Finite Number Of Observations Research Articles

Invariance transformations in wave-optics lensing: Implications for gravitational-wave astrophysics and cosmology

Gravitational lensing offers unique opportunities to learn about the astrophysical origin of distant sources, the abundance of intervening objects acting as lenses, and gravity and cosmology in general. However, all this information can only be retrieved as long as one can disentangle each effect from the finite number of observables. In the geometric optics regime, typical of electromagnetic radiation, when the wavelength of the lensed signal is small compared to the size of the lens, there are invariance transformations that change the mass of the lens and the source-lens configuration but leave the observables unchanged. Neglecting this “mass-sheet degeneracy” can lead to biased lens parameters or unrealistic low uncertainties, which could then transfer to an incorrect cosmography study. This might be different for gravitational waves as their long wavelengths can be comparable to the lens size and lensing enters into the wave-optics limit. We explore the existence of invariance transformations in the wave-optics regime of gravitational-wave lensing, extending previous work and examining the implications for astrophysical and cosmological studies. We study these invariance transformations using three different methods of increasing level of complexity: template mismatch, Fisher Matrix, and Bayesian parameter estimation. We find that, for a sufficiently loud signal, the degeneracy is partially broken and the lens and cosmological parameters, e.g. H0, can be retrieved independently and unbiased. In current ground-based detectors, though, considering also population studies, a strong constraint on these parameters seems quite remote and the prevailing degeneracy implies a larger uncertainty in the lens model reconstruction. However, with better sensitivity of the third-generation ground-based detectors, a meaningful constraint on H0 is possible to obtain. Published by the American Physical Society 2024

Supervisor synthesis for asynchronous diagnosability enforcement in labeled Petri nets

The asynchronous fault diagnosis problem involves detecting the occurred faults in a system in the scenario that one asynchronously initiates a diagnosis agent with the system. This research investigates the asynchronous fault diagnosis problem of a system represented by labeled Petri nets (LPNs). An important property named asynchronous diagnosability of an LPN plant is formulated and studied which indicates that the detection of a fault can be achieved within a finite number of observations in the context of asynchronous diagnosis. This paper first introduces a structure named an asynchronous basis diagnoser, based on which the asynchronous diagnosis problem of an LPN plant can be addressed. Moreover, an approach is presented which leverages the asynchronous basis diagnoser to identify the asynchronous diagnosability. If an LPN plant is not asynchronously diagnosable, a supervisor is then designed which constraints the plant's behavior such that the controlled system satisfies the requirement of asynchronous diagnosability.

Learning stability guarantees for constrained switching linear systems from noisy observations

We present a data-driven framework based on Lyapunov theory to provide stability guarantees for a family of hybrid systems. In particular, we are interested in the asymptotic stability of switching linear systems whose switching sequence is constrained by labeled graphs, namely constrained switching linear systems. In order to do so, we provide chance-constrained bounds on stability guarantees, that can be obtained from a finite number of observations with bounded noise. We first present a method providing stability guarantees from sampled trajectories in the hybrid state space of the system. We then study the harder situation where one only observes the continuous part of the hybrid states. We show that in this case, one may still obtain formal chance-constrained stability guarantees. For this latter result we provide a new upper bound of general interest, also for model-based stability analysis.

Support Recovery and Parameter Identification of Multivariate ARMA Systems with Exogenous Inputs

Focusing on identification, support recovery, and parameter estimation of multivariate ARMA systems with exogenous inputs (ARMAX) is considered. It is assumed that sparsity appears not only in the unknown parameter matrix of the system inputs and outputs but also in the coefficients of system noises, and with exogenous inputs the system admits feedback control and thus the observation of the ARMAX system may be nonstationary. A two-stage sparse identification algorithm is introduced. First, the extended least squares (ELS) algorithm is applied to obtain an initial estimate for the unknown parameter matrix of the ARMAX system. Second, a convex optimization criterion is introduced based on norm with regularization, where adaptive weights generated from ELS algorithms are adopted in the optimization variables in the term. Third, it is proved that by optimizing the criterion, the sets of the zero and the nonzero entries of the parameter matrix, i.e., the support set of the ARMAX system, can be correctly identified with a finite number of observations, and estimates of the nonzero parameters converge to the true values with probability one. Fourth, the strictly positive realness (SPR) condition required by the ELS algorithm is relaxed by an overparametrization technique. The performance of the algorithm is testified through simulation examples.

