Continuous Random Process Research Articles

Sparrow search algorithm enhanced multi-output regression for predicting rock fracture shear displacements: A metaheuristic-hybridized model

This study presents a metaheuristic-hybridized model based on sparrow search algorithm (SSA) and multi-output least-squares support vector regression machines (SSA-MLS-SVR) to predict the continuous shear displacements of rock fractures, which is closely related to the geo-structure stability and safety. Based on the database including 258 continuous shear samples of rock fractures, two subdatasets recording continuous shear displacements in three (216 samples) and four steps (141 samples) are used to develop the proposed model respectively. In the model improvement phase, three kinds of nonlinear transformations are utilized to eliminate the low sensitivity of SSA-MLS-SVR model caused by value scale and data distribution. The experimental results show that the nonlinear transformations can significantly improve the prediction accuracy of SSA-MLS-SVR. Additionally, some characteristics of the continuous shear process of rock fractures are also discovered via extended experiments. First, the prediction accuracy of the first-step shear displacement can be greatly improved using the peak shear displacement as an input. And it is inferred that the beginning of the post-peak shear process (the first-step shear) may be closely related to the peak shear of rock fractures. Second, the post-peak shear process is a continuous random process, in which the displacement of each shear step is only highly correlated with the most recent shear displacements in the time dimension.

On stationary random processes with fuzzy states

In this paper, continuous random processes with fuzzy states are studied. The properties of their numerical characteristics – fuzzy expectations, expected values and covariance functions – are established. The main attention is paid to the class of stationary fuzzy-random processes. For them, the ergodicity property and the spectral representation of covariance function (generalized Wiener–Khinchin theorem) are substantiated. The results obtained are based on the properties of fuzzy-random variables and numerical random processes. Triangular fuzzy-random processes are considered as examples.

Towards reliability of load-bearing beams of bridges (Tomsk)

One of the criteria for assessing the reliability of load-bearing elements of bridges, are peak values of the temporary mobile load significantly exceeding the standard level of loading. Peak loads are interpreted as a continuous random process in the form of a random pulse sequence manifested after a random period of time and having a random duration.Purpose: The aim of the paper is to show changes in temporary mobile load over time in the form of a simple Poissonian flow, where the bridge load is independent, and repeatability and duration of the random load are independent random variables.Research findings: Three cases are considered to determine sufficiently low probability of destructive temporary mobile loads during the given normative service life of bridges. Based on the failure types, the average failure-free operation of the beam is determined as well as the operating time for a possible failure with a prolonged or specific excess load.

SUBSTANTIATION OF A GENERALIZED MODEL OF THE FUNCTIONING OF A PROMISING INFORMATION AND MEASUREMENT SYSTEM FOR COLLECTING AND PROCESSING INFORMATION

The level of development of the defense sector directly depends on the degree of accumulation of knowledge in the latest technologies and the efficiency of this process, since the accumulation of the results of yesterday's scientific achievements cannot bring the state to new high technologies today or in the future.
 With the ever-increasing amount of information to be processed to make effective decisions, as well as the shrinking time to make a decision (this is especially true for systems operating in real time), the use of automation tools in the era of armament systems development is a necessity.
 The use of the mathematical theory of research planning allows us to develop a coherent logical scheme and develop optimal conditions for solving all stages of determining the movement of an air object.
 Based on the results of measurement processing and analysis, the correctness of the object's functioning is assessed. These operations are interdependent. If the experiment reveals a malfunction, an additional analysis is performed to localize the defects and determine their causes, after which appropriate modifications are made to the facility and, if necessary, changes are made to the test program and methodology.
 The concept of a mathematical model is now widely used to describe the processes occurring in complex technical systems. A mathematical model is built on the basis of a comprehensive analysis of the system's behavior and extensive use of statistical research results. It must be complete enough to adequately describe the system, but also simple enough to allow the resulting algorithms to be implemented on electronic computers.
 In addition, in order to perform measurement processing, i.e. to obtain estimates of measured values, and the experiment as a whole, to determine the estimation of characteristics describing the object, it is necessary to know the structure and statistical characteristics of the signal that is being processed. Measurement information can be a continuous random process, a discrete random process, a continuous or discrete random sequence. The measurement information of a multidimensional information and measurement system is described in the form of discrete random sequences.

On Continuous Random Processes with Fuzzy States

On Continuous Random Processes with Fuzzy States

Four Finite Dimensional (FD) Surrogates for Continuous Random Processes

Four Finite Dimensional (FD) Surrogates for Continuous Random Processes

Error detection and filtering of incompressible flow simulations for aeroacoustic predictions of human voice.

