Problem Of Stochastic Models Research Articles

Joint application of the Monte Carlo method and computational probabilistic analysis in problems of numerical modeling with data uncertainties

Abstract In this paper, we suggest joint application of computational probabilistic analysis and the Monte Carlo method for numerical stochastic modeling problems. We use all the capabilities of computational probabilistic analysis while maintaining all the advantages of the Monte Carlo method. Our approach allows us to efficiently implement a computational hybrid scheme. In this way, we reduce the computation time and present the results in the form of distributions. The crucial new points of our method are arithmetic operations on probability density functions and procedures for constructing on the probabilistic extensions. Relying on specific numerical examples of solving systems of linear algebraic equations with random coefficients, we present the advantages of our approach.

Comparison of Statistical Approaches for Reconstructing Random Coefficients in the Problem of Stochastic Modeling of Air–Sea Heat Flux Increments

This paper compares two statistical methods for parameter reconstruction (random drift and diffusion coefficients of the Itô stochastic differential equation, SDE) in the problem of stochastic modeling of air–sea heat flux increment evolution. The first method relates to a nonparametric estimation of the transition probabilities (wherein consistency is proven). The second approach is a semiparametric reconstruction based on the approximation of the SDE solution (in terms of distributions) by finite normal mixtures using the maximum likelihood estimates of the unknown parameters. This approach does not require any additional assumptions for the coefficients, with the exception of those guaranteeing the existence of the solution to the SDE itself. It is demonstrated that the corresponding conditions hold for the analyzed data. The comparison is carried out on the simulated samples, modeling the case where the SDE random coefficients are represented in trigonometric form, which is related to common climatic models, as well as on the ERA5 reanalysis data of the sensible and latent heat fluxes in the North Atlantic for 1979–2022. It is shown that the results of these two methods are close to each other in a quantitative sense, but differ somewhat in temporal variability and spatial localization. The differences during the observed period are analyzed, and their geophysical interpretations are presented. The semiparametric approach seems promising for physics-informed machine learning models.

Modeling of nonlinear and nonstationary stochasticity for atomic ensembles

This paper addresses the problem of stochastic modeling of atomic ensembles under multi-source noise and makes the model interpretable. First, based on Itô’s lemma and Allan variance analysis (ITÔ-AVAR), an approach is proposed to model nonstationary stochastic submodels of atomic ensembles. On this basis, the variance decomposition and nonlinear optimization algorithms are utilized to hybridize modeling atomic ensembles with nonlinear and nonstationary properties. Second, an Itô’s lemma dynamic allan variance analysis (ITÔ-DAVAR) approach is developed for online modeling of atomic ensembles. Further, an atomic ensembles sensitivity enhancement scheme based on the proposed approach is given, which effectively promotes the progress of quantum instrument engineering. Finally, the proposed scheme are deployed in the optical pumping magnetometer and spin-exchange relaxation-free comagnetometer, respectively, while experimentally verifying the sensitivity of the spin-exchange relaxation-free comagnetometer reaches 5.36×10−6degs−1Hz−1/2.

Stochastic model updating method for estimates of arbitrary distributed parameters using resampling technique

A stochastic model updating method is developed to quickly and comprehensively estimate the inherent uncertainties of parameters with the arbitrary distribution. In this methodology, the uncertainty propagation based on the point estimation method provides fast and accurate power for the model response computation. Then the resampling technique for robustness improving is devised to deal with the imprecise observations resulting from the imprecision of experimental measurement. Subsequently, as a quantitative metric, the Bhattacharyya distance is adopted to construct the objective function in stochastic model updating. Next, the classical mass-spring system and the 23-bar truss are presented to illustrate the feasibility and superiority of the proposed method, as well as the role of the resampling technique on the stochastic model updating containing multiple ill samples. To gain insights into the proposed method, the experiment of the beam structure with additional masses is designed, and a real engineering problem of the balance vehicle is implemented. Compared with the previous studies, which can only deal with a small proportion of ill samples, the advantage of the proposed method is that it can solve the problem of stochastic model updating of up to 20% ill samples, and each iterative step of updating process requires only a few calculations. As revealed from the case studies, the proposed method is suitable for coping with the stochastic model updating problems containing multiple imprecise observations in engineering.

Empirical Stochastic Model of Multi-GNSS Measurements.

The stochastic model, together with the functional model, form the mathematical model of observation that enables the estimation of the unknown parameters. In Global Navigation Satellite Systems (GNSS), the stochastic model is an especially important element as it affects not only the accuracy of the positioning model solution, but also the reliability of the carrier-phase ambiguity resolution (AR). In this paper, we study in detail the stochastic modeling problem for Multi-GNSS positioning models, for which the standard approach used so far was to adopt stochastic parameters from the Global Positioning System (GPS). The aim of this work is to develop an individual, empirical stochastic model for each signal and each satellite block for GPS, GLONASS, Galileo and BeiDou systems. The realistic stochastic model is created in the form of a fully populated variance-covariance (VC) matrix that takes into account, in addition to the Carrier-to-Noise density Ratio (C/N)-dependent variance function, also the cross- and time-correlations between the observations. The weekly measurements from a zero-length and very short baseline are utilized to derive stochastic parameters. The impact on the AR and solution accuracy is analyzed for different positioning scenarios using the modified Kalman Filter. Comparing the positioning results obtained for the created model with respect to the results for the standard elevation-dependent model allows to conclude that the individual empirical stochastic model increases the accuracy of positioning solution and the efficiency of AR. The optimal solution is achieved for four-system Multi-GNSS solution using fully populated empirical model individual for satellite blocks, which provides a 2% increase in the effectiveness of the AR (up to 100%), an increase in the number of solutions with errors below 5 mm by 37% and a reduction in the maximum error by 6 mm compared to the Multi-GNSS solution using the elevation-dependent model with neglected measurements correlations.

