Bayesian Reconstruction of Gravitational-wave Signals from Binary Black Holes with Nonzero Eccentricities

We present a comprehensive study on how well gravitational-wave signals of binary black holes with nonzero eccentricities can be recovered with state of the art model-independent waveform reconstruction and parameter estimation techniques. For this we use BayesWave, a Bayesian algorithm used by the LIGO-Virgo Collaboration for unmodeled reconstructions of signal waveforms and parameters. We used two different waveform models to produce simulated signals of binary black holes with eccentric orbits and embed them in samples of simulated noise of design-sensitivity Advanced LIGO detectors. We studied the network overlaps and point estimates of central moments of signal waveforms recovered by BayesWave as a function of $e$, the eccentricity of the binary at 8 Hz orbital frequency. BayesWave recovers signals of near-circular ($e\lesssim0.2$) and highly eccentric ($e\gtrsim0.7$) binaries with network overlaps similar to that of circular ($e=0$) ones, however it produces lower network overlaps for binaries with $e\in[0.2,0.7]$. Estimation errors on central frequencies and bandwidths (measured relative to bandwidths) are nearly independent from $e$, while estimation errors on central times and durations (measured relative to durations) increase and decrease with $e$ above $e\gtrsim0.5$, respectively. We also tested how BayesWave performs when reconstructions are carried out using generalized wavelets with linear frequency evolution (chirplets) instead of sine-Gaussian wavelets. We have found that network overlaps improve by $\sim 10-20$ percent when chirplets are used, and the improvement is the highest at low ($e<0.5$) eccentricities. There is however no significant change in the estimation errors of central moments when the chirplet base is used.Comment: 16 pages, 9 figures, accepted for publication in CQ

On Practical Algorithms for Entropy Estimation and the Improved Sample Complexity of Compressed Counting

Estimating the p-th frequency moment of data stream is a very heavily studied problem. The problem is actually trivial when p = 1, assuming the strict Turnstile model. The sample complexity of our proposed algorithm is essentially O(1) near p=1. This is a very large improvement over the previously believed O(1/eps^2) bound. The proposed algorithm makes the long-standing problem of entropy estimation an easy task, as verified by the experiments included in the appendix

Estimating Entropy of Data Streams Using Compressed Counting

The Shannon entropy is a widely used summary statistic, for example, network traffic measurement, anomaly detection, neural computations, spike trains, etc. This study focuses on estimating Shannon entropy of data streams. It is known that Shannon entropy can be approximated by Reenyi entropy or Tsallis entropy, which are both functions of the p-th frequency moments and approach Shannon entropy as p->1. Compressed Counting (CC) is a new method for approximating the p-th frequency moments of data streams. Our contributions include: 1) We prove that Renyi entropy is (much) better than Tsallis entropy for approximating Shannon entropy. 2) We propose the optimal quantile estimator for CC, which considerably improves the previous estimators. 3) Our experiments demonstrate that CC is indeed highly effective approximating the moments and entropies. We also demonstrate the crucial importance of utilizing the variance-bias trade-off

Testing the multipole structure and conservative dynamics of compact binaries using gravitational wave observations: The spinning case

In an earlier work [S. Kastha et al., PRD {\bf 98}, 124033 (2018)], we developed the {\it parametrized multipolar gravitational wave phasing formula} to test general relativity, for the non-spinning compact binaries in quasi-circular orbit. In this paper, we extend the method and include the important effect of spins in the inspiral dynamics. Furthermore, we consider parametric scaling of PN coefficients of the conserved energy for the compact binary, resulting in the parametrized phasing formula for non-precessing spinning compact binaries in quasi-circular orbit. We also compute the projected accuracies with which the second and third generation ground-based gravitational wave detector networks as well as the planned space-based detector LISA will be able to measure the multipole deformation parameters and the binding energy parameters. Based on different source configurations, we find that a network of third-generation detectors would have comparable ability to that of LISA in constraining the conservative and dissipative dynamics of the compact binary systems. This parametrized multipolar waveform would be extremely useful not only in deriving the first upper limits on any deviations of the multipole and the binding energy coefficients from general relativity using the gravitational wave detections, but also for science case studies of next generation gravitational wave detectors.Comment: 16 pages, 8 figures, Mathematica readable supplemental material file for all the inputs to calculate the parametrized waveform is with the sourc

Modelling network travel time reliability under stochastic demand

A technique is proposed for estimating the probability distribution of total network travel time, in the light of normal day-to-day variations in the travel demand matrix over a road traffic network. A solution method is proposed, based on a single run of a standard traffic assignment model, which operates in two stages. In stage one, moments of the total travel time distribution are computed by an analytic method, based on the multivariate moments of the link flow vector. In stage two, a flexible family of density functions is fitted to these moments. It is discussed how the resulting distribution may in practice be used to characterise unreliability. Illustrative numerical tests are reported on a simple network, where the method is seen to provide a means for identifying sensitive or vulnerable links, and for examining the impact on network reliability of changes to link capacities. Computational considerations for large networks, and directions for further research, are discussed

Spectral identification of networks with inputs

We consider a network of interconnected dynamical systems. Spectral network identification consists in recovering the eigenvalues of the network Laplacian from the measurements of a very limited number (possibly one) of signals. These eigenvalues allow to deduce some global properties of the network, such as bounds on the node degree. Having recently introduced this approach for autonomous networks of nonlinear systems, we extend it here to treat networked systems with external inputs on the nodes, in the case of linear dynamics. This is more natural in several applications, and removes the need to sometimes use several independent trajectories. We illustrate our framework with several examples, where we estimate the mean, minimum, and maximum node degree in the network. Inferring some information on the leading Laplacian eigenvectors, we also use our framework in the context of network clustering.Comment: 8 page