Discrete Stochastic Models Research Articles

Highly accelerated kinetic Monte Carlo models for depolymerisation systems

Kinetic Monte Carlo (kMC) models are a well-established modelling framework for the simulation of complex free-radical kinetic systems. kMC models offer the advantage of discretely monitoring every chain sequence in the system, providing full accounting of the chain molecular weight distribution. These models are marred by the necessity to simulate a minimum number of molecules, which confers significant computational burden. This paper adapts and creates a highly generalizable methodology for scaling dilute radical populations in discrete stochastic models, such as Gillespie's Stochastic Simulation Algorithm (SSA). The methodology is then applied to a kMC simulation of polystyrene (PS) pyrolysis, using a modelling framework adapted from literature. The results show that the required number of simulated molecules can be successfully reduced by up to three orders of magnitude with minimal loss of convergent behaviour, corresponding to a wall-clock simulation speed reduction of between 95.2 to 99.6 % at common pyrolysis temperatures.

A novel stochastic ProFiVaS model based on decomposition of stochastic Vasicek differential equation for modeling and simulating of financial indicators

The main purpose of this study is to develop a new discrete time stochastic ProFiVaS model based on decomposition of stochastic Vasicek differential equation which follows bouncing autoregressive AR1, with fixed variance and trend for modeling and simulating of international financial time series forecasting.This study offers the four discrete time stochastic models. The first model is determined as autoregressive integrated moving average-Arima1,1,1, and the parameter estimates of this model are found using the maximum likelihood method (MLM). Monte Carlo simulation is applied to the obtained parameters. This model is named MoCArima for short. Second, the MoCArima model whose parameters are estimated with particle swarm optimization (PSO) is determined as Proposed Monte Carlo (MC) Arima1,1,1, and is briefly named as ProMoCArima. The third model is determined as Fixed Variance Stochastic Model which has a bouncing autoregressive AR1 structure with constant variance and trend component. This model is briefly named as FiVaS. The parameter estimates are found using the MLM. The fourth model is determined as the Proposed Fixed Variance Stochastic Model. This model is briefly named as ProFiVaS. Structurally the same as the third model. The parameter estimates of both trend and stochastic parts of the FiVaS model are found by the PSO.Proposed models are applied to forecasting international different financial indicators to show its superiority and applicability. The performance and usage efficiency of the proposed four stochastic models are examined on a total of six different international financial data types, five of which are stock index data BIST100 (Türkiye), S&P500 (USA), DAX (Germany), FTSE100 (UK), NIKKEI225 (Japan), and one of which is the exchange rate EURO/USD. The ProFiVaS among all models provides the best performance on the all indices. The ProFiVaS can easily be extended to any real-world time series prediction requirements.The proposed models can be extended to all stationary and non-stationary linear or nonlinear autoregression models opening the door to modeling data types obtained from different sector fields. A part of the future plan of this work is to design a decision support system software that assigns a value of 1 when an investor needs to invest in a financial indicator, and 0 otherwise.

Observer-based SMC for discrete semi-Markov switching models

This study focuses on the observer-based sliding mode control (SMC) for discrete stochastic switching models, where the semi-Markov kernel information is incomplete. Because the sensors do not always have direct access to the state vector in a complex environment, an observer-based output measurement is considered to estimate the system state. In contrast to previous works, the observer-based strategy is first investigated in studying the SMC for semi-Markov switching systems under the discrete framework. The sufficient condition for achieving mean-square stability is proposed to solve the desired observer and controller gains by utilizing the upper bound of sojourn time, along with the Lyapunov function that pertains to the system mode and the elapsed time. A mode-dependent SMC law is also constructed to ensure that the system state can reach the specified sliding region. Finally, the efficiency of the proposed method is demonstrated through the single-link robot arm model.

Proactive decision making by incorporation of discrete random sums

Adoption of proactivity provides theorists and practitioners with very significant information for undertaking research work in a wide range of scientific fields. In particular, the presence of proactivity substantially supports the implementation of activities under conditions of uncertainty. It constitutes a general recognition that proactive thinking facilitates strategic acting on the provision of operations for the protection of systems evolving in uncertain environments. Clearly, proactive decision making constitutes the most valuable factor for improving the resilience of systems threatened by a variety of risks. The present paper is primarily concerned with the formulation, investigation, and interpretation in proactive decision processes of two discrete stochastic models. The definitions and the representations as discrete random variables of many structural concepts are significant reasons for implementing discrete stochastic models in proactive treatment of future situations.

