Analysis Of Random Processes Research Articles

STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE

The application of statistical methods for the analysis of random processes to obtain quantitative estimates for the performance indicators of photovoltaic plants as a stochastic energy source is considered. The power generation process is represented by a set of daily random functions with their respective trends and stochastic components. One hour was taken as the minimum stationari-ty interval of trend’s statistical characteristics. A time period of 16 years was studied to obtain statistically stable estimates. The meteorological database SARAH2 was used as a source of hourly data on the density of solar irradiance, ambient temperature and wind speed in the middle part of the Azov-Black Sea region of Ukraine. For mathematical modelling of power generation and electricity production processes, specialized software PVGIS version 5.2 of the European Commission was used. Hourly quantitative estimates of expected daily power generation trends for each month of the year, as well as their correlation functions, were computed. Algorithms were developed and levels of probabilistic assurance for hourly generated power were evaluated. Statistical estimates of the expected daily, monthly, and annual volumes of electricity production by photovoltaic power plants were studied, taking into account the meteorological conditions in the said region.

Innovative development of music education model in universities under stochastic process model analysis

Abstract Based on the analysis of random phenomena, random variables, and random processes, this paper constructs a Markov chain-based model for evaluating the effectiveness of music teaching using Markov chains. Aiming at the core literacy of college music education, a new education model is proposed to cultivate students’ comprehensive music literacy and transform the method of college music education. Traditional evaluation methods and the Markov chain model are used to analyze data examples for the new model of music education. The probability of transferring the number of students whose first score was on the 89.9-80 scale to the 100-90 scale was 0.5115, greater than the probability value of 0.3650 for those still on the 89.9-80 scale. The transfer matrix of the experimental class was shifted to the left as a whole, indicating a high degree of student progress.

Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information

We study a reliability system subject to occasional random shocks of random magnitudes W0,W1,W2,… occurring at times τ0,τ1,τ2,…. Any such shock is harmless or critical dependent on Wk≤H or Wk>H, given a fixed threshold H. It takes a total of N critical shocks to knock the system down. In addition, the system ages in accordance with a monotone increasing continuous function δ, so that when δT crosses some sustainability threshold D at time T, the system becomes essentially inoperational. However, it can still function for a while undetected. The most common way to do the checking is at one of the moments τ1,τ2,… when the shocks are registered. Thus, if crossing of D by δ occurs at time T∈τk,τk+1, only at time τk+1, can one identify the system’s failure. The age-related failure is detected with some random delay. The objective is to predict when the system fails, through the Nth critical shock or by the observed aging moment, whichever of the two events comes first. We use and embellish tools of discrete and continuous operational calculus (D-operator and Laplace–Carson transform), combined with first-passage time analysis of random walk processes, to arrive at fully explicit functionals of joint distributions for the observed lifetime of the system and cumulative damage to the system. We discuss various special cases and modifications including the assumption that D is random (and so is T). A number of examples and numerically drawn figures demonstrate the analytic tractability of the results.

A resilience evaluation method of natural gas pipeline system based on uncertainty analysis

The natural gas pipeline system (NGPS) is a complex network of units that is susceptible to potential random failures. Moreover, the environment in which the NGPS operates is characterized by complexity and uncertainty, including market volatility, natural disasters, and deliberate sabotage. While research on the NGPS has mainly focused on supply reliability through methods such as statistical analysis and random simulation, there is a lack of emphasis on the system's supply resilience, which encompasses its robustness and recovery abilities. To address the resilience of the NGPS under uncertainty, this study proposes a probabilistic analysis model that evaluates supply resilience based on failure mechanisms and reasonable data. The model takes into account the integrated evaluation indexes framework and analyzes the process of failure and recovery, using MCMC random process analysis and complex networks theory. The results indicate that both gas transportation capacity and user satisfaction should be considered in the evaluation and strategy analysis of NGPS resilience. Furthermore, the study of resilience can provide guidance for NGPS operation, pre-warning during the design and operational stages.

