Random Process Model Research Articles

Methods of mathematical modeling of non-Gaussian random variables quantities and processes

Mathematical methods allowing to model non-Gaussian random variables and processes are considered. The models and description of non-Gaussian correlated processes in the form of generated Gaussian noise are analyzed, as well as the methods of formation of stationary random processes defined by the one-dimensional density distribution of Vero -abilities and the autocorrelation function. Examples of formation of non-Gaussian random variables and processes are given.

Improving Equipment Reliability and System Maintenance and Repair Efficiency

Mean time to failure of modern machinery and equipment, their individual parts and components can be calculated over the years. Methods for determining the optimal frequency of maintenance and repair, based on the collection and processing of information about the reliability of industrial facilities, during their testing in laboratories and at special sites, as well as through long, operational tests require considerable time and become expensive. The purpose of this work is to develop methods for processing information about the reliability of equipment in automated systems for maintenance and repair, which will reduce the time to collect information on equipment failures and improve the cost-effectiveness of maintenance and repair. Small, multiple-censored right-side samples of equipment operating time for failure are formed as a result of failure data collection in an automated system for equipment maintenance and repair. Calculation of reliability indicators for such samples is performed using the maximum likelihood estimation method. The article presents experimental studies of the accuracy of the maximum likelihood estimates of the parameter of the exponential distribution law for small, multiple right-censored samples. The studies were carried out by computer modeling of censored samples, similar to samples that are formed when monitoring equipment during operation. Methods of simulation modeling of random processes on a computer and methods of regression analysis were used. Analysis show that most of the maximum likelihood estimates obtained from small, multiple-censored right-side samples have significant deviations from the true values. A technique for improving the accuracy of maximum likelihood estimates is proposed. The scientific novelty is regression models are constructed that establish the relationship between the deviation of the maximum likelihood estimate from the true value and the parameters characterizing the sample structure. These models calculate and introduce corrections to maximum likelihood estimates. The use of the developed regression models will reduce the time to collect information about the reliability of the equipment, while maintaining the reliability of the results.

Investigation into the contact characteristics of rough surfaces with surface tension

Surface tension is a kind of force that exists between the molecules of the material surface, which has a significant influence on the contact characteristics of rough surfaces. Therefore, the Nayak’s random process model is introduced to describe the height and curvature distribution of asperity on isotropic rough surfaces, and a single conical asperity system considering surface tension is established. Then, the single conical asperity system is extended to the entire rough surface, and a new contact model of the rough surface is presented based on statistical methods. The contact stiffness from the proposed model is added to the finite element model for comparison with experimental results. The effects of surface tension and semi-cone angle of conical asperity on the contact load, the contact area, and the contact stiffness of rough surfaces are revealed. The results show that the new model has larger normal load and contact stiffness, and a smaller real contact area than the classical elastic model where the dimensionless separation is constant. In addition, the contact load and contact stiffness between rough surfaces increase with increasing surface tension, but the real contact area decreases. When the surface tension is constant, the larger the semi-cone angles of the conical asperity, the greater the load, the real contact area, and the stiffness will be.

Terminal Orientation in OFDM-Based LiFi Systems

Light-fidelity (LiFi) is a wireless communication technology that employs both infrared and visible light spectra to support multiuser access and user mobility. Considering the small wavelength of light, the optical channel is affected by the random orientation of user equipment (UE). In this paper, a random process model for changes in the UE orientation is proposed based on the data measurements. We show that the coherence time of the random orientation is in the order of hundreds of milliseconds. Therefore, an indoor optical wireless channel can be treated as a slowly varying channel as its delay spread is typically in the order of nanoseconds. A study of the orientation model on the performance of direct-current-biased orthogonal frequency-division multiplexing (DC-OFDM) is also presented. The performance analysis of the DC-OFDM system incorporates the effect of a diffuse link due to reflection and blockage by the user. The results show that the diffuse link and the blockage have significant effects, especially if the UE is located relatively far away from an access point (AP). It is shown that the effect is notable if the horizontal distance between the UE and the AP is greater than 1.5 m in a typical $5\times 3.5\times 3\,\,\text{m}^{3}$ indoor room.

Comparing Random Forest and Gaussian Process Modeling in the Gp-demo Algorithm

Comparing Random Forest and Gaussian Process Modeling in the Gp-demo Algorithm

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.

A comparative example of cyclostationary description of a non-stationary random process

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.