Experimental methods and mathematical modeling quantify and predict DUSP1 RNA expression dynamics in response to dexamethasone.

Overactivation of mitogen-activated protein kinase (MAPK) signaling pathways is key to multiple inflammatory responses, and synthetic glucocorticoids (GC), such as Dexamethasone (Dex), have long been used to treat inflammatory pathologies. Upon cell entry, Dex binds to the glucocorticoid receptor (GR), which initiates translocation to the nucleus. Nuclear GR then interacts with Glucocorticoid Regulatory Elements (GRE's) to promote transcription of anti-inflammatory genes including dual-specificity phosphatase 1 (DUSP1). In turn, DUSP1 encodes for the protein mitogen-activated protein kinase phosphatase 1 (MKP-1), which regulates the cell's anti-inflammatory response through dephosphorylation of the P38 and JUN N-terminal kinase (JNK) MAPK pathways. In this presentation, we combine single-cell measurements and discrete stochastic models to elucidate the spatiotemporal mechanisms by which Dex modulates GR localization and DUSP1 transcription. Using single-molecule inexpensive fluorescent in situ hybridization, we quantify the nascent transcription, spatial localization, and cellular heterogeneity of DUSP1 mRNA for hundreds of individual HeLa cells at 20 different time points following application of continuous 100 nM Dex stimulation. We observe a rapid increase in DUSP1 transcription sites (TS's) as early as 10-20 minutes, followed by a four-fold increase in expression of mature DUSP1 mRNA that peaks at 75-90 minutes and a temporary decrease in DUSP1 expression at 120-150 minutes post stimulation. We found that expression levels stay elevated during longer time course experiments, and high cellular heterogeneity is observed throughout the post-stimuli response. We used these experimental data to infer parameters and mechanisms for a discrete stochastic model that quantitatively reproduces the time-varying probability distributions for single-cell DUSP1 expression at all time points following Dex stimulation. Finally, we use this model inferred from DUSP1 expression to predict GR translocation dynamics and compare to direct measurement of GR localization dynamic using single-cell immunocytochemistry.

Analysis of methods for modeling human daily thermometry data

Abstract. Mathematical and computer modeling of daily thermometry allows to study processes of human thermal homeostasis more deeply. In practice, thermometry data is obtained using a digital thermometer, which autonomously reads the temperature of human skin in certain time intervals. The aim of present work is to analyse the methods of modeling and processing of human daily thermometry data. The first method consists in applying linear discrete stochastic models in the state space with Gaussian noises and known vector of input actions, while the estimation of the state vector is performed by discrete covariance Kalman filter. The second method assumes that the vector of input actions is unknown, and the S. Gillijns and B.D. Moor algorithm is used to process daily thermometry data. An alternative option is to use a model with an extended state vector and a Kalman filtering algorithm. The third method takes into account the presence of anomalous measurements (outliers) in the measurement data, and correntropy filter is proposed for their effective filtering. Numerical experiments for modeling and processing of daily thermometry data in MATLAB were carried out in order to compare the quality of discrete filtering algorithms. Modeling of thermometry data was carried out using a three-dimensional model 3dDRCM (3-dimension Discrete-time Real-valued Canonical Model). The results obtained can be used in the study of human daily thermometry processes, for example, to study the reaction of the athlete’s body to the received load.

Discrete stochastic models of SELEX: Aptamer capture probabilities and protocol optimization.

Antibodies are important biomolecules that are often designed to recognize target antigens. However, they are expensive to produce and their relatively large size prevents their transport across lipid membranes. An alternative to antibodies is aptamers, short (∼15-60 bp) oligonucleotides (and amino acid sequences) with specific secondary and tertiary structures that govern their affinity to specific target molecules. Aptamers are typically generated via solid phase oligonucleotide synthesis before selection and amplification through Systematic Evolution of Ligands by EXponential enrichment (SELEX), a process based on competitive binding that enriches the population of certain strands while removing unwanted sequences, yielding aptamers with high specificity and affinity to a target molecule. Mathematical analyses of SELEX have been formulated in the mass action limit, which assumes large system sizes and/or high aptamer and target molecule concentrations. In this paper, we develop a fully discrete stochastic model of SELEX. While converging to a mass-action model in the large system-size limit, our stochastic model allows us to study statistical quantities when the system size is small, such as the probability of losing the best-binding aptamer during each round of selection. Specifically, we find that optimal SELEX protocols in the stochastic model differ from those predicted by a deterministic model.