The presented filtering technique is proposed to detect errors and correct outliers inside the acoustic sources, respectively, the first time derivative of the incompressible pressure obtained from large eddy simulations with prescribed vocal fold motion using overlay mesh methods. Regarding the perturbed convective wave equation, the time derivative of the incompressible pressure is the primary sound source in the human phonation process. However, the incompressible pressure can be erroneous and have outliers when fulfilling the divergence-free constraint of the velocity field. This error is primarily occurring for non-conserving prescribed vocal fold motions. Therefore, the method based on a continuous stationary random process was designed to detect rare events in the time derivative of the pressure. The detected events are then localized and treated by a defined window function to increase their probability. As a consequence, the data quality of the non-linearly filtered data is enhanced significantly. Furthermore, the proposed method can also be used to assess convergence of the aeroacoustic source terms, and detect regions and time intervals, which show a non-converging behavior by an impulse-like structure.

Enhanced Facilitated Diffusion of Membrane-Associating Proteins under Symmetric Confinement.

The facilitated surface diffusion of transiently adsorbing molecules in a planar confined microenvironment (i.e., slit-like confinement) is highly relevant to biological phenomena, such as extracellular signaling, as well as numerous biotechnology systems. Here, we studied the surface diffusion of individual proteins confined between two symmetric lipid bilayer membranes, under a continuum of confinement heights, using single-molecule tracking and convex lens-induced confinement as well as hybrid, kinetic Monte Carlo simulations of a generalized continuous time random walk process. Surface diffusion was observed to vary non-monotonically with confinement height, exhibiting a maximum at a height of ∼750 nm, where diffusion was nearly 40% greater than that for a semi-infinite system. This demonstrated that planar confinement can, in fact, increase surface diffusion, qualitatively validating previous theoretical predictions. Simulations reproduced the experimental results and suggested that confinement enhancement of surface diffusion for symmetric systems is limited to cases where the adsorbate exhibits weak surface sticking.

Dynamic Response Simulation of the Slope Triggered by Fluctuating Pressure during High Dam Flood Discharge

The problem of slope vibration induced by flood discharge of high dams cannot be ignored. In this paper, this problem is simplified to be the relationship among the fluctuating pressures, the inherent properties of the slope system, and its dynamic responses. The characteristics of fluctuating pressures in the plunge pool during flood discharge are analyzed by the hydraulic model test. And a numerical slope model consisting of the finite element domain and the PML domain is established to study the inherent characteristics and vibration responses by numerical simulation. The results show that the flow fluctuation in plunge pool is a continuous and stable random vibration process. And the fluctuating pressures on the bottom area of the plunge pool are greater than that of the sidewall, but the fluctuation phenomenon at the sidewall is more intense. In addition, the slope displacement vibration intensity caused by fluctuating pressures is small, only at the micron level. The fluctuating pressures on the bottom area of the plunge pool are the dominant factor affecting the kinetic energy density distribution of the slope. This work provides a simple and efficient method for the study of slope vibration induced by high dam flood discharge.

Development of a Methodology for Optimal Control of the Reliability of Highways at the Stages of Their Design and Stages of Construction, Reconstruction, Repair, and Maintenance

A methodological approach to the problem of research and selection of specific optimal procedures while ensuring indicators of the reliability of highways is stated, oriented in relation to both their temporal and spatial coordinates. In general, their mathematical models are presented in the form of continuous or discrete random processes. A specific methodological approach to the choice of rational intervals is proposed, based on statistical modeling of processes that affect the numerical values of the time variables of the defining parameters of highways using elements of the general correlation theory of random processes. The main calculation ratios are presented, which are necessary for the implementation of optimization procedures while ensuring the numerical values of the defining parameters of highways in their characteristic control points, which affect the reliability indicators of highways. A meaningful analysis of individual approaches to the choice of rational control intervals from the point of view of their practical use, in relation to the tasks of ensuring reliability indicators of highways at the stages of their design and stages of construction, reconstruction, repair, and maintenance is carried out.

