Markovian Random Process Research Articles

Learning mechanism for a collective classifier based on competition driven by training examples

The purpose of this work is to modify the learning mechanism of a collective classifier in order to provide learning by population dynamics alone, without requiring an external sorting device. A collective classifier is an ensemble of non-identical simple elements, which do not have any intrinsic dynamics neither variable parameters; the classifier admits learning by adjusting the composition of the ensemble, which was provided in the preceding literature by selecting the ensemble elements using a sorting device. Methods. The population dynamics model of a collective classifier is extended by adding a “learning subsystem”, which is controlled by a sequence of training examples and, in turn, controls the strength of intraspecific competition in the population dynamics. The learning subsystem dynamics is reduced to a linear mapping with random parameters expressed via training examples. The solution to the mapping is an asymptotically stationary Markovian random process, for which we analytically find asymptotic expectation and show its variance to vanish in the limit under the specified assumptions, thus allowing an approximate deterministic description of the coupled population dynamics based on available results from the preceding literature. Results. We show analytically and illustrate it by numerical simulation that the decision rule of our classifier in the course of learning converges to the Bayesian rule under assumptions which are essentially in line with available literature on collective classifiers. The implementation of the required competitive dynamics does not require an external sorting device. Conclusion. We propose a conceptual model for a collective classifier, whose learning is fully provided by its own population dynamics. We expect that our classifier, similarly to the approaches taken in the preceding literature, can be implemented as an ensemble of living cells equipped with synthetic genetic circuits, when a mechanism of population dynamics with synthetically controlled intraspecific competition becomes available.

ALGORITHMS AND SOFTWARE SUITE FOR RELIABILITY ASSESSMENT OF COMPLEX TECHNICAL SYSTEMS

Context. One of the most essential properties of technical systems is their reliability, i.e. the ability of the system to perform intended functions, preserving with time the values of operation indicators within the predefined boundaries. The failure cost for modern complex technical system can be very high, which can result in events of different severity ranging from economic losses to harm to human life and health. Hence, the requirements for their reliability constantly increase. The reliability assessment of complex technical systems can be simplified by the combination of analytical research methods with computational capabilities of modern computers. The most widely used analytical methods are based on the theory of Markov processes which in turn provide the possibility to determine the time dependencies of probabilities of the system to be in defined states (operating, recovering, failure etc.), and thus the values and time dependencies of the reliability indices needed. These methods can be successfully used for the reliability analysis of different kinds of technical systems: both non-recovered and recovered; non-redundant and redundant of different redundancy types, maintenance priorities etc. However, the application of these methods for complex technical systems containing large number of elements meets the high dimensional calculation problem, which makes it impossible to perform these tasks manually. Hence the problem of automation of complex technical system reliability modeling using modern computational systems is very relevant research topic. To solve this problem, one can use specific algorithmic and software techniques described in this paper.Objective. The goal of this article is to develop the algorithms for automated RBD processing and reliability indices assessment of complex technical systems along with the software suite for automated reliability assessment.Method. To perform the reliability analysis the RBD approach is used which allows one to represent and visualize each element of the system in the form of a rectangle, joined by the lines in parallel or in series with other elements of the system. To obtain the reliability indices values the mathematical model of technical system reliability behavior using Markovian random process was suggested. The algorithm of RBD processing and automatic determination of operability conditions of a technical system was further considered. To calculate the minimum and maximum number of operational and failure states for the system of n elements and r recoveries the paper introduces a mathematical model based on combinatorial approach. To develop the software suite the objectoriented approach was used.Results. The algorithms and software suite allows us to easily construct RBD for a technical system, to automatically determine the operability condition with execution time of about 10 sec for 1,000 elements with mixed type of connection, to form automatically a state-and-transition matrix along with the corresponding differential equation system and solve it with total execution time of about 35 sec for 109 states and, thus to obtain the numerical values of reliability indices for the technical system studied. A case study of the reliability assessment for the system consisting of 22 elements using RBD shows that the total time of software execution is 36.712 sec. During executing of this test case the most time (35.168 sec) was spent for execution of the algorithm for construction of a state-and-transition graph consisting of 52,694 states.Conclusions. The algorithms and methods for automated reliability indices assessment of complex technical systems based on RBD approach, as well as model for estimating the number of total and working system states are presented. The modular structure of the developed software suite makes it flexible and gives an opportunity to add and make modifications of modules fast and without significant program changes.