Hybrid kalman filtering algorithm with wavelet packet data processing for linear dynamical systems

The paper develops a hybrid algorithm for predicting a linear dynamic system based on a combination of an adaptive Kalman filter with preprocessing using a wavelet packet analysis of the initial data of the background of the system under study.
 Being based on Fourier analysis, wavelet analysis and wavelet packet analysis are quite acceptable for time-frequency analysis of a signal, but they cannot be performed recursively and in real time and, therefore, cannot be used for dynamic analysis of random processes. In combination with the Kalman filter, a combination of the characteristics of the multiple-resolution wavelet transform and the recurrent formulas of the Kalman filter in real time is obtained.
 Since the original signal is usually given in the form of discrete measurements, to implement their convolution used in the Kalman filter, it is necessary to use cyclic convolutions with periodic continuation of the signal for any time interval. In the case of different values of the original signal at the ends of the considered time interval [0,T], the periodized signal can have large values and sharp different amplitude at the ends of the periodization interval.
 To smooth out the values of the periodized signal at the ends of the periodization interval, a cascade decomposition and recovery algorithm was used using Dobshy boundary wavelets with a finite number of moments. Signal recovery is performed in a series of operations comparable to the duration of the time interval under consideration.
 The smoothed signal obtained in this way is used as a Kalman filter platform for predicting the dynamic system under study.
 Taking into account that the correlation functions of the noise in the observation equation and the phase state of the system are usually unknown, the adaptation of the Kalman filter to these noises (interference) is carried out on the basis of a zeroing sequence. The manuscript does not contain related data

Computer Technology for Construction ε-Solution of the Problem

Introduction. Issues of selection and construction of computing resources and methods of their effective use to calculate an approximate solution of the problem with the given accuracy in a limited processor time are considered. The purpose of the article. Based on the analysis of the total error of the computational algorithm, to develop a computer technology (CT) for the construction and solution of the ε-problem and illustrate it on the problems of integrating rapidly oscillating functions and digital signal processing. The results. The general scheme for solving problems of applied and computational mathematics using computer technology is presented. Computer technology includes the formation of a certain set of computing resources, the method and conditions of their use to build a computing process, the result of which should be the solution of the problem. The main attention is paid to the choice of a computer calculations model (input data about the problem, the class of problems of computational mathematics, the class of computational algorithms for calculating the solution, evaluation of quality characteristics and parameters of the computational process, computer architecture, software, restrictions on the value of quality characteristics), the construction of quality characteristic estimates, the possibility of adjusting the mathematical model, the construction of the computational process and the performance of calculations. The issues of determining the optimal parameters of computational algorithms-programs for finding ε-solution are investigated, which are demonstrated on two classes of problems – integration of rapidly oscillating functions and correlation analysis of random processes. Conclusions. The issues of selection and construction of computing resources and methods of their effective use to calculate an approximate solution of the problem with the given accuracy in a limited processor time are considered. The elements of computer technology for solving specific problems on the examples of problems of integration of rapidly oscillating functions and correlation analysis of random processes are applied. Keywords: total error, computer technology, rounding error, rapidly oscillating functions.

Methodology for Maintaining the Dynamic Characteristics of Passenger Cars

The work presents the results of a statistical analysis of random processes of the coefficient of vertical dynamics of a passenger car variation. To determine the dynamic performance of cars the tests were running on the tracks of JSC "Ukrzaliznytsia". The main statistical characteristics of random processes are determined for various modes of car movement along curved and straight sections of the track. As a result of statistical processing of experimental data, correlation functions have been constructed that characterize the degree of connection between different time sections of the process of changing the coefficient of vertical dynamics. The obtained indicators of the law of distribution of the vertical dynamics coefficient values, approximating the models of correlation functions and the numerical values of their parameters, make it possible to further simulate the process of loading a passenger car without carrying out lengthy expensive tests.