On extremes of PSI-processes and gaussian limits of their normalized independent identical distributed sums

We define PSI-process — Poisson Stochastic Index process, as a continuous time random process which is obtained by a manner of a randomization for the discrete time of a random sequence. We consider the case when a double stochastic Poisson process generates this randomization, i. e. such Poisson process has a random intensity. Under condition of existence of the second moment the stationary PSI-processes possess a covariance which coincides with the Laplace transform of the random intensity. In our paper we derive distributions of extremes for a one PSI-process, and these extremes are expressed in terms of Laplace transform of the random intensity. The second task that we solve is a convergence of the maximum of Gaussian limit for normalized sums of i. i. d. stationary PSI-processes. We obtain necessary and sufficient conditions for the intensity under which, after proper centering and normalization, this Gaussian limit converges in distribution to the double Exponential Law. For solution this task we essentially base on the monograph: M.R.Leadbetter, Georg Lindgren, Holder Rootzen (1986) “Extremes and Relative Properties of Random Sequences and Processes”, end essentially use the Tauberian theorem in W. Feller form.

Data-Driven Continuous-Time Modeling of Power System Uncertainty Using a Multi-Dimensional Stochastic Differential Equation

This paper proposes a generalized continuous random process modeling framework for imposing any desired probability distribution, autocorrelation and power spectral density of measured data of power system uncertainty by developing a multi-dimension stochastic differential equation (SDE) together with a quadratic output equation. An algorithm for determining appropriate parameters of the proposed SDE model is presented. The effectiveness of derived random modeling scheme is verified by carrying out comparative simulations using actual time series data of wind speed, solar radiation, PJM dynamic regulation, and wind power. The proposed SDE model can be readily integrated in various electrical component models for reliable operation, stability assessment and control of renewable power systems.

ON THE KOLMOGOROV-WIENER FILTER FOR RANDOM PROCESSES WITH A POWER-LAW STRUCTURE FUNCTION BASED ON THE WALSH FUNCTIONS

Context. We investigate the Kolmogorov-Wiener filter weight function for the prediction of a continuous stationary random process with a power-law structure function.
 Objective. The aim of the work is to develop an algorithm of obtaining an approximate solution for the weight function without recourse to numerical calculation of integrals.
 Method. The weight function under consideration obeys the Wiener-Hopf integral equation. A search for an exact analytical solution for the corresponding integral equation meets difficulties, so an approximate solution for the weight function is sought in the framework of the Galerkin method on the basis of a truncated Walsh function series expansion.
 Results. An algorithm of the weight function obtaining is developed. All the integrals are calculated analytically rather than numerically. Moreover, it is shown that the accuracy of the Walsh function approximations is significantly better than the accuracy of polynomial approximations obtained in the authors’ previous papers. The Walsh function solutions are applicable in wider range of parameters than the polynomial ones.
 Conclusions. An algorithm of obtaining the Kolmogorov-Wiener filter weight function for the prediction of a stationary continuous random process with a power-law structure function is developed. A truncated Walsh function expansion is the basis of the developed algorithm. In opposite to the polynomial solutions investigated in the previous papers, the developed algorithm has the following advantages. First of all, all the integrals are calculated analytically, and any numerical calculation of the integrals is not needed. Secondly, the problem of the product of very small and very large numbers is absent in the framework of the developed algorithm. In our opinion, this is the reason why the accuracy of the Walsh function solutions is better than that of the polynomial solutions for many approximations and why the Walsh function solutions are applicable in a wider range of parameters than the polynomial ones. The results of the paper may be applied, for example, to practical traffic prediction in telecommunication systems with data packet transfer.

Realizability of the rapid distortion theory spectrum: The mechanism behind the Kelvin–Townsend equations

In the context of homogeneous turbulence, we prove that the Rapid Distortion Theory (RDT) model for the spectral tensor preserves the symmetry, positive semidefiniteness, and integrability properties required in Cramér’s characterization of the spectral tensor of a continuous homogeneous random process. From this, a statistically valid correlation tensor is obtained that returns a Reynolds stress tensor model that satisfies realizability conditions. The number of hypotheses used is kept to a minimum, which allows a flexible use of the model in the applications. The Kelvin–Townsend equations allow us to construct the solution and prove its properties by means of a factorization approach. Since the RDT spectral tensor model is a system of transport equations plus an algebraic restriction due to incompressibility, we deal with the existence, uniqueness, and persistence of solutions in a specific set of functions by using DiPerna–Lions renormalization techniques.

Examination of Interest-Growth Differentials and the Risk of Sovereign Insolvency

The objective of this research was to demonstrate the (nonlinear) risks of sovereign insolvency and explore the applicability of stochastic modeling in public debt management, given a structural economic model of stochastic government debt dynamics. A stochastic optimal control model was developed to model public debt dynamics based on the debt accounting identity, where the interest-growth differential obeys a continuous random process. This stochasticity represents both the interest rate risk of public debt and the variability of the growth rate of the nominal Gross Domestic Product combined. The optimal fiscal policy was analyzed in terms of the model parameters. The model was simulated, and results were visualized. The insolvency risk was demonstrated by examining the variance of the optimal process. The model was amended with hidden credit risk premia and fiscal multipliers, which forces the debt dynamics to be nonlinear in the debt ratio. The results, on the other hand, confirm that the volatility of the interest-growth differential is crucial in terms of sovereign solvency and in addition, it demonstrates the large risks stemming from the multiplier effect, which underlines the need for prudent debt management and fiscal policy.