A unifying nonlinear probabilistic epidemic model in space and time

Covid-19 epidemic dramatically relaunched the importance of mathematical modelling in supporting governments decisions to slow down the disease propagation. On the other hand, it remains a challenging task for mathematical modelling. The interplay between different models could be a key element in the modelling strategies. Here we propose a continuous space-time non-linear probabilistic model from which we can derive many of the existing models both deterministic and stochastic as for example SI, SIR, SIR stochastic, continuous-time stochastic models, discrete stochastic models, Fisher–Kolmogorov model. A partial analogy with the statistical interpretation of quantum mechanics provides an interpretation of the model. Epidemic forecasting is one of its possible applications; in principle, the model can be used in order to locate those regions of space where the infection probability is going to increase. The connection between non-linear probabilistic and non-linear deterministic models is analyzed. In particular, it is shown that the Fisher–Kolmogorov equation is connected to linear probabilistic models. On the other hand, a generalized version of the Fisher–Kolmogorov equation is derived from the non-linear probabilistic model and is shown to be characterized by a non-homogeneous time-dependent diffusion coefficient (anomalous diffusion) which encodes information about the non-linearity of the probabilistic model.

Accelerated regression-based summary statistics for discrete stochastic systems via approximate simulators

BackgroundApproximate Bayesian Computation (ABC) has become a key tool for calibrating the parameters of discrete stochastic biochemical models. For higher dimensional models and data, its performance is strongly dependent on having a representative set of summary statistics. While regression-based methods have been demonstrated to allow for the automatic construction of effective summary statistics, their reliance on first simulating a large training set creates a significant overhead when applying these methods to discrete stochastic models for which simulation is relatively expensive. In this τ work, we present a method to reduce this computational burden by leveraging approximate simulators of these systems, such as ordinary differential equations and τ-Leaping approximations.ResultsWe have developed an algorithm to accelerate the construction of regression-based summary statistics for Approximate Bayesian Computation by selectively using the faster approximate algorithms for simulations. By posing the problem as one of ratio estimation, we use state-of-the-art methods in machine learning to show that, in many cases, our algorithm can significantly reduce the number of simulations from the full resolution model at a minimal cost to accuracy and little additional tuning from the user. We demonstrate the usefulness and robustness of our method with four different experiments.ConclusionsWe provide a novel algorithm for accelerating the construction of summary statistics for stochastic biochemical systems. Compared to the standard practice of exclusively training from exact simulator samples, our method is able to dramatically reduce the number of required calls to the stochastic simulator at a minimal loss in accuracy. This can immediately be implemented to increase the overall speed of the ABC workflow for estimating parameters in complex systems.

A note on calculating expected shortfall for discrete time stochastic volatility models

In this paper we consider the problem of estimating expected shortfall (ES) for discrete time stochastic volatility (SV) models. Specifically, we develop Monte Carlo methods to evaluate ES for a variety of commonly used SV models. This includes both models where the innovations are independent of the volatility and where there is dependence. This dependence aims to capture the well-known leverage effect. The performance of our Monte Carlo methods is analyzed through simulations and empirical analyses of four major US indices.

Numerical solution of fluid queueing models for communication networks

Discrete stochastic models of queues are commonly employed in communication networks for their analysis and control. Fluid models, largely stochastic, are being used recently due to their simple character. However, asymptotic analysis of queueing models leads to deterministic fluid models and efficient methods for computing their solution are needed. This paper develops procedures for numerical solution of fluid queueing models whose arrival processes with rates that are piecewise constant. Solutions for fluid queue with single input, queue with two classes under priority and a model with two fluid queues in tandem are developed and implemented. The piecewise linear nature of the dynamics are employed to develop these efficient solution procedures by determining the solution at time instances of rate change of the input. This aviods using numerical solution of differential equations and also greatly reduces the computational complexity. Computer programs are developed implementing the numerical solutions. Motivated by their application to communication network sources, ON/OFF fluid inputs are considered for analysis. The developed numerical solution procedures are applied to the ON/OFF sources as special cases of the piecewise constant rate inputs.

