Strong Consistency of the Over- and Underdetermined LSE of 2-D Exponentials in White Noise

Abstract—We consider the problem of least squares estimation of the parameters of two–dimensional (2-D) exponential signals observed in the presence of an additive noise field, when the assumed number of exponentials is incorrect. We consider both the case where the number of exponential signals is underestimated, and the case where the number of exponential signals is overestimated. In the case where the number of exponential signals is underestimated, we prove the almost sure convergence of the least squares estimates (LSE) to the parameters of the dominant exponentials. In the case where the number of exponential signals is overestimated, the estimated parameter vector obtained by the least squares estimator contains a subvector that converges almost surely to the correct parameters of the exponentials. Index Terms—Least squares estimation, model-order selection, random fields, strong consistency, two–dimensional (2-D) exponentials, 2-D parameter estimation. I

Proceedings of the Fifth European Workshop on Probabilistic Graphical Models

Peer reviewe

Putting reaction-diffusion systems into port-Hamiltonian framework

Reaction-diffusion systems model the evolution of the constituents distributed in space under the influence of chemical reactions and diffusion [6], [10]. These systems arise naturally in chemistry [5], but can also be used to model dynamical processes beyond the realm of chemistry such as biology, ecology, geology, and physics. In this paper, by adopting the viewpoint of port-controlled Hamiltonian systems [7] we cast reaction-diffusion systems into the portHamiltonian framework. Aside from offering conceptually a clear geometric interpretation formalized by a Stokes-Dirac structure [8], a port-Hamiltonian perspective allows to treat these dissipative systems as interconnected and thus makes their analysis, both quantitative and qualitative, more accessible from a modern dynamical systems and control theory point of view. This modeling approach permits us to draw immediately some conclusions regarding passivity and stability of reaction-diffusion systems. It is well-known that adding diffusion to the reaction system can generate behaviors absent in the ode case. This primarily pertains to the problem of diffusion-driven instability which constitutes the basis of Turing’s mechanism for pattern formation [11], [5]. Here the treatment of reaction-diffusion systems as dissipative distributed portHamiltonian systems could prove to be instrumental in supply of the results on absorbing sets, the existence of the maximal attractor and stability analysis. Furthermore, by adopting a discrete differential geometrybased approach [9] and discretizing the reaction-diffusion system in port-Hamiltonian form, apart from preserving a geometric structure, a compartmental model analogous to the standard one [1], [2] is obtaine