Modeling of random processes based on Karhunen-Loeve decomposition

3 - Time varying signals

Statistical properties of several models of fractional random point processes

The research article demonstrates that agricultural transport and technological systems are susceptible to external random influences during their operation. By accurately describing and modeling these effects, the performance of the tool can be enhanced during the research and design stages. (Research purpose) The objective of this research is to generate random processes with predefined correlation functions that simulate the profile of the supporting surface under the wheels of agricultural transport and technological systems during operations. Additionally, the study aims to visually represent the resulting random implementations of the support surface profile height. (Materials and methods) The proposed approach considers the variation in support surface profile height along the trajectory of agricultural transport and technological systems, while maintaining a constant speed, as a stationary random process. The selection of the correlation function type and its parameters is dependent on the specific characteristics of the support surface. Subsequently, using the Matlab environment, the model for the support surface profile height under the wheels is developed. (Results and discussion) Successful implementations of support surface profile height have been achieved for two distinct scenarios: asphalt and wheat stubble. A comparative analysis is conducted between the correlation function of the simulated implementation of the random process and the original correlation function. (Conclusions) The proposed algorithm has demonstrated its capability to generate a random process with specified correlation function properties, accurately describing the height fluctuations of the support surface profile under the wheels of agricultural transport and technological systems. The adequacy of the algorithm has been confirmed through a comprehensive analysis of at least 50 implementations, highlighting its effectiveness and reliability.

A simplified quasi-Gaussian random process model based on non-linearity

This chapter introduces the terminology of random processes and their properties; stationarity, correlation function, power spectral density, and the Wiener–Khinchin theorem. The theory of linear, causual, and time-invariant systems is presented, as is and their characterization in terms of impulse response and frequency response function. Response to random input is next examined. The correlation functions and power spectral densitites of output in terms of those of input are given.

The box bearing unit of the freight car is a complex mechanical system, which is influenced by various factors. Under normal operation, the parameters characterizing the functional state of the components of the axle box should be within the prescribed limits throughout the standard service life. Mathematically, this corresponds to the placement of the axle box elements in the space of admissible states. The railroad model consists of a set of submodules: track and elastic ballast layer. The model allows simulate the microgeometry of the track, the characteristics of the inequalities, and the elasticity of the base under the track. Subsequently, the main parameters characterizing random processes were determined: the magnitude of the mathematical expectation, the variance, as well as the minimum and maximum values of the effort. The results indicate that random processes are distributed according to normal law. According to the results of the research, for each of the random loading processes of the axle boxes, appropriate correlation functions were constructed and the space of “admissible states” for different operating modes was con-structed.

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.

This article proposes a technique for assessing risks related to operational reliability levels of systems of electric interlocking of track switches and signals at railroad stations. The assessment relies on the mathematical model of such random processes as failures and recoveries of the system of electric interlocking of track switches and signals and on using this system to fulfill train traffic control functions at a given station. The risk related to the operational safety of the electric interlocking system is calculated for the railroad station in general, not for a specific scenario of events. The statistics of safety violation events is based on a small sample. The risk related to the operational reliability of the electric interlocking system is assessed using risk matrices in the form of tables, where the likelihood of occurrence and the number of train delays at the station due to failures in the electric interlocking system are used together. The train delay likelihood found from the matrix is used as the frequency of events to assess operational reliability level risks. The number of trains delayed per calculated period and the average time of train delay at the considered station are used as event aftereffects. The assessment results can be used to make valid managerial decisions.

In this book, modeling and estimation problems of random processes are treated in a unified geometric framework. For this, we need some basic facts about the Hilbert space theory of stochastic vector processes that have finite second order moments and are stationary in the wide sense. Such a process \(\{y(t)\}_{t\in \mathbb{Z}}\) is a collection of real or complex-valued random variables \(y_{k}(t),k = 1,2,\ldots,m,t \in \mathbb{Z}\), which generate a Hilbert space H with inner product $$\displaystyle{\langle \xi,\eta \rangle =\mathop{ \mathrm{E}}\nolimits \{\xi \bar{\eta }\},}$$ where bar denotes conjugation. This Hilbert space is endowed with a shift, i.e., a unitary operator \(\mathcal{U}:\, \mathbf{H} \rightarrow \mathbf{H}\) with the property that $$\displaystyle{y_{k}(t + 1) = \mathcal{U}y_{k}(t),\quad k = 1,2,\ldots,t \in \mathbb{Z}.}$$ In this chapter we introduce some basic geometric facts for such Hilbert spaces. Although we shall assume that the reader has some knowledge of elementary Hilbert space theory, for the benefit of the reader, some relevant facts are collected in Appendix B.2.

