A stochastic template placement algorithm for gravitational wave data analysis

This paper presents an algorithm for constructing matched-filter template banks in an arbitrary parameter space. The method places templates at random, then removes those which are "too close" together. The properties and optimality of stochastic template banks generated in this manner are investigated for some simple models. The effectiveness of these template banks for gravitational wave searches for binary inspiral waveforms is also examined. The properties of a stochastic template bank are then compared to the deterministically placed template banks that are currently used in gravitational wave data analysis.Comment: 14 pages, 11 figure

Asymptotic distinguishability measures for shift-invariant quasi-free states of fermionic lattice systems

We apply the recent results of F. Hiai, M. Mosonyi and T. Ogawa [arXiv:0707.2020, to appear in J. Math. Phys.] to the asymptotic hypothesis testing problem of locally faithful shift-invariant quasi-free states on a CAR algebra. We use a multivariate extension of Szego's theorem to show the existence of the mean Chernoff and Hoeffding bounds and the mean relative entropy, and show that these quantities arise as the optimal error exponents in suitable settings.Comment: Results extended to higher dimensional lattices, title changed. Submitted versio

Applications of distance between probability distributions to gravitational wave data analysis

We present a definition of the distance between probability distributions. Our definition is based on the $L_1$ norm on space of probability measures. We compare our distance with the well-known Kullback-Leibler divergence and with the proper distance defined using the Fisher matrix as a metric on the parameter space. We consider using our notion of distance in several problems in gravitational wave data analysis: to place templates in the parameter space in searches for gravitational-wave signals, to assess quality of search templates, and to study the signal resolution.Comment: 18 pages, 5 figure

Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination

We consider the multiple hypothesis testing problem for symmetric quantum state discrimination between r given states \sigma_1,...,\sigma_r. By splitting up the overall test into multiple binary tests in various ways we obtain a number of upper bounds on the optimal error probability in terms of the binary error probabilities. These upper bounds allow us to deduce various bounds on the asymptotic error rate, for which it has been hypothesised that it is given by the multi-hypothesis quantum Chernoff bound (or Chernoff divergence) C(\sigma_1,...,\sigma_r), as recently introduced by Nussbaum and Szko{\l}a in analogy with Salikhov's classical multi-hypothesis Chernoff bound. This quantity is defined as the minimum of the pairwise binary Chernoff divergences min_{j<k}C(\sigma_j,\sigma_k). It was known already that the optimal asymptotic rate must lie between C/3 and C, and that for certain classes of sets of states the bound is actually achieved. It was known to be achieved, in particular, when the state pair that is closest together in Chernoff divergence is more than 6 times closer than the next closest pair. Our results improve on this in two ways. Firstly, we show that the optimal asymptotic rate must lie between C/2 and C. Secondly, we show that the Chernoff bound is already achieved when the closest state pair is more than 2 times closer than the next closest pair. We also show that the Chernoff bound is achieved when at least $r-2$ of the states are pure, improving on a previous result by Nussbaum and Szko{\l}a. Finally, we indicate a number of potential pathways along which a proof (or disproof) may eventually be found that the multi-hypothesis quantum Chernoff bound is always achieved.Comment: 50 pages. v3: Slightly restructured, main results unchanged, connection to Barnum and Knill's result (arXiv:quant-ph/0004088) clarified. Accepted for JM

Bayesian Bounds on Parameter Estimation Accuracy for Compact Coalescing Binary Gravitational Wave Signals

A global network of laser interferometric gravitational wave detectors is projected to be in operation by around the turn of the century. Here, the noisy output of a single instrument is examined. A gravitational wave is assumed to have been detected in the data and we deal with the subsequent problem of parameter estimation. Specifically, we investigate theoretical lower bounds on the minimum mean-square errors associated with measuring the parameters of the inspiral waveform generated by an orbiting system of neutron stars/black holes. Three theoretical lower bounds on parameter estimation accuracy are considered: the Cramer-Rao bound (CRB); the Weiss-Weinstein bound (WWB); and the Ziv-Zakai bound (ZZB). We obtain the WWB and ZZB for the Newtonian-form of the coalescing binary waveform, and compare them with published CRB and numerical Monte-Carlo results. At large SNR, we find that the theoretical bounds are all identical and are attained by the Monte-Carlo results. As SNR gradually drops below 10, the WWB and ZZB are both found to provide increasingly tighter lower bounds than the CRB. However, at these levels of moderate SNR, there is a significant departure between all the bounds and the numerical Monte-Carlo results.Comment: 17 pages (LaTeX), 4 figures. Submitted to Physical Review

Correlator Bank Detection of GW chirps. False-Alarm Probability, Template Density and Thresholds: Behind and Beyond the Minimal-Match Issue

The general problem of computing the false-alarm rate vs. detection-threshold relationship for a bank of correlators is addressed, in the context of maximum-likelihood detection of gravitational waves, with specific reference to chirps from coalescing binary systems. Accurate (lower-bound) approximants for the cumulative distribution of the whole-bank supremum are deduced from a class of Bonferroni-type inequalities. The asymptotic properties of the cumulative distribution are obtained, in the limit where the number of correlators goes to infinity. The validity of numerical simulations made on small-size banks is extended to banks of any size, via a gaussian-correlation inequality. The result is used to estimate the optimum template density, yielding the best tradeoff between computational cost and detection efficiency, in terms of undetected potentially observable sources at a prescribed false-alarm level, for the simplest case of Newtonian chirps.Comment: submitted to Phys. Rev.

Data analysis of gravitational-wave signals from spinning neutron stars. III. Detection statistics and computational requirements

We develop the analytic and numerical tools for data analysis of the gravitational-wave signals from spinning neutron stars for ground-based laser interferometric detectors. We study in detail the statistical properties of the optimum functional that need to be calculated in order to detect the gravitational-wave signal from a spinning neutron star and estimate its parameters. We derive formulae for false alarm and detection probabilities both for the optimal and the suboptimal filters. We assess the computational requirements needed to do the signal search. We compare a number of criteria to build sufficiently accurate templates for our data analysis scheme. We verify the validity of our concepts and formulae by means of the Monte Carlo simulations. We present algorithms by which one can estimate the parameters of the continuous signals accurately.Comment: LaTeX, 45 pages, 13 figures, submitted to Phys. Rev.

On Estimation of the Post-Newtonian Parameters in the Gravitational-Wave Emission of a Coalescing Binary

The effect of the recently obtained 2nd post-Newtonian corrections on the accuracy of estimation of parameters of the gravitational-wave signal from a coalescing binary is investigated. It is shown that addition of this correction degrades considerably the accuracy of determination of individual masses of the members of the binary. However the chirp mass and the time parameter in the signal is still determined to a very good accuracy. The possibility of estimation of effects of other theories of gravity is investigated. The performance of the Newtonian filter is investigated and it is compared with performance of post-Newtonian search templates introduced recently. It is shown that both search templates can extract accurately useful information about the binary.Comment: 34 pages, 118Kb, LATEX format, submitted to Phys. Rev.