Sub- and super-diffusion on Cantor sets: Beyond the paradox

There is no way to build a nontrivial Markov process having continuous trajectories on a totally disconnected fractal embedded in the Euclidean space. Accordingly, in order to delineate the diffusion process on the totally disconnected fractal, one needs to relax the continuum requirement. Consequently, a diffusion process depends on how the continuum requirement is handled. This explains the emergence of different types of anomalous diffusion on the same totally disconnected set. In this regard, we argue that the number of effective spatial degrees of freedom of a random walker on the totally disconnected Cantor set is equal to nsp=[D]+1, where [D] is the integer part of the Hausdorff dimension of the Cantor set. Conversely, the number of effective dynamical degrees of freedom (ds) depends on the definition of a Markov process on the totally disconnected Cantor set embedded in the Euclidean space En(n≥nsp). This allows us to deduce the equation of diffusion by employing the local differential operators on the Fα-support. The exact solutions of this equation are obtained on the middle-ϵ Cantor sets for different kinds of the Markovian random processes. The relation of our findings to physical phenomena observed in complex systems is highlighted.

Spatially constrained forest cover dynamics using Markovian random processes

Essential to analyses of forest-cover change is application of geospatial empirical or semi-empirical models of transition potentials based on the likelihood that forest land would change to non-forested land or vice versa depending on prevailing conditions of land-use change. Modeling land-cover as a function of land-use aids in understanding pertinent land-cover dynamics. This can enable forecasting of ramifications of current conversion processes on land designated for agriculture or development. National Agricultural Statistics Service (NASS) grid data published by USDA for years 1997–2002 were used as preliminary inputs. Two prevalent transition probabilities were derived: probability of a pixel changing (a) from forested to non-forested, Pfnf and (b) from non-forested to forested, Pnff. These probabilities were determined for the years (i) 1999–2000, (ii) 2000–2001, (iii) 1999–2001, and (iv) 1999–2009 using decade stratification. The maximal transition probability ranges for forest to non-forest transition were higher for 1999–2000 and 1999–2001 transiting periods compared to 2000–2001. The maximal transition probability ranges for non-forest to forest transition were lower for 1999–2000 transiting period compared to 2000–2001, and 1999–2001 transiting periods. The analysis provided a glimpse on areas deemed prone to forest conversion and those that would immensely benefit from federally funded programs.

Optimal reassignment-based complex nonlinear processing of vector discrete-continuous Markovian random processes

Methods of the Markovian estimation theory are applied to the synthesis of optimal and quasi-optimal algorithms that involve time-stepped reassignment of the values of the parameters of conditional a posteriori probability densities of filtered continuous processes. The algorithms are intended for nonlinear processing of a vector discrete-continuous Markovian random process whose continuous component is a vector diffusion Markovian process and whose discrete component is a two-component vector, each component of which is characterized by a simple several-position Markovian chain. It is assumed that one group of simultaneously processed signals is observed against the background of colored (time-correlated) noise, while the other group is observed in the presence of white Gaussian noise. Basic analytic relationships and the corresponding block diagram of a system for quasi-optimal complex nonlinear processing of signals typical of the GLONASS, GPS, Galileo, and QZSS satellite navigation systems and such satellite mobile-communications systems as the Globalstar system are presented.

Center-of-mass tomographic approach to quantum dynamics

In the tomography representation we propose a new approach, which describes the dynamics of quantum particles by the Kolmogorov equations for non-negative propagators. To solve the Kolmogorov equations we use a diffusive Markovian random processes described by the related nonlinear stochastic Langevin equations. As a result the dynamics of quantum particles is described by the proposed numerical scheme combining both Langevin dynamics and Monte Carlo methods. We test the developed approach by applying it to the wave packet dynamics in harmonic potentials and to particle tunneling through a barrier.

Majorant frequency principle for an approximate solution of a nonlinear spatially inhomogeneous coagulation equation by the Monte Carlo method

We propose an efficient algorithm in the direct statistical simulation method for an approximate solution to the Cauchy problem for a nonlinear spatially inhomogeneous coagulation equation that describes the coagulation of particles together with their diffusion transfer. The use of the majorant frequency principle allows one to attain a linear dependence of the computational costs of the algorithm on the initial number of test particles. The main properties of the algorithm include a special markovian random process and a splitting scheme with respect to physical processes.

Majorant frequency principle for an approximate solution of a nonlinear spatially inhomogeneous coagulation equation by the Monte Carlo method

We propose an efficient algorithm in the direct statistical simulation method for an approximate solution to the Cauchy problem for a nonlinear spatially inhomogeneous coagulation equation that describes the coagulation of particles together with their diffusion transfer. The use of the majorant frequency principle allows one to attain a linear dependence of the computational costs of the algorithm on the initial number of test particles. The main properties of the algorithm include a special markovian random process and a splitting scheme with respect to physical processes.

Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes

Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes

A method for testing first-order Markovian property of noise phenomena including 1/ ƒ noise

In this paper, we present a method to investigate the first-order Markovian property of noise processes. The first-order transition probability matrix of a signal was determined numerically and a new Markovian random process was generated from this probability matrix. The method is demonstrated on computer generated and analog random signals. It is concluded that the 1/ ƒ noises under consideration are not first-order Markovian processes, which means that the knowledge of the first-order transition matrix is not sufficient for the description of the time-behaviour of those signals.