Paths of adaptation to climate change in major Italian agricultural areas: Effectiveness and limits in supporting the profitability of farms

Climate change requires Mediterranean farms to maintain adequate profitability, while adapting to the increase in water needs of crops, the growth of water demand for non-agricultural users and the reduction of resource availability. We use discrete stochastic programming models of three Italian farming areas to simulate the adaptation paths of their farm types to changes in yields and irrigation needs of crops that climate change could generate at 2030. This is done in a context of progressive decrease of water availability. We condense the adaptation choices attributed by the optimization process to the farm types by means of indicators of value produced per water cubic meter and per hectare. We use those indicators in a quadratic regression analysis to explain the change in the hourly wages of family labour, which represents the farm resources’ profitability. The examination of the resulting elasticity indices outlines the adaptation paths impact on farm profitability. This integrated analysis of optimization models and econometric regression shows that the core of Italian cow milk production, based on medium-high intensity forage systems, adapts better. Instead, available technologies and structural features provide much narrower margins to livestock farms based on intensive forage production and, conversely, to more extensive livestock systems. Crop farms adapt by reducing labour use in less profitable activities, which boosts family hourly wages but increases unemployment. Furthermore, the surface of some crops can be expanded because their cycle takes place in months in which the availability of water decreases less and their irrigation requirement increases less; however, the low productivity of these crops does not allow to increase wages for family labour. The results on farms overall profitability derive from an integrated analysis of changes in water resources productivity, soil and labor. Mathematical programming outlines the adaptation paths that maximize farming incomes; the econometric analysis indicates whether those paths support profitability, favouring farm economic sustainability.

Quantifying epigenetic stability with minimum action paths.

Chromatin can adopt multiple stable, heritable states with distinct histone modifications and varying levels of gene expression. Insight on the stability and maintenance of such epigenetic states can be gained by mathematical modeling of stochastic reaction networks for histone modifications. Analytical results for the kinetic networks are particularly valuable. Compared to computationally demanding numerical simulations, they often are more convenient at evaluating the robustness of conclusions with respect to model parameters. In this communication, we developed a second-quantization-based approach that can be used to analyze discrete stochastic models with a fixed, finite number of particles using a representation of the SU(2) algebra. We applied the approach to a kinetic model of chromatin states that captures the feedback between nucleosomes and the enzymes conferring histone modifications. Using a path-integral expression for the transition probability, we computed the epigenetic landscape that helps to identify the emergence of bistability and the most probable path connecting the two steady states. We anticipate the generalizability of the approach will make it useful for studying more complicated models that couple epigenetic modifications with transcription factors and chromatin structure.

THE FORECASTING OF THE TECHNICAL STATE OF THE TRACTION GEARBOXES OF ELECTRIC TRAINS WITH DISCRETE STOCHASTIC MODELS

The article deals with the forecasting of the technical state of the traction gearboxes of electric trains with discrete stochastic models. During every 3rd level maintenance, the vibration signals of the previously selected fourteen traction gearboxes were recorded, and the box counting dimension was estimated. For the forecasting of the technical state, among the broadcast deterministic methods of exponential smoothing and trend analysis, the most effective ARIMA model was selected. This model in certain applications has higher precision than GRNN and BPNN neural networks. The implementation of the ARIMA model requires accomplishment of the identification, evaluation, fitting and practical application stages for the forecasting model. The identification was performed due to the one-step differentiation with a further stationarity check according to the autocorrelation function and partial autocorrelation function. The evaluation of the ARIMA model with the different orders of the autoregression component and moving average component was done. For the model fitting, the Akaike’s information criterion and Bayes information criterion were calculated, and the autoregression component of the ARIMA model having the minimum values of these criteria was selected. The forecasting of the fitted ARIMA models with an 80 % confidence interval was done for the period since the 71st 3rd level maintenance up to the next 2nd level current repair. After disassembling 14 traction gearboxes during the 2nd level current repair, the insufficient amount of a lubricant in the first gearbox, wear of a roller bearing of the front cover in the second gearbox, a crack of a bearing ring of the front cover in the third gearbox, a tooth break of a gear in the fourth traction gearbox, destruction of a bearing in the fifth gearbox and a tooth crack of its gear were detected. It was established that the ARIMA model is difficult to upgrade for the evaluation of other data, which requires the implementation of measures for a resimulation. Despite high complexity, the absence of the automatic process for the calculation and the necessity of performing several iterative procedures, the calculated minimum value precision of the ARIMA model forecasting is equal to 91.4 %

