BIG DATA STOCHASTIC SEMANTICS

First of all the paper considers the problem of verification or formalization of the online visualization and parallel computing system from the point of view of dynamic systems as a development of the theory of computational complexity for random processes. Considering problems involving truly big data inevitably leads to the use of a block approach which is also used in both information theory and stochastic differential equations. As a natural metaphor the graph signals were chosen. This is a graph in nodes, of which a spectral function is defined in the examples considered this is a function of color (RGB), height or amount of data. In parallel computing, a block can be associated with a computing unit (processor) and consider the problem of entropy (performance) maximization. In the developed on-line visualization and concurrent computing system for geometric parallelization, it is possible to implement and compare a stationary random process (equiprobable messages implemented using broadcasting and mixins) and a steady-state random process (point-to-point messages), which have different analytical solutions. Together, this allows concluding that the proposed implementation of a stationary process has a certain novelty; in addition, it was intended to be more convenient for automated parallelization. The problems of automatic load balancing (interpolation problem) and optimal scalability of parallel computing (extrapolation problem) are also considered. Not much has been done in the field of visualization verification for example a mesh visualization has been proposed to be considered as a parameterized model of a white-noise random process. Of course, this work cannot be considered complete, but the direction that the authors called stochastic semantics is obviously promising. The authors intend to take a close look at the established perturbed processes in the field of visualizations including those that take into account the human factor (the sketches of the formalization in the form of a discussion are given).

: Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.

3 - Time varying signals

The problem of digital modeling of random processes with given either correlation function or spectral density of the process is considered. These functions of a random process are interconnected by the Wiener–Khinchin theorem. The solution of one function can be used to solve another. The development of a mathematical representation of a stationary random process with a given correlation function based on the Karhunen-Loeve transformation, which is most often used to decorrelate the original process in order to describe it more concisely (data compression problem), has been completed. It is proposed to use the Karhunen–Loev transformation to impart the required correlation properties to the original uncorrelated random process by inverting (converting) this transformation. The form of the required transformation for a discrete (in time) representation of input and output processes of various lengths and methods for ensuring the required modeling accuracy are substantiated. A procedure for obtaining a correlation function from a given spectral density of a simulated random process is presented. An experimental study of the proposed method was carried out in the course of computer simulation in the Mathcad package which simplified the solution of the required computational problems. The initial random process was obtained as a sequence of independent (and, therefore, uncorrelated) random numbers, and the output process, as a result of the transformation, was obtained in the work. The calculated approximate correlation function is compared with the given one and the error variance is determined. The results of modeling random processes with given correlation functions and a homogeneous Markov process with a given transition probability are given as well as an example of the transition from a given spectral density of a random process to its correlation function. The results obtained confirm the effectiveness and feasibility of the developed modeling methods which will allow them to be used in computer research and design of various systems.

Wide-sense stationary discrete-time random processes: Analyses in the discrete-Fresnel and discrete-frequency domains

This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.

Goal. To improve the efficiency of processing and presenting the results of experimental studies of random oscillatory processes of agricultural machines and machine units. Methods. The basic provisions of higher mathematics, in particular mathematical statistics and variance-correlation analyzes of statistically random processes, were used during the research. Results. To achieve this goal, it is proposed to return the coordinate system with the random process presented in it by a non-stationary mathematical expectation to an angle equal to the angle of inclination of the line approximating this random process. The further algorithm of actions provides recalculation of ordinates of non-stationary random processes in the new (returned) coordinate system. A comparison of the statistical characteristics of non-stationary and stationary processes shows that at the statistical level of 0.05 the null hypothesis about the equality of their mean values (according to Student’s t-test) and variances of oscillations (according to Fisher’s F-test) is rejected. The normalized correlation function of a stationary random process at the same time, in contrast to a non-stationary one, is characterized by its normal course. Conclusions. Based on the conducted theoretical research the new analytical dependence (5) is obtained, the use of which allows lead to a stationary array the linearly non-stationary by mathematical expectation array of data of random oscillating process of agricultural machines and machine units by statistical processing of experimental research results. The adequacy of the developed analytical dependence (5) is confirmed by the acceptable nature of the normalized correlation function of a random oscillatory process reduced to a stationary one.

Intelligent diagnosis methods based on big data has been widely used in the field of machinery, which plays an important role. However, the application process of intelligent diagnostic methods in airborne electronic products is relatively slow. Aiming the requirements of fault diagnosis and the typical problems in engineering practice, a diagnosis system architecture of multi-intelligence model is defined, which is realized by constructing support vector machine (SVM), deep belief network (DBN), long short-term memory neural network (LSTM), unscented particle filter (UPF) and random process (RP) models with big data. The architecture can support condition based maintenance and meet the needs of online status analysis and remaining life prediction. According to the case of airborne electronic products, software development was implemented to prove the application effect of this framework in the field of airborne electronic product maintenance.