Innovative technologies in construction and Building Information Modeling

The paper presents the results of research in the field of the use of innovative technologies in the design of buildings. The modern technologies of information modeling, in particular, BIM, which makes it possible to represent the building not only in three-dimensional space, but also allows designing it in the dynamic system « design - construction – operation » are considered. The experience of using BIM technologies is based on identifying errors at the initial design stage and the possibility of their elimination at the same stage. The development of mechanisms for detecting design errors and defects formed the basis of the task set in the work. A model for signaling about the detection of defects has been developed, and the problem of stochastic modeling of the qualitative parameters of the designed object has been solved.

A Tree‐Based Direct Sampling Method for Stochastic Surface and Subsurface Hydrological Modeling

AbstractDirect sampling (DS) is a versatile multiple‐point statistics method for generating spatial‐temporal geostatistical models. DS is known for being able to address a variety of training images and hence spatiotemporal stochastic modeling problems. One limitation of DS is the central processing unit (CPU) time, mostly attributed to the use of a random search for patterns in the training image. To improve CPU performance, we propose a tree‐based direct sampling (TDS) method. In our method, training patterns are grouped according to their similarities combined with a clustering tree for fast lookup. Rather than patterns, we store locations in our database. During the simulation, TDS applies a tree‐driven search approach. Two objectives, similarity and diversity, are used to rapidly retrieve patterns and prevent trapping into local optima. We also introduce a way to speed up simulation by means of pasting patterns with adaptive size. The performance of our TDS is investigated using a 2‐D benchmark training image. Moreover, we apply the proposed method to two real cases including gap filling the bedrock topography in Antarctica from radar to better understand subglacial hydrology and creating 3‐D groundwater models in the Danish aquifer system. Based on several quantitative evaluations, we find the proposed TDS is comparable to DS in terms of simulation quality, while significantly saves CPU time.

Auto-thermal reforming (ATR) of natural gas: An automated derivation of optimised reduced chemical schemes

A fully automated strategy is discussed to construct reduced chemistry suitable for the numerical simulation of stationary combustion systems of large dimension, such as auto-thermal reforming (ATR) of natural gas for syngas production. Because of computing limitations in terms of space and time resolution, three-dimensional simulations of an ATR unit cannot be addressed with detailed chemistry. A procedure is proposed to automatically derive optimised and reduced chemical schemes under specific ATR operating conditions. A stochastic model problem is first designed to probe the dynamical response of a detailed chemical scheme, over a large range of chemical compositions of the mixture. Reference composition space trajectories are built, featuring turbulent micro-mixing and reactions. Following these trajectories, the chemical response is analysed using directed relation graphs with error propagation combined with quasi-steady state hypothesis, to reduce the number of species and elementary reactions. Then, the time evolution of the model problem is coupled with a genetic algorithm, to optimise on the fly the chemical rates of the reduced kinetics, according to the reference composition space trajectories. The accuracy of the reduced scheme is monitored with a fitness function and the results are tested against the reference detailed chemistry.

Analytical Stochastic Model of Traffic

The problem of stochastic modeling of trafic of abstract agents on a graph is considered. The key function to be estimated is the Probability Distribution function (PDF) that reflects the probability that a given number of agents are located at a given node at a given moment. Once this function is constructed, other statistical characteristics of trafic can be easily derived. A methodology for recurrent estimation of the PDF is proposed and analytically justified. A numerical comparison with agent-based models confirms the accuracy of the model.

A comparison of two stochastic model updating methods using the DLR AIRMOD test structure

The problem of stochastic model updating is addressed by means of the application of two methods (covariance and interval model updating) to the DLR AIRMOD structure which is repeatedly disassembled and reassembled to provide a database of modal variability due to uncertainty in joint and support stiffnesses and masses of cables and screws. The covariance method is based on an assumption of small uncertainty and implemented at each step of an iterative approach by forward propagation of uncertain parameters using a multivariate normal distribution. The interval approach is based on a Kriging meta-model, thereby providing a very efficient surrogate to replace the expensive full finite element model. This allows a mapping from multiple output measurements to define a hypercube bounded by intervals of parameter uncertainty. It is shown that the measured data is fully enclosed by the hyper-ellipses and hypercubes of the covariance and interval methods respectively. As expected, the interval method is found to be more conservative than the covariance approach but still provides useful estimates without restriction by any assumption of probability distribution.

