Approximation Of Random Processes Research Articles

Stability-preserving model order reduction for linear stochastic Galerkin systems

Mathematical modeling often yields linear dynamical systems in science and engineering. We change physical parameters of the system into random variables to perform an uncertainty quantification. The stochastic Galerkin method yields a larger linear dynamical system, whose solution represents an approximation of random processes. A model order reduction (MOR) of the Galerkin system is advantageous due to the high dimensionality. However, asymptotic stability may be lost in some MOR techniques. In Galerkin-type MOR methods, the stability can be guaranteed by a transformation to a dissipative form. Either the original dynamical system or the stochastic Galerkin system can be transformed. We investigate the two variants of this stability-preserving approach. Both techniques are feasible, while featuring different properties in numerical methods. Results of numerical computations are demonstrated for two test examples modeling a mechanical application and an electric circuit, respectively.

The Behavior of the Generator Normalization Factor in Approximation of Random Processes

Markov random evolutions and their approximations are analyzed. The main object of study is generators of random processes with independent increments. These processes are considered in Poisson approximation and Levy approximation schemes. Generators of random processes are normalized by parameters that are nonlinear functions. The explicit form of such normalization parameters is shown. The asymptotic representation of generators in both approximation schemes is shown. Normalizing factors of random evolution are presented.

A versatile technique for the optimal approximation of random processes by Functional Quantization

This paper presents the mathematical foundation of a novel and versatile technique for the approximation of random processes using the Functional Quantization concept. In this approach, random processes are represented by optimal quantizers; that is, finite collections of deterministic functions and corresponding probability masses that are carefully constructed to attain certain optimal properties. Although the computational cost of obtaining such optimal quantizers is not negligible, their use in applications is simple and relatively inexpensive. The paper has two objectives. First, we present an accessible overview of Functional Quantization theory. Second, we introduce a novel methodology to compute optimal quantizers, which is based on the classical Lloyd’s Method and Monte Carlo Simulation. For validation purposes, the proposed methodology is tested using two fundamental Gaussian random processes (Brownian motion and fractional Brownian motion) for which optimal quantizers are known. The results obtained compare very well with previous results reported in the literature. Then, to demonstrate the unique versatility of the methodology, the application to a non-Gaussian process is presented.

Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs.

Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Galán recently introduced a novel stochastic shielding approximation as a fast, accurate method for generating approximate sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion-channel models, such as the Hodgkin–Huxley or other conductance-based neural models, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. We consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion-channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Galán’s approximation is in fact optimal in a specific sense, we use recent results from random matrix theory to provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of random graphs. Moreover, we provide a novel quantitative measure of the contribution of individual transitions within the reaction graph to the accuracy of the approximate process.

On support vector machines and sparse approximation for random processes

After reviewing the role played by support vector machines and other sparse approximation instruments in studies about random processes, this work aims to emphasize some of their properties and proposes a space dimensionality reduction approach which is conceptually simple and computationally feasible. The stochastic analysis of random phenomena is often required in a context characterized by non-standard probabilistic assumptions. In other terms, real-life problems offer statistical model conditions such as strong forms of dependence, non-stationarity, non-gaussianity, for instance; it often appears that these features prevent from using classical statistical inference procedures or model-building strategies, and instead require computational techniques based on simulations and numerical approximation. One initial problem, here investigated, is how to cast these instruments in a certain theoretical framework, for then achieving suitable model structure and interpretation. This contribution, which is of an introductory nature, suggests further advances into the domain of real life applications, in particular time series.

Spline approximation of random processes and design problems

Abstract We consider the spline approximation of a continuous (continuously differentiable) random process with finite second moments based on n observations of the process (and its derivatives). The performance of the approximation is measured by mean errors (e.g., integrated or maximal quadratic mean errors). For Hermite interpolation splines, an optimal rule sets n observation locations (i.e., a design , a mesh ). While, for a fixed n , an optimal rule is difficult to construct, we find the sequence of designs with asymptotically optimal properties as n →∞. We investigate the class of locally stationary random processes whose local behavior is like m -fold integrated fractional Brownian motion for a given non-negative m in the quadratic mean sense.

Multidimensional Random Process Synthesis and Simulation

In the paper, wide-sense stationary M-D (multidimensional) random processes, defined by their power spectral densities, are synthesised and simulated by using as an approximation M-D multisine random processes consisting of a sum of M-D sine components with deterministic amplitudes and random phase shifts. On the basis of the power spectral density function and phase shifts chosen with some well defined random properties the finite discrete Fourier transform of the M-D multisine random process is synthesised. The corresponding M-D multisine random process approximation is simulated by performing an inverse M-D finite Fourier transform of the synthesised spectrum. Realisations of the synthesised and simulated random process approximations may be obtained by using an M-D fast Fourier transform algorithm. The properties of synthesised M-D multisine random processes are discussed including M-D white multisine random process.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Propagation of a beam of light in a medium made up of large correlated black disks

We consider the passage of a beam of light through a layer or a monolayer of a medium consisting of randomly distributed optically rigid scatterers which are large compared to a wavelength. Of the shadow-forming and refracted components of the field scattered from an isolated scatterer, we consider only the former. We obtain as a result a model of a medium consisting of black disks which are oriented with their planes perpendicular to the axis of the light beam. We take all order of mutual correlation between disks into acocunt, without any limitation on their spatial density distribution. With the aid of a parabolic equation, we show that when the disks are 6-correlated in the direction of the light-beam axis, the statistical moments of the wave field can be calculated exactly in two ways: via a Markov random process approximation and via a single-group approximation. We give a graphic geometric/probabilistic interpretation of quantities related to the coefficient of the equation for the statistical moment of the field.