Vibration fatigue by spectral methods—A review with open-source support

Vibration fatigue by spectral methods relates the theory of structural dynamics to high-cycle vibration fatigue. An ideal spectral method should perform well and consistently, regardless of the response spectrum and the material being analyzed. The primary aim of this review is to develop a common theoretical and open-source-code framework for a side-by-side comparison of more than 20 spectral methods, which will help a future spectral-domain vibration-fatigue research. The reviewed spectral methods are structured in terms of the damage-estimation concept: narrowband approximation, narrowband correction factor (Wirsching–Light, Ortiz–Chen, α0.75, Tovo–Benasciutti (two versions)), rainflow probability-density-function approximation (Dirlik, Zhao–Baker, Park, Jun–Park) and combined fatigue damage, where the damage is combined according to the cycle types (Jiao–Moan, Sakai–Okamura, Fu–Cebon, modified Fu–Cebon, Low’s bimodal, Low 2014, Gao–Moan) and the narrowband damage combination criterion (Lotsberg, Huang–Moan, single moment, bands method). All the reviewed methods are implemented in the supporting open-source Python package FLife, with the comparison being fully reproducible using the package documentation. The accuracy of the spectral methods is investigated in terms of a time-domain rainflow analysis, where three different materials are considered: steel, aluminium and spring steel. The comparison is based on typical PSD defined signals, with the focus on: spectral width, background noise, close modes, number of modes and typical vibration profiles used in accelerated automotive tests. In addition, a bimodal spectrum is formulated to examine a specific group of spectral methods that are developed for bimodal random processes. This research shows that other methods, besides the well-established ones, such as the Dirlik and Tovo-Benasciutti methods, should be considered when the fatigue load is broadband: Ortiz–Chen, α0.75, Park, Jun–Park and Huang–Moan methods. Furthermore, as the fatigue analysis of bimodal random processes has become well established, the applicability of bimodal methods is inspected. Among the reviewed spectral methods, Low’s bimodal and Low 2014 method show exceptional accuracy that can be attained using the bimodal formulation.

ТЕОРЕТИЧНІ ДОСЛІДЖЕННЯ ВИЗНАЧЕННЯ КРИТИЧНИХ УМОВ МІКРОКЛІМАТУ ПРИ АВАРІЙНИХ ВІДКЛЮЧЕННЯХ ТЕПЛОПОСТАЧАННЯ З УРАХУВАННЯМ КОНСТРУКТИВНИХ ОСОБЛИВОСТЕЙ БУДИНКІВ

Purpose. Analysis and research of determining critical microclimate conditions in emergency situations in heat supply systems, taking into account external climatic conditions and structural features of buildings, which are caused by the acceleration and cheapening of construction processes thanks to the introduction of the latest technologies. Methodology. Theoretical and experimental research based on fundamental knowledge in the field of thermal processes and methods of solving heat exchange problems, modelling of dynamic processes, methods and analysis of random processes, methods of mathematical statistics and forecasting. Results. On the basis of the conducted research, the regularities of the influence of the environment on the maintenance of the microclimate conditions of the premises, depending on the structural features of buildings and structures in emergency situations in heat supply systems, have been established. The limit conditions for achieving critical conditions of the microclimate of premises as a result of accidents in heat supply systems have been determined, which makes it possible to predict the limit conditions for eliminating the consequences of emergency situations and to optimize the activities of repair and rescue services, which are especially relevant in connection with the failure of outdated heat supply systems and combat operations on territory of the country. Scientific novelty. Determining the patterns of environmental influence on the maintenance of indoor microclimate conditions in connection with the structural features of buildings and structures in emergency situations in heat supply systems, which allows controlling the heat supply mode to ensure certain indoor microclimate conditions. Practical significance. Improving the forecasting of the limit conditions for liquidation of the consequences of emergency situations in heat supply systems allows optimizing the activities of repair and rescue services, especially in connection with the failure of outdated heat supply systems and combat operations on the territory of the country.