Extreme value modeling for environmental data analysis

This article deals with the problem of stochastic modeling the total maximum amount of flushed rainfall in a watershed when data come from daily rainfall measurements in various locations inside a watershed. The annual total maximum \(Y\) of a random number \(N\) of independent total extremes in a given year is of great interest, for instance when dealing with the prediction of the largest flood that might occur in a given year in order to construct hydraulic protections in populated areas. Here, we propose a joint probability model for \(Y\) and \(N\) on the basis of the peaks over a threshold method. From this model, it is possible to obtain the marginal distribution of \(Y\), which includes information contained in the \(N\) variable. It is also possible to obtain the conditional distribution of \(N\) given that \(Y\) is less than a given value \(y\), which provides information about predicting the mean number of extreme events \(N\) in a year. Some discussion is included on two methods for estimating the parameters involved in the distribution of \(Y\). An application of this model is provided where daily rainfall data coming from some basins located in a southeast region of Mexico are analyzed.

Human Activity as a Manifold-Valued Random Process

Most of previous shape based human activity models were built with either a linear assumption or an extrinsic interpretation of the nonlinear geometry of the shape space, both of which proved to be problematic on account of the nonlinear intrinsic geometry of the associated shape spaces. In this paper we propose an intrinsic stochastic modeling of human activity on a shape manifold. More importantly, within an elegant and theoretically sound framework, our work effectively bridges the nonlinear modeling of human activity on a nonlinear space, with the classic stochastic modeling in a Euclidean space, and thereby provides a foundation for a more effective and accurate analysis of the nonlinear feature space of activity models. From a video sequence, human activity is extracted as a sequence of shapes. Such a sequence is considered as one realization of a random process on a shape manifold. Different activities are then modeled as manifold valued random processes with different distributions. To address the problem of stochastic modeling on a manifold, we first construct a nonlinear invertible map of a manifold valued process to a Euclidean process. The resulting process is then modeled as a global or piecewise Brownian motion. The mapping from a manifold to a Euclidean space is known as a stochastic development. The advantage of such a technique is that it yields a one-one correspondence, and the resulting Euclidean process intrinsically captures the curvature on the original manifold. The proposed algorithm is validated on two activity databases [15], [5] and compared with the related works on each of these. The substantiating results demonstrate the viability and high accuracy of our modeling technique in characterizing and classifying different activities.

In This Issue

In This Issue

Inverse Problems in Stochastic Modeling of Mixed-Mode Power-Law and Diffusional Creep for Distributed Grain Size

We seek to develop analytical methods through which the high-temperature deformation behavior of polycrystals can be explained in terms of the statistical distribution of the grain size. Changes in the stress exponent and grain size exponent with the strain rate are related to mixed-mode deformation in which the large grains deform by power-law creep and the small grains by diffusional creep. Two results are obtained. The first is an expression (Eq. [13]) that relates the experimental values of the power-law exponent and the grain size exponent to the values predicted from the classical models for uniform grain size. This equation is independent of the standard deviation of the grain size distribution, the average grain size, and the temperature. In a second result, it is shown that measurements of the change in the stress exponent with the strain rate can be analyzed to estimate the standard deviation and the median value of the grain size. The possible significance of these results is tested against experiments on the superplastic deformation of aluminum, drawn from the literature.

Guest Editors' Introduction: Stochastic Modeling of Complex Systems

This special issue of CiSE highlights various aspects of and outstanding problems in stochastic modeling of complex systems. The techniques covered include stochastic spectral methods, the information-theoretic approach, and sensitivity analysis for uncertainty quantification. Two application articles-one in the field of ecology and the other in the field of manufacturing design-emphasize the diversity of research disciplines in which the ability to handle uncertainty is crucial to obtaining reliable science-based predictions

A Review on Phase-type Distributions and their Use in Risk Theory

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

A Review on Phase-type Distributions and their Use in Risk Theory

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

６．地震の力学問題 地震現象の数値シミュレーションのための確率モデルと解析理論について

The group of the author have been proposing stochastic modeling for the simulation of earthquake phenomena. This paper presents the analysis theory of a stochastic model. While the stochastic model problem is formulated as a boundary value problem, the proposed analysis theory transforms the boundary value problem into an equivalent variational problem, in which the correlation between the input uncertainty of the model and the output variability of the responses needs not to be explicitly considered. This paper gives detailed explanations of the analysis theory and presents efficient numerical computation methods of a stochastic model problem based on the present analysis theory.

Analysis of Sample-Path Optimization

Sample-path optimization is a method for optimizing limit functions occurring in stochastic modeling problems, such as steady-state functions in discrete-event dynamic systems. It is closely related to retrospective optimization techniques and to M-estimation. The method has been computationally tested elsewhere on problems arising in production and in project planning, with apparent success. In this paper we provide a mathematical justification for sample-path optimization by showing that under certain assumptions—which hold for the problems just mentioned—the method will almost surely find a point that is, in a specified sense, sufficiently close to the set of optimizers of the limit function.

Analysis of Sample-Path Optimization

Sample-path optimization is a method for optimizing limit functions occurring in stochastic modeling problems, such as steady-state functions in discrete-event dynamic systems. It is closely relate...