Distributed sparse identification for stochastic dynamic systems under cooperative non-persistent excitation condition

This paper considers the distributed sparse identification problem over wireless sensor networks such that all sensors cooperatively estimate the unknown sparse parameter vector of stochastic dynamic systems by using the local information from neighbors. A distributed sparse least squares algorithm is proposed by minimizing a local information criterion formulated as a linear combination of accumulative local estimation error and L1-regularization term. The upper bound of the estimation error of the proposed algorithm is presented. Furthermore, by designing a suitable adaptive weighting coefficient based on the local observation data, the set convergence of zero elements with a finite number of observations is obtained under a cooperative non-persistent excitation condition. It is shown that the proposed distributed algorithm can work well in a cooperative way even though none of the individual sensors can fulfill the estimation task. Our theoretical results are obtained without relying on the independency assumptions of regression signals that have been commonly used in the existing literature. Thus, our results are expected to be applied to stochastic feedback systems. Finally, the numerical simulations are provided to demonstrate the effectiveness of our theoretical results.

On Quasiconvex Functions Which are Convexifiable or Not

A quasiconvex function f being given, does there exist an increasing and continuous function k which makes \(k\circ f\) convex? How to build such a k? Some words on least convex (concave) functions. The ratio of two positive numbers is neither locally convexifiable nor locally concavifiable. Finally, some considerations on the approximation of a preorder from a finite number of observations and on the revealed preference problem are discussed.

Analysis of strong and strong periodic detectability of bounded labeled Petri nets

Detectability describes the property of a system to uniquely determine, after a finite number of observations, the current and the subsequent states. Different notions of detectability have been proposed in the literature. In this paper, we formalize and analyze strong detectability and strong periodic detectability for systems that are modeled as labeled Petri nets with partial observation on their transitions. We provide three new approaches for the verification of such detectability properties using three different structures. The computational complexity of the proposed approaches is analyzed and the three methods are compared. The main feature of all the three approaches is that they do not require the calculation of the entire reachability space or the construction of an observer. As a result, they have lower computational complexity than other methods in the literature.

Optimal Spectrum Sensing in MIMO-Based Cognitive Radio Wireless Sensor Network (CR-WSN) Using GLRT With Noise Uncertainty at Low SNR

Noise uncertainty can severely deteriorate a primary user (PU) detector’s sensing performance, so robustness against the noise uncertainty is of fundamental significance in cognitive radio networks. In this paper, we investigate robust detection schemes in the presence of noise uncertainty for multiple-input multiple-output (MIMO) based cognitive radio wireless sensor networks (CR-WSN) with multiple spectrum sensing scenarios. In practice, it is very seldom that accurate statistical characterization of noise is known a priori. In this context, we propose the generalized likelihood ratio test (GLRT) paradigm and estimator-correlator based optimal detectors to sense the unoccupied primary bands in the presence of noise uncertainty. First, we compute the estimator-correlator based detector (ECD) and generalized likelihood detector (GLD) for known noise uncertainty statistics. Next, a composite hypothesis based detector (CHD) is proposed using the GLRT framework for unknown noise uncertainty statistics. The proposed detection schemes provide robustness against uncertainty in the available noise power estimates using a finite number of observations (i.e., fast spectrum sensing) and particularly in critical areas of low Signal-to-Noise Ratio (SNR). Theoretical analysis is performed based on statistical theory. Closed-form expressions for detection and false-alarm probability, and analytic decision threshold are derived to demonstrate spectrum sensing performance using receiver operating characteristic (ROC) curves. Simulation results validate that proposed PU detection schemes are robust and achieve an improved sensing performance when there is noise uncertainty and corroborate our theoretical findings.

Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations

We consider the coefficient inverse problem for the first-order hyperbolic system, which describes the propagation of the 2D acoustic waves in a heterogeneous medium. We recover both the denstity of the medium and the speed of sound by using a finite number of data measurements. We use the second-order MUSCL-Hancock scheme to solve the direct and adjoint problems, and apply optimization scheme to the coefficient inverse problem. The obtained functional is minimized by using the gradient-based approach. We consider different variations of the method in order to obtain the better accuracy and stability of the appoach and present the results of numerical experiments.