The local time of a two-dimensional diffusion

When a markovian random process taking values in a continuous state-space, such as R, visits a particular point repeatedly, it is natural to seek some quantity which records how long it spends there. Typically, however, the number of visits made to the point is uncountably infinite, and the (Lebesgue) length of time spent there is zero. One interesting object to consider is the local time, sometimes thought of as the occupation density of the process, which at each point is a random Cantor function that increases only when the process visits the point. The review article by Rogers (1989) contains a good introduction to the local time of a one-dimensional brownian motion and its relevance to the excursions of brownian motion from zero. In two dimensions, a typical diffusion, such as brownian motion in the plane, never revisits a point, so it does not have a local time. In this paper we shall construct the local times of some particular two-dimensional diffusions on a one-dimensional subspace, and show that they are jointly continuous in both time and space.

Toward a theory of random semi-Markov processes

Matrix differences and integral equations are introduced for the description of finite-valued non markovian random processes. In a special case we deduce an integral equation of a semi-Markov process. Necessary conditions are found for a semi-Markov process to be markovian. A generalized understanding of a semi-Markov random process is offered.

Stochastic quantization study of the quantum-mechanical motional narrowing of molecular vibrational spectra.

The natural width of molecular vibrational spectra arises from anharmonic coupling with zero-point vibrations of other modes. The stochastic quantization of Nelson describes a quantum-mechanical state as a Markovian random process in which the position is a time-dependent random variable even in a pure stationary state. As a consequence, the zero-point fluctuation is shown to give rise to the quantum-mechanical motional narrowing of molecular vibrational spectra.

Contribution to the experiment of axiomatizing the theory of turbulent diffusion

The efforts to axiomatize the theory of turbulent diffusion, founded on the theory of Markovian random processes, are based on the principal logical algebraic aspects of the concept of modelling.

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.

Generation of a Synthetic Vertical Profile of a Fluvial Sandstone Body

Abstract Any stratigraphic section or bedding sequence can be synthesized if there is a transition procedure from one lithology or bedding type to another, and if thickness distributions of the different lithologies are known. Stratigraphic sections of a fluvial sandstone body were synthesized with five bedding types: cross-bedding, massive beds, parting lineation, ripple mark and mudstone. The transition procedure from one bedding type to another used dependent, Markovian random processes which have a memory that extends one step backward in the depositional process. As observed in nature, median grain size and sand wave thickness (cross-bedding and ripple mark) decline upward in the synthesized sections as proportions of the different bedding types change. Grain size and permeability were also incorporated into the sections. By changing the transition procedures, bed thickness distributions, rate of upward decline or sand wave height and length, different types of sections can be synthesized, thus making it possible to model many different sedimentation problems. Introduction This paper describes a general method for synthesizing stratigraphic sections and bedding sequences of sedimentary, metamorphic or igneous origin. Synthetic generation is of interest for several reasons. Close correspondence between real and synthetic sections suggest that the factors used in the synthesizing model may indeed be the correct ones, thus giving the investigator a check on his assumptions. Rapid, inexpensive simulation of many stratigraphic sections permits one to synthesize a rock body (sandstone or carbonate reservoir) or, on a larger scale, the fill of a sedimentary basin. Harbaugh gives an example of mathematical simulation of a carbonate basin. He simulated the basin in the hope that improved prediction would follow better understanding of the depositional processes. From the petroleum engineer's viewpoint it seems reasonable to believe that the synthetic generation of rock properties and their distribution in a reservoir should be relevant in the study of reservoirs. Any stratigraphic section or bedding sequence can be generated provided there is a transition procedure from one lithology or bedding type to another and provided the thickness distributions of the different units are known. The transition procedure involves random processes that are either independent or dependent. If the depositional process is independent, previous deposition will have no influence on present deposition. However, if it is dependent, past deposition will influence either present or future deposition. Such a dependent depositional process can be thought of as having a memory that extends backward in time through one or more pulses of deposition. A process with a memory can be described by a Markov process. Because the concept of memory or dependence appears to be in accord with our understanding of many depositional processes, Markov processes were used to synthesize the bedding sequences of this study (see Appendix). The above methods are perfectly general and are appropriate for any stratigraphic section or bedding sequence: bedding types in a beach deposit, an evolving carbonate bank or the changing lithologic fill of a thick geosyncline sequence. We chose to synthesize a vertical profile of a fluvial sandstone body because its characteristics were well documented, much was known about fluvial processes and fluvial-deltaic sandstone bodies constitute an important class of petroleum reservoirs. CHARACTERISTICS AND ORIGIN OF FLUVIAL CYCLE The fluvial cycle has been well documented in recent years by Bersier, Allen and Visher. Deposits from fluvial cycles range from 10 to 150 ft or more in thickness and are characterized by a "fining upwards": coarse sandstones with occasional conglomerates grade upward into medium- to fine-grained sandstone, and hence into siltstone and mudstone. SPEJ P. 243ˆ