Algorithm for Calculating Primary Spectral Density Estimates Using FFT and Analysis of its Accuracy

Introduction. Fast algorithms for solving problems of spectral and correlation analysis of random processes began to appear mainly after 1965, when the algorithm of fast Fourier transform (FFT) entered computational practice. With its appearance, a number of computational algorithms for the accelerated solution of some problems of digital signal processing were developed, speed-efficient algorithms for calculating such estimates of probabilistic characteristics of control objects as estimates of convolutions, correlation functions, spectral densities of stationary and some types of non-stationary random processes were built. The purpose of the article is to study a speed-efficient algorithm for calculating the primary estimate of the spectral density of stationary ergodic random processes with zero mean. Most often, the direct Fourier transform method using the FFT algorithm, is used to calculate it. The article continues the research and substantiation of this method in the direction of obtaining better estimates of rounding errors. Results. The research and substantiation of the method in the direction of obtaining more qualitative estimates of rounding errors, taking into account the errors of the input information specification, has been continued. The main characteristics of the given algorithm for calculating the primary estimate of the spectral density are accuracy and computational complexity. The main attention is paid to obtaining error estimates accompanying the process of calculating the primary estimate of the spectral density. The estimates of the rounding error and ineradicable error of the given algorithm for calculating the primary estimate of the spectral density, which appear during the implementation of the algorithm for the classical rounding rule for calculation in floating-point mode with τ digits in the mantissa of the number, taking into account the input error, are obtained. Conclusions. The obtained results make it possible to diagnose the quality of the solution to the problem of calculating the primary estimate of the spectral density of stationary ergodic random processes with a zero mean value by the described method and to choose the parameters of the algorithm that will ensure the required accuracy of the approximate solution of the problem. Keywords: primary estimation of spectral density, fast Fourier transform, discrete Fourier transform, rounding error, input error.

A Novel Efficient Method for Simulating Non-Stationary Random Processes Combining Generalized Harmonic Wavelet and Stochastic Harmonic Function

Integrating the general expression of the generalized harmonic wavelet (GHW)-based spectral representation method (SRM) and the idea of stochastic harmonic function (SHF), a novel stochastic generalized harmonic wavelet (SGHW) method for fully nonstationary stochastic processes, is established. The advantages of the proposed method are (1) a stochastic process with accurate probability information can be obtained by retaining fewer components. A smaller number of components greatly reduces the number of random variables. Fewer random variables also reduce the difficulty of random process analysis. The superior fitting effect between evolutionary power spectral density (EPSD) obtained from the samples using the SGHW method and the EPSD model proves the validity and effectiveness of the SGHW method. (2) The proposed method contains more accurate probability information and has a higher computational efficiency. The comparisons of relative errors and computational time between the SGHW method and the SRM demonstrate the accuracy and efficiency of the proposed method.

Property analysis of multivariate conditional linear random processes in the problems of mathematical modelling of signals

The object of research is the process of mathematical modelling of a multidimensional random signal, which in the structure of its generation is the sum of a large number of random impulses that occur at random times. Examples of stochastic signals of this type can be, in particular, electroencephalographic and cardiographic signals, photoplethysmography signals, resource consumption processes (electricity, gas, water consumption), radar signals, vibrations of bearings of electric machines and others. A common mathematical model (especially in the multidimensional case) of this type of signal is a linear random process that allows the signal to be represented as the sum of a large number of stochastically independent random impulses that occur at Poisson moments. If the impulses are stochastically dependent (or the moments of time of their appearance are not Poisson), then the mathematical model is a conditional linear random process. The definition and analysis of the probabilistic properties of such processes for the multidimensional case have not been conducted. The paper defines a multidimensional conditional linear random process, each component of which is represented as a stochastic integral of a random kernel driven by a process with independent increments. Expressions for the characteristic function and moment functions of the specified process are obtained. The approach used was to use the mathematical apparatus of conditional characteristic functions, as well as the known representation of an infinitely divisible characteristic function of a linear random process as a functional of a process with independent increments. The obtained results provide a possibility for theoretical analysis of probabilistic properties of multichannel stochastic signals, the mathematical model of which is a multidimensional conditional linear random process. Justification of their properties of stationarity or cyclostationarity, which are the consequence of corresponding properties of the kernel and process with independent increments, can be carried out.

