1,721,050 research outputs found
Graphical model selection for a particular class of continuous-time processes
Full text linksummary:Graphical models provide an undirected graph representation of relations between the components of a random vector. In the Gaussian case such an undirected graph is used to describe conditional independence relations among such components. In this paper, we consider a continuous-time Gaussian model which is accessible to observations only at time . We introduce the concept of infinitesimal conditional independence for such a model. Then, we address the corresponding graphical model selection problem, i. e. the problem to estimate the graphical model from data. Finally, simulation studies are proposed to test the effectiveness of the graphical model selection procedure
A Second-Order Generalization of TC and DC Kernels
Full text linkKernel-based methods have been successfully introduced in system identification to estimate the impulse response of a linear system. Adopting the Bayesian viewpoint, the impulse response is modeled as a zero mean Gaussian process whose covariance function (kernel) is estimated from the data. The most popular kernels used in system identification are the tuned-correlated (TC), the diagonal-correlated (DC) and the stable spline (SS) kernel. TC and DC kernels admit a closed-form factorization of the inverse. The SS kernel induces more smoothness than TC and DC on the estimated impulse response, however, the aforementioned property does not hold in this case. In this paper we propose a second-order extension of the TC and DC kernel, which induces more smoothness than TC and DC, respectively, on the impulse response and a generalized-correlated kernel, which incorporates the TC and DC kernels and their second order extensions. Moreover, these generalizations admit a closed-form factorization of the inverse and thus they allow to design efficient algorithms for the search of the optimal kernel hyperparameters. We also show how to use this idea to develop higher oder extensions. Interestingly, these new kernels belong to the family of the so called exponentially convex local stationary kernels: such a property allows to immediately analyze the frequency properties induced on the estimated impulse response by these kernels
A New Family of High-Resolution Multivariate Spectral Estimators
No full textIn this paper, we extend the Beta divergence family to multivariate power spectral densities. Similarly to the scalar case, we show that it smoothly connects the multivariate Kullback-Leibler divergence with the multivariate Itakura-Saito distance. We successively study a spectrum approximation problem, based on the Beta divergence family, which is related to a multivariate extension of the THREE spectral estimation technique. It is then possible to characterize a family of solutions to the problem. An upper bound on the complexity of these solutions will also be provided. Finally, we will show that the most suitable solution of this family depends on the specific features required from the estimation problem
Rational approximations of spectral densities based on the Alpha divergence
No full textWe approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by selecting the spectral density having minimum “distance” from under the constraint corresponding to imposing the given second-order statistics. We analyze the structure of the optimal solutions as the minimized “distance” varies in the Alpha divergence family. We show that the corresponding approximation problem leads to a family of rational solutions. Secondly, such a family contains the solution which generalizes the Kullback–Leibler solution proposed by Georgiou and Lindquist in 2003. Finally, numerical simulations suggest that this family contains solutions close to the non-rational solution given by the principle of minimum discrimination information
Empirical Bayesian learning in AR graphical models
No full textWe address the problem of learning graphical models which correspond to high dimensional autoregressive stationary stochastic processes. A graphical model describes the conditional dependence relations among the components of a stochastic process and represents an important tool in many fields. We propose an empirical Bayes estimator of sparse autoregressive graphical models and latent-variable autoregressive graphical models. Numerical experiments show the benefit to take this Bayesian perspective for learning these types of graphical models
Generalized Moment Problems for Estimation of Spectral Densities and Quantum Channels
Full text linkThis thesis is concerned with two generalized moment problems arising in the estimation of stochastic models.
Firstly, we consider the THREE approach, introduced by Byrnes Georgiou and Lindquist, for estimating spectral densities. Here, the output covariance matrix of a known bank of filters is used to extract information on the input spectral density which needs to be estimated. The parametrization of the family of spectral densities matching the output covariance is a generalized moment problem. An estimate of the input spectral density is then chosen from this family. The choice criterium is based on the minimization of a suitable divergence index among spectral densities. After the introduction of the THREE-like paradigm, we present a multivariate extension of the Beta divergence for solving the problem. Afterward, we deal with the estimation of the output covariance of the filters bank given a finite-length
data generated by the unknown input spectral density.
Secondly, we deal with the quantum process tomography. This problem consists in the estimation of a quantum channel which can be thought as the quantum equivalent of the Markov transition matrix in the classical setting. Here, a quantum system prepared in a known pure state is fed to the unknown channel. A measurement of an observable is performed on the output state. The set of the employed pure states and observables represents the experimental setting. Again, the parametrization of the family of quantum channels matching the measurements is a generalized moment problem. The choice criterium for the best estimate in this family is based on the maximization of maximum likelihood functionals. The corresponding estimate,
however, may not be unique since the experimental setting is not "rich" enough in many cases of interest. We characterize the minimal experimental setting which guarantees the uniqueness of the estimate. Numerical simulation evidences that experimental settings richer than the minimal one do not lead to better performance
Convergence analysis of a family of robust Kalman filters based on the contraction principle
No full textIn this paper, we analyze the convergence of a family of robust Kalman filters. For each filter of this family, the model uncertainty is tuned according to the so-called tolerance parameter. Assuming that the corresponding state-space model is reachable and observable, we show that the N-fold composition of the corresponding Riccati-like mapping is strictly contractive provided that the tolerance is suficiently small and, accordingly, the filter converges
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