ДІАГНОСТУВАННЯ СУДНОВОГО УСТАТКУВАННЯ ПРИ ВИКОРИСТАННІ РІЗНИХ МОДЕЛЕЙ ВІБРАЦІЇ

Розглянуті моделі діагностики суднового устаткування та доцільність їх використання. Найбільш ефективними методами діагностування суднових машин та механізмів є віброакустичні методи, які використовують ту або іншу модель сигналу вібрації. Проаналізовано конкретні моделі вібрації в якості вібродіагностичних моделей. Проведено аналіз властивостей розглянутих моделей випадкових процесів, що характеризують різні процеси в теорії і при проведенні експериментальних досліджень. На підставі експериментальних досліджень знайшли підтвердження різні моделі, які дозволяють розробити і отримати практичні методи діагностування.

The use of random process models and machine learning to analyze the operation of a taxi order service

The possibilities of constructing an effective forecast of the number of taxi service orders based on mathematical models are considered. A comparative analysis of the variances of forecasting errors for various stochastic models and models based on fuzzy logic is carried out. It is shown that the best estimates are provided by the doubly stochastic model, as well as by the fuzzy Sugeno model.

Ensuring the effectiveness of the taxi order service by mathematical modeling and machine learning

We can’t imagine modern science without mathematical modeling. At the same time, ideas related to mathematical modeling find themselves in various fields. One of the interesting applications of mathematical modeling of random processes is the work of the taxi order service. The article describes the description of the operation of the taxi order service using autoregressive models with multiple roots of characteristic equations of different multiplicity. Also attempts were made to approximate and predict the time series of various important parameters of the taxi order service. Algorithms for identifying model parameters are proposed. Besides, we try to introduce machine learning methods to optimize work of taxi service.

Assessment of Operational Risks of Electric Interlocking Systems

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.

Estimation and increase of information measuring systems’ metrological reliability on the designing stage

The paper deals with mathematical modelling application aimed at the estimation and increase of the metrological reliability of the information measuring systems on designing stage. The algorithm of metrological reliability estimation is based on mathematical modeling of non-stationary random processes of temporal variation of the metrological characteristics of the measuring means. There is a discussion of two ways to realize the developed method of increasing metrological reliability of the information measuring systems in the design phase based on mathematical modeling and use of parametric optimization methods. The obtained results make it possible for customer to develop information measuring systems with counted value of metrological reliability and also to form the design decisions in order to make information measuring system with maximum value of metrological resource.

Probability Analysis of Crane Load and Load Combination Actions

This study presents probability models of crane load and load combination actions for reliability analysis of the industry buildings. Crane load values are simplified here only varying in time, and load combination actions only vary in space. With the feasible survey program and K-S test, the Gumbel distribution is chosen as the probability distribution of the arbitrary point-in-time crane load values with the statistical investigation data. With the simple stationary binomial random process model hypothesis and the block maxima method, it determined that the maximum value of crane load during the design reference period also follows the Gumbel distribution, with the peaks-over-threshold method evaluation. For multiple crane load combination actions, the modified Turkstra rule is improved to determine the probability distribution of the actions as the Gumbel distribution by considering the occurrence probability, numbers, location, and values of each crane load acting on the influence line, evaluated by the Monte Carlo simulations. Design standard values for crane loads and load combination actions, as specified in building codes, are evaluated based on these distributions in probability significance. The calculation results illustrate that the design standard values for crane loads are relatively safe and conservative.

Application of an Additive Mathematical Model of Cyclic Random Processes and a Deterministic Trend Function to the Analysis of Fracture of the Gas Pipeline

We propose an approach to the mathematical analysis of the crack resistance of steel of a main gas pipeline, which enables one to give quantitative estimates of the kinetics of dynamic fracture of the pipe with regard for the stochastic and cyclic nature of this process. It opens the possibility of modeling of the process of crack propagation in the pipe of a gas main and the analysis of separate stages of its growth. Special attention is given to the analysis of reliability of the proposed approach on the basis of comparison with the available data of full-scale experiments.

The Statistical Characteristics and Calculated Values for Air Temperature in Building`S Cladding Design

The article demonstrates the methodology and the results of statistical characteristics determination and territorial distribution as well as free air winter temperature calculated values required for the designing the building`s cladding.Daily temperatures of free air have been described by a probabilistic model of quasi-stationary differentiable random process. The expectation function is set by the Fourier series or sequence of monthly temperatures, the standard and asymmetry are determined by the mathematical expectation. The Ordinates Distribution Law is described by a linear combination of Gauss and Gumbel distributions. The frequency structure is determined by the time-constant value of the effective frequency. The required statistical characteristics are determined by the results of systematic measurements of average daily air temperature at 485 observation points inUkraine.The calculated values of winter air temperature are determined by using probabilistic approach with accounting for return period equal to building working life, as well as temperature averaging interval, reflecting building`s cladding thermal inertia. The map of territorial distribution of Ukraine has been created based on free air temperature values and dependencies which can lead to calculated values taking into account given building operating life, temperature averaging interval and altitude in mountainous regions.The developed methodology ensures the possibility of minimal calculated values determination for free air temperatures including all the factors mentioned above, and the suggestions as for introduction of norms enabling accounting for building`s cladding thermal inertia within the building working life when making thermotechnical characteristics calculations.

