address this challenge and demonstrate its effectiveness on a simplified problem involving 100 synthetic sinusoidal signals in noise. We use a reversible jump Markov chain Monte Carlo technique to infer simultaneously the number of signals present, the parameters of each identified signal, and the noise level. Our approach therefore tackles the detection and parameter estimation problems simultaneously, without the need to evaluate formal model selection criteria, such as the Akaike Information Criterion or explicit Bayes factors. The method does not require a stopping criterion to determine the number of signals and produces results which compare very favorably with classical spectral techniques.

Figure 1

The evidences for models ${\mathcal{M}}_1$ (solid line, one parameter fit) and ${\mathcal{M}}_2$ (dashed line, two parameter fit) as a function of the datum $d$. Note that the evidence ratio always favors the simpler model ${\mathcal{M}}_1$ when the datum is equally consistent with either.

Figure 2

(a) shows the model probabilities deduced from our analysis. The model corresponding to 95 sinusoids has highest probability. Each model has a different noise level, and (b) shows the estimated noise standard deviation and their confidence intervals for the different models. The dotted line indicates the real noise level. As model 100 has a very low probability only two MCMC were obtained during the run of $3\times {10}^8$ iterations, making it poorly determined.

Figure 3

Comparison of (a) the real signal energies, (b) the Bayesian spectral density, and (c) the classical periodogram of the test data. Note that (a) shows component energies, whereas (b) and (c) show energy densities (inferred energy per unit frequency). The peaks generated by the Bayesian method are higher than those of the periodogram, reflecting its greater ability to constrain the frequency.

Figure 4

Joint posteriors for the amplitudes and frequency in Regions-1, -2 and -3. The circles show the true values of the parameters comprising signal pairs.

Figure 5

Bayesian spectral estimates, showing 95% confidence bounds in spectral density, for the three regions considered. Each regions contain two distinct sinusoids, one shown with solid lines and the other with dashed lines. The three lines for each sinusoid show the median value (thick) and 95% bounds (thin) of the spectral estimate at each frequency. The vertical lines show the true frequency values of these components. The dotted line traversing the entire width of the plots is the corresponding classical periodogram.

Figure 6

95% posterior confidence contours of energy and frequency for the six sinusoids considered. The empty circles show the median value of the posterior peak, and the filled circles show the true value.

Figure 7

Top: A section of the (energy, frequency) joint posterior probability density containing eleven 95% posterior confidence areas. The bottom plot shows the classical periodogram for the same frequency range. Twelve sinusoids are actually present in this region, indicated by gray dots and one sinusoid, at $f=0.3387209$, was too weak to be identified by our method ($\gamma =0.11$).

Figure 8

Model probabilities for different pairs of equal-amplitude sinusoids in Gaussian noise ($\sigma =1$). Each plot displays two sinusoids with varying frequency gaps expressed as a percentage of one $(1/N)$ step (with $N=1000$). Each row corresponds to a different phase shift between the sinusoids: from top to bottom the phase shift is $0,\pi /4,\pi /2,3\pi /4,\pi$. The two columns display results with different signal strengths, as given in the title of the plots.