Causality and stochastic realization

In this paper we consider a problem that follows directly from realization problem: how to find Markovian representations , even minimall, for a given family of Hilbert spaces, understood as outputs of a stochastic dynamic system S1, provided it is in a certain causality relationship with another family of Hilbert spaces , i. e. with some informations about states of a stochastic dynamic system S2. This paper is continuation of the papers Gill and Petrovic [7] and Petrovic [16,17].

We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length.

: Given a mean-square continuous stochastic vector process y with stationary increments and a rational spectral density Phi such that Phi(infinity) is finite an nonsingular, consider the problem of finding all minimal Gauss-Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal Stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman-Bucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families of external realizations defined on the same probability space and totally ordered with respect to state covariances. (Author)

Previous work on the stochastic realization and approximation problem has cast this problem in the framework of theRV-coefficient, a measure of correlation recently introduced in the multivariate statistical literature. This allowed the introduction of a common measure for the “goodness of fit” for the different realization algorithms. This paper explores the deeper geometrical and logical foundation for this common measure in a unified theory for the data-driven and the exact covariance approaches.

This paper attempts to view texture analysis and synthesis in image processing as a problem of realization and subspace identification in terms of reciprocal processes defined on a finite interval. We discuss modeling of textures in the framework of acausal stochastic systems and stochastic realization theory.

Systemrealization is the construction of a state-space model given input-output data of a system. One approach, briefly summarized here, is the subspace method. In the deterministic realization problem, the data are used in alinear fashion, whereas the stochastic realization problem usesquadratic forms in the data. This dichotomy is related to the basic assumptions of repeatability or nonrepeatability of the input-output experiments performed on the system. In particular, the logic of the system is constructed, closely following the axiomatic foundations of physics. It is shown that this logic is Boolean in the deterministic and quantal in the stochastic case. The system dynamics is obtained from the data-induced measures one can define on the lattice.

This paper uses noncommutative dilation theory to develop a theory of stochastic bilinear systems. A certain stochastic bilinear realization and partial realization problem is solved. A new Caratheodory interpolation problem is presented and solved.

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M2 -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

We present a solution of the extended Phillips-Kato extension problem about existence and parametrization of all accretive (*)-extensions (with the exit into triplets of rigged Hilbert spaces) of a densely defined non-negative operator.I n particular, the analogs of the von Neumann and Friedrichs theorems for existence of non-negative self-adjoint (*)-extensions are obtained. Relying on these results we introduce the extremal classes of Stieltjes and inverse Stieltjes functions and show that each function from these classes can be realized as the impedance function of an L-system.I t is proved that in this case the realizing L-system contains an accretive operator and, in case of Stieltjes functions, an accretive (*)-extension.M oreover, we establish the connection between the above-mentioned classes and the Friedrichs and Kre”n-von Neumann extremal non-negative extensions.

Greedy algorithms are central to sparse approximation and stage-wise learning methods such as matching pursuit and boosting. It is known that the Power-Relaxed Greedy Algorithm with step sizes m−α may fail to converge when α>1 in general Hilbert spaces. In this work, we revisit this phenomenon from a sparse learning perspective. We study realizable regression problems with controlled feature coherence and derive explicit lower bounds on the residual norm, showing that over-decaying step-size schedules induce structural stagnation even in low-dimensional sparse settings. Numerical experiments confirm the theoretical predictions and illustrate the role of feature coherence. Our results provide insight into step-size design in greedy sparse learning.

On the Nevanlinna-Pick interpolation problem: Analysis of the McMillan degree of the solutions

A solution of the nonlinear stochastic realization problem

A counterexample is presented to show that the solution to the stochastic realization problem given in the above-titled letter (ibid., vol.75, no.9, p.1333-4, Sept. 1987) does not apply, in general, to the infinite impulse response systems. The author expands on the comments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>