Optimal Bayesian Kalman Filtering With Prior Update

In many practical filter design problems, the exact statistical information of the underlying random processes is not available. One robust filtering approach in these situations is to design an intrinsically Bayesian robust filter that provides optimal solution relative to the prior distribution governing the uncertainty class of all possible joint random process models. In this context, the intrinsically Bayesian robust Kalman filter has been recently introduced for the case that the second-order statistics of the observation and process noise in the state-space model are unknown. However, such a filter does not utilize the additional information embedded in the data being observed. In this paper, we derive the optimal Bayesian Kalman filter, which is optimal over posterior distribution obtained from incorporating data into the prior distribution. This filter has the same recursive structure as that of the classical Kalman filter, except that it is designed relative to the posterior effective noise statistics , which are found by employing the method of factor graphs through formulating the problem of computing the likelihood function as a message passing algorithm.

Track Extraction With Hidden Reciprocal Chains

This technical note (TN) develops Bayesian track extraction algorithms for targets modeled as hidden reciprocal chains (HRC). HRC are a class of finite-state random process models that generalize the familiar hidden Markov chains (HMC). HRC are able to model the “intention” of a target to proceed from a given origin to a destination, behavior that cannot be properly captured by an HMC. While Bayesian estimation problems for HRC have previously been studied, this TN focuses principally on the problem of track extraction, that is, confirming target existence in a set of observations of uncertain origin. Simulation examples are presented, which show that the additional model information contained in an HRC, measured in terms of Kullback–Leibler divergence, improves detection performance when compared to HMC models.

Zależna od czasu niezawodność układu przekładni zębatej jako funkcja procesu stopniowego zużycia

To study dynamic evolution law of mechanical reliability caused by wear, gear transmission system is taken as a research object. Considering the effect of clearance caused by wear on gear teeth load in double meshing area, the formula of dynamic distribution load which is undertaken by two adjacent teeth is deduced. And the distributed pressure and meshing speed, which should be taken into account while calculating gear wear, are obtained based on the Winkler surface model and principle of tooth mesh. Based on the Archard’s wear model, numerical simulation model for wear in spur gear is deduced, and the wear depth of each meshing points on teeth outline with different wear cycles are obtained. The calculation wear model is replaced with a surrogate model with Neural Network and Kriging method to overcome time-consuming defect. Random process model is integrated with the surrogate model, and dynamic reliability for nonlinear stochastic structure with unknown distribution characteristic is obtained with Neural Network-based Edgeworth series technique and four moment methods, which is compared with Kriging-based Monte Carlo simulation method. The computational efficiency and accuracy are also demonstrated.

Local linear estimation for spatial random processes with stochastic trend and stationary noise.

We consider the problem of estimating the trend for a spatial random process model expressed as Z(x) = μ(x) + ε(x) + δ(x), where the trend μ is a smooth random function, ε(x) is a mean zero, stationary random process, and {δ(x)} are assumed to be i.i.d. noise with zero mean. We propose a new model for stochastic trend in by generalizing the notion of a structural model for trend in time series. We estimate the stochastic trend nonparametrically using a local linear regression method and derive the asymptotic mean squared error of the trend estimate under the proposed model for trend. Our results show that the asymptotic mean squared error for the stochastic trend is of the same order of magnitude as that of a deterministic trend of comparable complexity. This result suggests from the point of view of estimation under stationary noise, it is immaterial whether the trend is treated as deterministic or stochastic. Moreover, we show that the rate of convergence of the estimator is determined by the degree of decay of the correlation function of the stationary process ε(x) and this rate can be different from the usual rate of convergence found in the literature on nonparametric function estimation. We also propose a data dependent selection procedure for the bandwidth parameter which is based on a generalization of Mallow's C p criterion. We illustrate the methodology by simulation studies and by analyzing a data on surface temperature anomalies.

Analysis of Fracture Characteristic of a Gas Main Pipe on the Basis of the Additive Mathematical Model of the Cyclic Random Process and Polynomial Function

The use of a mathematical model to describe the phenomenon of dynamic fracture of the gas main in a broad range of crack propagation rates is substantiated, taking into account the stochastic and cyclic nature of this process. Particular attention is paid to the reliability of the proposed approach, which is evaluated based on comparison with known natural experiments.