On the Approximation of Random Processes by Convolution Processes

AbstractThe smoothness in the mean of a weak sense stationary process is closely connected with that of its autocorrelation function. This enables one to prove results on the order of convergence of certain convolution processes to the given random process. On the other hand, it is possible to conclude smoothness properties, in case the order is known.

The maximum mutual information between two random processes

Given two discrete random processes, what is the largest possible average mutual information between them? An application of the Ornstein—Sinai theorem of ergodic theory is used to show that if the processes are ergodic, then there exists a pair process with the given processes as coordinates such that the average mutual information between the coordinates is the maximum possible value, the smaller of the two entropy rates. As an application and interpretation, the information—theoretic analog of a recently developed distance measure on random processes is defined and the topologies induced by the two distance measures on spaces of random processes are compared and contrasted. This demonstrates a fundamental relation between mutual information and mutual approximation of random processes.

Derivation of an exact spectral density transport equation for a nonstationary scattering medium

Within the framework of the quasioptical description and the pure Markovian random process approximation, an exact kinetic equation is derived for the spectral density function in the case of wave propagation in a nondispersive medium characterized by large-scale space–time fluctuations. Also, a quantity, called the degree of coherence function, is defined as a quantitative measure of the irreversible effects of randomness.

Diffusive random process approximation in certain nonstationary statistical problems of physics

The review considers, on the basis of a unified approach, the problem of Brownian motion in nonlinear dynamic systems, including a linear oscillator acted upon by random forces, parametric resonance in an oscillating system with random parameters, turbulent diffusion of particles in a random-velocity field, and diffusion of rays in a medium with random inhomogeneities of the refractive index. The same method is used to consider also more complicated problems such as equilibrium hydrodynamic fluctuations in an ideal gas, description of hydrodynamic turbulence by the method of random forces, and propagation of light in a medium with random inhomogeneities. The method used to treat these problems consists of constructing equations for the probability density of the system or for its statistical moments, using as the small parameter the ratio of the characteristic time of the random actions to the time constant of the system (in many problems, the role of the time is played by one of the spatial coordinates). The first-order approximation of the method is equivalent to replacement of the real correlation function of the action by a δ function; this yields equations for the characteristics in closed form. The method makes it possible to determine also higher approximations in terms of the aforementioned first-order small parameter.

Приближение диффузионного случайного процесса в некоторых нестационарных статистических задачах физики

Приближение диффузионного случайного процесса в некоторых нестационарных статистических задачах физики

On Stochastic Approximation for Random Processes with Continuous Time

Previous article Next article On Stochastic Approximation for Random Processes with Continuous TimeT. P. KrasulinaT. P. Krasulinahttps://doi.org/10.1137/1116073PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Herbert Robbins and , Sutton Monro, A stochastic approximation method, Ann. Math. Statistics, 22 (1951), 400–407 MR0042668 0054.05901 CrossrefGoogle Scholar[2] J. Kiefer and , J. Wolfowitz, Stochastic estimation of the maximum of a regression function, Ann. Math. Statistics, 23 (1952), 462–466 MR0050243 0049.36601 CrossrefGoogle Scholar[3] Miloslav Driml and , J. Nedoma, Stochastic approximations for continuous random processes, Trans. 2nd Prague Conf. Information Theory, Publ. House Czechoslovak Acad. Sci., Prague, Academic Press, New York, 1960, 145–158 MR0126877 0098.32202 Google Scholar[4] Jerome Sacks, Asymptotic distribution of stochastic approximation procedures, Ann. Math. Statist., 29 (1958), 373–405 MR0098427 0229.62010 CrossrefGoogle Scholar[5] David J. Sakrison, A continuous Kiefer-Wolfowitz procedure for random processes, Ann. Math. Statist., 35 (1964), 590–599 MR0161445 0131.35905 CrossrefGoogle Scholar[6] V. A. Volkonskii˘ and , Yu. A. Rozanov, Some limit theorems for random functions. I, Theor. Probability Appl., 4 (1959), 178–197 MR0121856 0092.33502 LinkGoogle Scholar[7] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Prob. Applications, 7 (1962), 349–382 10.1137/1107036 0119.14204 LinkGoogle Scholar[8] S. G. Mikhlin, Lectures on Linear Integral Equations, Fizmatgiz, Moscow, 1959, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Gradient procedures for stochastic approximation with dependent noise and their asymptotic behaviourInternational Journal of Systems Science, Vol. 16, No. 8 | 10 May 2007 Cross Ref Some Limit Theorems for Processes with Random TimeA. N. BorodinTheory of Probability & Its Applications, Vol. 24, No. 4 | 17 July 2006AbstractPDF (1390 KB)Sufficient conditions for convergence of stochastic approximation algorithms for random processes with continuous timeCybernetics, Vol. 15, No. 6 | 1 Nov 1979 Cross Ref A Stochastic Approximation Procedure in the Case of Weakly Dependent ObservationsA. N. BorodinTheory of Probability & Its Applications, Vol. 24, No. 1 | 17 July 2006AbstractPDF (1321 KB)Recursive stochastic gradient procedures in the presence of dependent noiseStochastic Optimization Cross Ref Volume 16, Issue 4| 1971Theory of Probability & Its Applications575-746 History Submitted:07 July 1969Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1116073Article page range:pp. 674-682ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On Linear Approximation of Random Processes in the Case of Random Observation

On Linear Approximation of Random Processes in the Case of Random Observation

On best approximation of random processes et al. (Corresp.)

On best approximation of random processes et al. (Corresp.)