Finite-Length Bounds on Hypothesis Testing Subject to Vanishing Type I Error Restrictions

A central problem in Binary Hypothesis Testing (BHT) is to determine the optimal tradeoff between the Type I error (referred to as false alarm) and Type II (referred to as miss) error. In this context, the exponential rate of convergence of the optimal miss error probability -- as the sample size tends to infinity -- given some (positive) restrictions on the false alarm probabilities is a fundamental question to address in theory. Considering the more realistic context of a BHT with a finite number of observations, this paper presents a new non-asymptotic result for the scenario with monotonic (sub-exponential decreasing) restriction on the Type I error probability, which extends the result presented by Strassen in 2009. Building on the use of concentration inequalities, we offer new upper and lower bounds to the optimal Type II error probability for the case of finite observations. Finally, the derived bounds are evaluated and interpreted numerically (as a function of the number samples) for some vanishing Type I error restrictions.

Sparse system identification for stochastic systems with general observation sequences

Focusing on identification, this paper develops techniques to reconstruct zero and nonzero elements of a sparse parameter vector θ of a stochastic dynamic system with general observation sequences, including stationary time series and feedback control, for which the current input may depend on the past inputs and outputs, system noises as well as exogenous dithers. First, a sparse parameter identification algorithm is introduced based on L2 norm with L1 regularization, where the adaptive weights are adopted in the optimization variables of L1 term. Second, estimates generated by the algorithm are shown to have both set and parameter convergence. That is, sets of the zero and nonzero elements in the parameter θ can be correctly identified with probability one using a finite number of observations, and further estimates of the nonzero elements converge to the true values almost surely. Third, it is shown that the results are applicable to a large number of applications, including variable selection, open-loop identification, and closed-loop control of stochastic systems. Finally, numerical examples are given to support the theoretical analysis.

An optimization problem based on a Bayesian approach for the 2D Helmholtz equation

Elastography is a ill-posed inverse problem that aims at recovering the Lame and density of the domain of interest from finite number of observations, we consider as model the Helmoltz equation. We present an implementation for solving the Helmholtz inverse problem in two dimensions via an optimization problem based on Bayesian approach. In addition, the accuracy of the method is also investigated with respect to the amount of information taken from the generalized Hermitian Eigenvalue problem and by comparing the maximum a posterior estimate to the true parameter distribution in simulated experiments.

Verification of Detectability for Unambiguous Weighted Automata

In this article, we deal with the detectability problem for unambiguous weighted automata (UWAs). The problem is to determine if, after a finite number of observations, the set of possible states is reduced to a singleton. Four types of detectabilities, namely, strong detectability, detectability, strong periodic detectability, and periodic detectability are defined in terms of different requirements for current state estimation. We first construct a deterministic finite state automaton (called observer) over a weighted alphabet and prove that it can be used as the current-state estimator of the studied UWA. Finally, necessary and sufficient conditions based on the observer are proposed to verify detectabilities of a UWA.

Verification of C-detectability using Petri nets

Detectability describes the property of a system to uniquely determine, after a finite number of observations, the current and the subsequent states. In this paper, we extend detectability to C-detectability that only requires that a given set of crucial states can be distinguished from other states. We define four types of C-detectability in the framework of labeled Petri nets: strong C-detectability, weak C-detectability, periodically strong C-detectability, and periodically weak C-detectability. Moreover, we propose efficient approaches to verify such properties in the case of bounded labeled Petri net systems. The proposed approaches use the notion of basis marking and thus do not require an exhaustive enumeration of the reachability space.

Deciding detectability for labeled Petri nets

Detectability of discrete event systems is a property to decide whether the current and subsequent states can be determined based on observations. We investigate the existence of algorithms for checking strong and weak detectability for systems modeled as labeled Petri nets. Strong detectability requires that we can always determine, after a finite number of observations, the current and subsequent markings of the system, while weak detectability requires that we can determine, after a finite number of observations, the current and subsequent markings for some trajectories of the system. We show that there is an algorithm to check strong detectability requiring exponential space, and that there is no algorithm to check weak detectability.