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).

For perfect surface roughness description is not enough to know characteristics of surface profile. It is necessary to use topography methods, so called microtopography. Thereby, surface roughness in microtopographycal understanding must be described with three coordinates, whose in Cartesian coordinates system compose point under consideration height h, abscissa and ordinate, determines point position in the plane. Most efficient methods in irregular surface roughness research are random function theory methods. Therefore, microtopography, analogically to profile, may consider as random function, but two dimensional function, i.e. two variable x and y random field h(x,y). From analogy with random process, random field can be normal – ordinates are distributed by normal (Gaussian) distribution. Moreover, random field can be homogeneous and heterogeneous. Random field is deemed homogeneous if its mean value is discretionary and correlation function depends only from distance between surface points. Important characteristic of random field is correlation function, whose depends of two variables t1 and t2 – orthogonal Cartesian coordinates of vector t. Random field is homogeneous and isotropic when its characteristics are equivalent in any direction. There are three types of surface anisotropy: • General event of surface anisotropy. Characteristics of this event roughness parameters are depend of surface split direction. • Surface roughness with direct anisotropy. Those surfaces are with typical traces of tool and they proper two mutually perpendicular surface roughness directions. • Extended anisotropy area – special event of anisotropy roughness. Of analytical opinion, gainfully anisotropy roughness see as extended occasional isotropy area. This let easy cross from anisotropy surface to isotropy and contrariwise, thereby embrace amount class of surface roughness. Let’s formulate microtopography model of rough surface [1]. Surface roughness is described with homogeneous normal random field h(x,y) that has uninterrupted correlation function and uninterrupted deriviates. We may consider that E{h(x,y)}=0. The mean random field value is plane called mean plane. For describing random field we must know mathematical expectation and field correlation function, what in fact reduces on determining dispersion and rationed correlation function r(t1, t2). Homogeneous random field dispersion D{h} doesn’t depends of direction and can be founded in any surface split. Given model of rough surface let inspect surfaces produced by abrasive instruments and friction surfaces after wear-in period.

The basic tools for describing and analyzing random processes have all been developed in the preceding chapters along with a variety of examples of random processes with and without memory. The goal of this chapter is to use these tools to describe a menagerie of useful random processes, usually by taking a simple random process and applying some form of signal processing such as linear filtering in order to produce a more complicated random process. In Chapter 5 the effect of linear filtering on second-order moments was considered. In this chapter we look in more detail at the resulting output process and consider other forms of signal processing as well. In the course of the development a few new tools and several variations on old tools for deriving distributions are introduced. Much of this chapter can be considered as practice of the methods developed in the previous chapters, with names often being given to the specific examples developed. In fact several processes with memory have been encountered previously: the binomial counting process and the discrete time Wiener process, in particular. The goal now is to extend the techniques used in these special cases to more general situations and to introduce a wider variety of processes. The development of examples begins with a continuation of the study of the output processes of linear systems with random process inputs.

The paper deals with cyclostationarity as a natural extension of stationarity as the key property in designing the widely-used models of random processes. The comparative example of two processes, one is wide-sense stationary and the other is wide-sense cyclostationary, is given in the paper and reveals the lack of the conventional stationary description based on one-dimensional autocorrelation functions. It is shown that two significantly different random processes appear to be characterized by exactly the same autocorrelation function while their two-dimensional autocorrelation functions provide outlook where the difference between processes of two above-mentioned classes becomes much clearer. More concise representation by expanding the two-dimensional autocorrelation function to its Fourier series where the cyclic frequency appears as the transform parameter is illustrated. The closed-form expression for the components of the cyclic autocorrelation function is also given for the random process which is an infinite pulse train made of rectangular pulses with randomly varying amplitudes.

Considering problems involving truly big data inevitably leads to the use of a block approach, which is 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 concurrent 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 - mesh visualization has been proposed, 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.

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.