The chapter contains elements of theory of random processes required for time series analysis and for understanding its results. The time series discussed in this book belong to the class of stationary random processes, which means that their statistical properties averaged over an ensemble of realizations of the process do not depend upon the time origin. If averaging over an ensemble of realizations is equivalent to averaging over time of any single realization, the process is ergodic. If a stationary process is Gaussian, it is also ergodic. In our research, we mostly have just one sample realization (one time series) of finite length and by extending the properties estimated from a single time series to the entire process under the study, we assume that the process is ergodic. The most important characteristics of any time series include the covariance function and spectral density, which describe the time series properties in the time and frequency domains, respectively. The two stationary discrete random processes typical for climatology and other disciplines are white noise and Markov chains. Examples of the true and estimated correlation functions and spectral densities including their sampling variability are briefly discussed for simulated time series of length \( N = 100 \).

A new method for representing and generating realizations of a wide-sense stationary non-Gaussian random process is described. The representation allows one to independently specify the power spectral density and the first-order probability density function of the random process. The only proviso is that the probability density function must be symmetric and infinitely divisible. The method proposed models the sinusoidal component frequencies as random variables, a key departure from the usual representation a of wide-sense stationary random process by the spectral theorem. Ergodicity in the mean and autocorrelation is also proven, under certain conditions. An example is given to illustrate its application to the K distribution, which is important in many physical modeling problems in radar and sonar.

Based on the mathematical tools of theory of runs of random processes it is developed the forecasting method of parametric reliability of electronic equipment (EE), which allows to predict the refusals with reasonable accuracy at stationary and quasi-stationary and quasi-determined processes of drift of deterministic parameters for bilateral restrictions. Parametric reliability characterizes the capacity of device to serve specifications and compliance with defined probabilistic parameters defining parameters of given value. Drifts of deterministic parameters are random processes that are different. In the case of quasi-stationary processes the EE reliability can be predicted using the method of quantile zones. Known methods for forecasting stationary or quasi-stationary processes to date are available. The obtaining method allows to predict parameters of runs of random drift process, namely their number, total duration and average duration of run. Simulations showed ability to predict emission parameters for different characters of drift random processes in the case of bilateral constraints.

The subject of random or stochastic processes analysis is a very important part of scientific inquire. The terms stochastic and random process are used interchangeably. Random processes are used as mathematical models for a large number of phenomena in physics, chemistry, biology, computer science, information theory, economics, environmental science and others. Each subject is described and problems are implemented in the R code, with real data collected in experiments performed by the authors or taken from the literature. With this intent, the reader can promptly apply the analysis to her or his own data, making the subject operational. Consistent with modern trends in university instruction, the book make readers active learners, with hands-on computer experiments directing readers through applications of random process analysis (RPA).

There is increasing evidence from in vivo recordings in monkeys trained to respond to stimuli by making left- or rightward eye movements, that firing rates in certain groups of neurons in oculo-motor areas mimic drift-diffusion processes, rising to a (fixed) threshold prior to movement initiation. This supplements earlier observations of psychologists, that human reaction-time and error-rate data can be fitted by random walk and diffusion models, and has renewed interest in optimal decision-making ideas from information theory and statistical decision theory as a clue to neural mechanisms. We review results from decision theory and stochastic ordinary differential equations, and show how they may be extended and applied to derive explicit parameter dependencies in optimal performance that may be tested on human and animal subjects. We then briefly describe a biophysically-based model of a pool of neurons in locus coeruleus, a brainstem nucleus implicated in widespread norepinephrine release. This neurotransmitter can effect transient gain changes in cortical circuits of the type that the abstract drift-diffusion analysis requires. We also describe how optimal gain schedules can be computed in the presence of time-varying noisy signals. We argue that a rational account of how neural spikes give rise to simple behaviors is beginning to emerge.

Fractional Brownian Motions, Fractional Noises and Applications

Background. The study of the asymptotic behavior of solutions of stochastic differential equations is one of the main places in many sections of insurance and financial mathematics, economics, management theory since stochastic differential equations, as an effective model of random process is the basis for the study of random phenomena. Objective. In this paper we consider the almost sure asymptotic behavior of the solution of the nonautonomous stochastic differential equation. Methods. We proposed a method to study the y-asymptotic properties of a solution of a stochastic differential equation by comparison with a solution of an ordinary differential equations obtained by dropping the stochastic part. We also use of the theory of pseudo-regularly varying functions. Results. We investigate the asymptotic behavior of solutions stochastic differential equations and establish sufficient conditions that provide different types of asymptotic behavior of a random process. Conclusions. Stochastic models approximate the real processes much better than deterministic ones, however, deterministic modelling has been preferred to stochastic one because of much greater ease of computability. The presented result enabled comparing properties of solution a stochastic differential equation with a solution of an ordinary differential equation.