On the choice of basic regression functions and machine learning

As is known, the regression analysis task is widely used in machine learning problems, whic hallows to establish relationship between observed data and compactly store of information. Most often, a regression function is described by a linear combination of some of the selected functions fj(X),j= 1, . . . , m,X∈D⊂Rs. If the observed data contains a random error, then the regression function restored from the observed data contains a random error and asystematic error depending on the selected functions fj. The article indicates the possibility of optimal selection of functions fj in the sense of a given functional metric, if it is known that the true dependence is consistent with some functional equation. In some cases (regular grids,s≤2), similar results can be obtained using the random process analysis method. The numerical examples given in this article illustrate much more opportunities for the task of constructing the regression function.

Complexity-based permutation entropies: From deterministic time series to white noise

This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this “ordinal” methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of allowed permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of allowed permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes.

Stochastic Processes under Constraints

The analysis of random processes under various constraints and conditions has been a central theme in the theory of stochastic processes, which links together several mathematical subdisciplines. The connection between potential theory and a certain type of conditioning of Markov processes via Doob’s h-transform can be seen as a classical highlight. The last decades have seen further exciting and highly interesting developments which are related to the title of the workshop such as the analysis of persistence exponents for various classes of processes and various types of penalization problems. Many of these problems are rooted in questions from statistical mechanics. The workshop aims to investigate the topic stochastic processes under constraints from all these different perspectives.

Оценка действий экипажа самолёта по данным видеоокулографии

Mathematical models and methods for crew training level assessing based on video oculography data are presented. The results obtained are based on comparing the studied fragments of oculomotor activity of pilots with comparable patterns of video oculography data of various types and performance quality contained in a pre-formed specialized database. To obtain estimates, a complex combination of random process analysis and multivariate statistical analysis is used. The “intelligence” of diagnostic tools is contained in empirical data and can flexibly change as they accumulate. The considered example of determining the flight mode and pilot qualification based on video oculography data allows us to talk about the possibility of significant discrimination of the gaze movement trajectories of pilots at different flight phases and significant discrimination of the gaze movement trajectories of experienced and inexperienced pilots at certain phases of flight. An important new component of the presented results is a discriminant analysis for solving the problem of flight exercises classification, based on the principles of quantum computing. The scope of the considered approach is not limited to aviation applications and can be extended to tasks that are similar in content.

The Choice of Sampling Frequency and Optimal Method of Signals Digital Processing in Problems of a Random Loading Process Treating to Assess Durability

Theoretical and practical issues of digital information processing in the problems of load assessment in the evaluation of durability are considered. Due to the specifics of the problem under consideration, in which the precise definition of the extremum values and their sequence is of paramount importance, a number of generally accepted recommendations are unacceptable. For example, the Kotelnikov theorem, which was originally proposed in relation to the problems of estimating the frequency composition of the process, can lead to significant errors. It is shown that the requirements for the analysis of random loading processes for the further purpose of assessing the durability, due to their specificity, are in contradiction with this indication, namely, when choosing a frequency according to this rule, it is very likely to make an error, and not into the safe side. It was also considered the digital filtering of hardware data. Alternative approaches to the selection of extrema of the random process are analyzed: (1) direct hardware selection of extrema and (2) discretization by the method of level crossings. The latter approach has an optimal algorithm for selecting extrema allowing selecting the extremes of the random process with less cost and greater accuracy. The natural transition to integer arithmetic allows to optimize this algorithm even more. The model and real examples demonstrate the gain in terms of speed and memory, which ultimately will help to increase the reliability of the information necessary to assess the durability. The savings in memory and performance will allow longer implementations to be processed, which ultimately allows for a more accurate estimate of the durability at the production stage and for the estimation of the residual life.