Continuous Random Process Modeling of AGC Signals Based on Stochastic Differential Equations

Reflecting the uncertainty of renewable energy generations and loads, power system AGC signals are essentially random processes. For the sake of the optimal operation and control of the AGC participant, such as energy storage systems (ESSs) in a performance/mileage-based regulation market, taking the uncertain nature, especially the temporal correlation, of the AGC signals into consideration can be beneficial; hence, random process models of the AGC signals are needed. However, a continuous random process model of the AGC signal that jointly considers the probability distribution and the temporal correlation is still lacking. To fill this gap, this paper first presents a systematic methodology for modeling the continuous random processes of AGC signals based on stochastic differential equations (SDEs). It is shown that AGC signals may have a saturated stationary probability density function and a biexponential temporal correlation, which are very different from the renewable generations. To capture these special characteristics, SDEs are then carefully constructed, which are easy to use in optimization and control. Using the PJM traditional and dynamic regulation (RegD and RegA) signals and the signal received from a battery ESS (BESS) plant in Jiangsu, China for example, simulation shows that the SDE is able to simultaneously capture the probability distribution and temporal correlation accurately.

Model of Continuous Random Cascade Processes in Financial Markets

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.

On bounded solutions of linear SDEs driven by convergent system matrix processes with Hurwitz limits

Linear time-varying stochastic differential equations with a.s. convergent continuous random system matrix processes are considered. It is shown that given the limit is known to be Hurwitz (i.e. asymptotically stable), the generated state solutions are a.s. bounded. This property is shown to hold by substantiating that, w.p.1, (i) no finite escape time exists and (ii) no divergence to infinity, as , may occur. An application to stochastic adaptive control is provided.

Simulation of continuous random process according to the specified sequence of extremes

A method has been developed for converting a discrete sequence of extrema into a continuous process. The relevancy of the problem is attributed to the necessity of an approximate estimation of spectral density in in testing materials and structures under random (irregular) loading. A great bulk of available experimental data thus can be used in development and validation of calculation methods for assessing durability in the multi-cycle region. Postulating the continuity of random stress processes and their first derivative we propose to connect piecewise the available starting points (namely, the extrema of the random process) with half-cosine functions under the condition of compatibility at the points of extrema. A distinctive feature of the method is the provision of 100% coincidence of the values and sequences of extrema in the initial discrete and simulated continuous processes. The issue of choosing the magnitude of half-periods for these half-cosine functions is addressed on the basis of information obtained from the analysis of real stress records in the form of a regression equation linking half-periods and half-ranges for some realizations of the random process for transport vehicles. The regression dependences of the half-periods and semi-ranges of bending stresses (part of a railway train) and torsion (torsion shaft of a tracked vehicle) are shown as an example. An analysis of the correlation of two random variables (half-periods and half-ranges) according to empirical data has shown that the correlation exists and is significant for the observed number of points thus providing the basis for using the regression formula for an approximate choice of the frequency composition of the process. Moreover, the lower restrictions are imposed on the number of points (at least 5) in the half-period. Since the extrema of the initial and simulated processes coincide in accordance with the principle of the proposed simulation, the distribution of the amplitudes of complete cycles, as well as the results of schematization by other known methods are identical, therefore, the estimate of the durability by hypotheses based on a linear one is also identical. The validation of the method consists in consideration of the chain: 1) the initial continuous process; 2) the discrete process of extrema; 3) simulated continuous process according to the proposed method. Auxiliary distributions, such as distributions of maximum, minimum and average cycle values also coincide in accordance with the principle of modeling. The method is proposed to be used in analysis of the comparability of two competing approaches in assessing the loading in the problems of assessing durability, namely: those that use cycle-counting methods and methods based on the spectral density of processes. Since the spectral densities of the processes can differ due to an approximate choice of the frequencies on the basis of a regression formula, methods on their base can give estimates of the durability that differ from those obtained by schematization methods. To study this phenomenon, further computational experiments are required. The developed method can be very useful for the experiment design.

Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound

We investigate the connection between the time-evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramer-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that this minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.