A new approach in non-parametric estimation of returns in mean-downside risk portfolio frontier

The DownSide Risk (DSR) model for portfolio optimization allows to overcome the drawbacks of the classical Mean-Variance model concerning the asymmetry of returns and the risk perception of investors. This optimization model deals with a positive definite matrix that is endogenous with respect to the portfolio weights and hence yields to a non standard optimization problem. To bypass this hurdle, Athayde (2001) developed a new recursive minimization procedure that ensures the convergence to the solution. However, when a finite number of observations is available, the portfolio frontier usually exhibits some inflexion points which make this curve not very smooth. In order to overcome these points, Athayde (2003) proposed a mean kernel estimation of returns to get a smoother portfolio frontier. This technique provides an effect similar to the case in which an infinite number of observations is available. In spite of the originality of this approach, the proposed algorithm was not neatly written. Moreover, no application was presented in his paper. Ben Salah et al (2015), taking advantage on the the robustness of the median, replaced the mean estimator in Athayde's model by a nonparametric median estimator of the returns, and gave a tidily and comprehensive version of the former algorithm (of Athayde (2001, 2003)). In all the previous cases, the problem is computationally complex since at each iteration, the returns (for each asset and for the portfolio) need to be re-estimated. Due to the changes in the kernel weights for every time, the portfolio is altered. In this paper, a new method to reduce the number of iterations is proposed. Its principle is to start by estimating non parametrically all the returns for each asset; then, the returns of a given portfolio will be derived from the previous estimated assets returns. Using the DSR criterion and Athayde's algorithm, a smoother portfolio frontier is obtained when short selling is or is not allowed. The proposed approach is applied on the French and Brazilian stock markets.

Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations

We consider an inverse problem of reconstructing two spatially varying coefficients in an acoustic equation of hyperbolic type using interior data of solutions with suitable choices of initial condition. Using a Carleman estimate, we prove Lipschitz stability estimates which ensure unique reconstruction of both coefficients. Our theoretical results are justified by numerical studies on the reconstruction of two unknown coefficients using noisy backscattered data.

Information inequalities for characteristics of group-sequential test with groups of observations of random size

We consider a group-sequential test for testing a simple hypothesis against a composite one-sided alternative, which defines the following sequential statistical procedure: At each stage a random number of independent identically distributed observations (a group of observations) is observed and, based on the collected data, the decision to accept or to reject the hypothesis or to continue the observation is made. For the tests with finite number of observations, we prove the existence of the derivative of the power function and establish the information-type inequalities relating that derivative to other characteristics of the test: the average number of observations and the type I error.

Perfect sampling for nonhomogeneous Markov chains and hidden Markov models

We obtain a perfect sampling characterization of weak ergodicity for backward products of finite stochastic matrices, and equivalently, simultaneous tail triviality of the corresponding nonhomogeneous Markov chains. Applying these ideas to hidden Markov models, we show how to sample exactly from the finite-dimensional conditional distributions of the signal process given infinitely many observations, using an algorithm which requires only an almost surely finite number of observations to actually be accessed. A notion of “successful” coupling is introduced and its occurrence is characterized in terms of conditional ergodicity properties of the hidden Markov model and related to the stability of nonlinear filters.

Mean and median-based nonparametric estimation of returns in mean-downside risk portfolio frontier

The DownSide Risk (DSR) model for portfolio optimisation allows to overcome the drawbacks of the classical Mean-Variance model concerning the asymmetry of returns and the risk perception of investors. This model optimization deals with a positive definite matrix that is endogenous with respect to portfolio weights. This aspect makes the problem far more difficult to handle. For this purpose, Athayde (2001) developed a new recursive minimization procedure that ensures the convergence to the solution. However, when a finite number of observations is available, the portfolio frontier presents some discontinuity and is not very smooth. In order to overcome that, Athayde (2003) proposed a Mean Kernel estimation of the returns, so as to create a smoother portfolio frontier. This technique provides an effect similar to the case in which continuous observations are available. In this paper, Athayde model is reformulated and clarified. Then, taking advantage on the robustness of the median, another nonparametric approach based on Median Kernel returns estimation is proposed in order to construct a portfolio frontier. A new version of Athayde's algorithm will be exhibited. Finally, the properties of this improved portfolio frontier are studied and analysed on the French Stock Market. Keywords DownSide Risk · Kernel Method · Mean Nonparametric Estimation · Median Nonparametric Estimation · Portefolio Efficient Frontier · Semi-Variance.