Operation mode optimization of centrifugal pumps of a water supply system booster pumping station with a reserve tank

A technique of the optimal selection of power units and their operation mode in the booster pumping station of the water supply system with a reserve tank is proposed. The optimization mathematical model of pumping station in the form of a non-linear, multi-criteria, partially integral mathematical programming problem is developed. Necessary data for optimization problem definition: the structure of daily water consumption by consumers and resistance of the water-supply system were found on the base of a full-scale experiment results. The adopted approach has provided both mathematical rigor of the problem formulation and solution, and real data recording of consumer behavior and water supply system conditions. The processing of the full-scale experiment data was carried out by statistical methods of analysis of non-stationary random processes. The optimization procedure was realized with the help of genetic algorithm method. Comparison of booster pumping station operation optimal plans on the base of different centrifugal pumps has allowed choosing such a type of pump, that when using it, maximum energy efficiency of the water supply system is achieved and operation of pumping station with minimum number of pumps with maximum possible efficiency is provided. Minimum essential dimensions of a reserve tank, optimal operation speed of the centrifugal pumps are determined.

The choice of the sampling frequency and optimal method of signal digital processing in the problems considering random loading process for assessing durability

Theoretical and practical issues of processing digital information in the problems related to durability assessment in conditions of loading are considered. Due to the specifics of the problem, in which the precise dedermination of the extreme values and their sequence is of primary importance, a number of generally accepted recommendations appeared invalid. For example, the Kotelnikov theorem, which was originally proposed in relation to the problems of estimating the frequency composition of the process, can lead to significant errors. It is shown that the requirements to the analysis of random loading processes with a goal of further assessment of the durability, due to their specificity, contradict with the specified requirements, namely, when choosing a frequency according to this rule, an error can be made, and not to the margin of the strength assessment. We considered the issue regarding digital filtering of hardware overshoots. Alternative approaches to the selection of extrema of the random process are analyzed: 1) direct hardware selection of extrema and 2) discretization by the method of level crossings. The latter approach has an optimal algorithm for isolating extremes and makes it possible to isolate extremes of a random process with lower costs and greater accuracy. The natural transition to integer arithmetic provides further optimization of the algorithm. The model and real examples demonstrate gains in terms of speed and memory, which ultimately promote an increase in the reliability of the information required for assessing the durability. The savings in memory and performance will allow processing of long-term implementations and result in a more accurate estimating of the remaining life and durability at the stage of production.

Probabilistic analysis of two-dimensional vector random processes on the phase plane

Introduction: The models of vector random processes are characterized by a variety of practical applications and, at the same time, by the complexity of studying the detailed probabilistic structure. Such features often lead to considering only separate components of vector functions, forcing the shift to scalar process analysis. All this significantly reduces the overall information content of the research, increasing the relevance of the search for promising approaches to probabilistic analysis of vector processes. Purpose: To present vector random processes in a phase space of states, and to apply the general theory of level crossings of random functions to the study of probabilistic structure of phase trajectories. Results: We have identified the characteristic features of displaying random processes on a phase plane, and introduced numerical characteristics for the description and analysis of probabilistic behavior of sample functions. A probabilistic analysis has been performed for the characteristics of the «level crossing» type for vector processes with various definitions of the areas of acceptable values. We used the typical models of two-dimensional Gaussian processes and «signal plus noise» models to demonstrate how the probabilistic structure of the phase trajectories depends on the given threshold levels, major distribution parameters and spectral-correlation properties of the studied processes. Practical relevance : The article contributes to the research information content and its visualization in the analysis of probabilistic behavior of vector random processes. It combines general methods of phase space of vector processes, methods of phase plane in the analysis of scalar processes and the theory of level crossings of random functions. The classical approaches to the visual description of phase trajectories are supplemented by the methods of quantitative analysis of detailed probabilistic structure of random functions.