The Impact of Rate Formulations on Stochastic Molecular Motor Dynamics

Cells are complex structures which require considerable amounts of organization via transport of large intracellular cargo. While passive diffusion is often sufficiently fast for the transport of smaller cargo, active transport is necessary to organize large structures on the short timescales necessary for biological function. The main mechanism of this transport is by cargo attachment to motors which walk in a directed fashion along intracellular filaments. There are a number of models which seek to describe the motion of motors with attached cargo, from detailed microscopic to coarse phenomenological descriptions. We focus on the intermediate-detailed discrete stochastic hopping models, and explore how cargo transport changes depending on the number of motors, motor interaction, system constraints and rate formulations, which are derived from common thermodynamic assumptions. We find that, despite obeying the same detailed balance constraint, the choice of rate formulation considerably affects the characteristics of the overall motion of the system, with one rate formulation exhibiting novel behavior of loaded motor groups moving faster than a single unloaded motor.

On random walk models for simulation of particle-laden turbulent flows

In this investigation, the accuracy of the discrete and continuous random walk (DRW, CRW) stochastic models for simulation of fluid (material) point particle, as well as inertial and Brownian particles, was studied. The corresponding dispersion, concentration, and deposition of suspended micro- and nano-particles in turbulent flows were analyzed. First, the DRW model used in the ANSYS-Fluent commercial CFD code for generating instantaneous flow fluctuations in inhomogeneous turbulent flows was evaluated. For this purpose, turbulent flows in a channel were simulated using a Reynolds-averaged Navier–Stokes (RANS) approach in conjunction with the Reynolds Stress Transport turbulence model (RSTM). Then spherical particles with diameters in the range of 30 µm to 10 nm were introduced uniformly in the channel. Under the assumption of one-way coupling, ensembles of particle trajectories for different sizes were generated by solving the particle equation of motion, including the drag and Brownian forces. The DRW stochastic turbulence model of the software was used to include the effects of instantaneous velocity fluctuations on particle motion, and the steady state concentration distribution and deposition velocity of particles of various sizes were evaluated. In addition, the improved CRW model based on the normalized Langevin equation was used in an in-house Matlab code. Comparisons of the predicted results of the DRW model of ANSYS-Fluent with the available experimental data and the DNS simulation results and empirical predictions showed that this model is not able to accurately predict the flow fluctuations seen by the particles in that it leads to unreasonable concentration profiles and time-varying deposition velocities. However, the predictions of the improved CRW model were in good agreement with the experimental data and the DNS results. Possible reasons causing the discrepancies between the DRW predictions and the experimental data were discussed. The improved CRW model was also implemented through user-defined functions into the ANSYS-Fluent code, which resulted in accurate concentration distribution and deposition velocity for different size particles.

Bayesian Estimation for Stochastic Gene Expression Using Multifidelity Models

The finite state projection (FSP) approach to solving the chemical master equation has enabled successful inference of discrete stochastic models to predict single-cell gene regulation dynamics. Unfortunately, the FSP approach is highly computationally intensive for all but the simplest models, an issue that is highly problematic when parameter inference and uncertainty quantification takes enormous numbers of parameter evaluations. To address this issue, we propose two new computational methods for the Bayesian inference of stochastic gene expression parameters given single-cell experiments. We formulate and verify an adaptive delayed acceptance Metropolis-Hastings (ADAMH) algorithm to utilize with reduced Krylov-basis projections of the FSP. We then introduce an extension of the ADAMH into a hybrid scheme that consists of an initial phase to construct a reduced model and a faster second phase to sample from the approximate posterior distribution determined by the constructed model. We test and compare both algorithms to an adaptive Metropolis algorithm with full FSP-based likelihood evaluations on three example models and simulated data to show that the new ADAMH variants achieve substantial speedup in comparison to the full FSP approach. By reducing the computational costs of parameter estimation, we expect the ADAMH approach to enable efficient data-driven estimation for more complex gene regulation models.

Adaptive estimation of an object motion parameters based on the hybrid stochastic model

This paper examines the problem of estimating parameters of the motion of an object moving along a complex trajectory in the presence of incomplete noisy measurements. This problem is of current interest because of the importance of its practical applications. Examples of such applications are the tracking problems of moving technical devices in robotics, navigation, ground vehicles. We propose a solution to the problem of adaptive estimation of the motion parameters of an object based on the hybrid stochastic model, which is a set of discrete linear stochastic models corresponding to various pieces of the object’s trajectory. The linearity of the models allows us to apply the optimal Kalman filtering to calculate the estimates of the motion parameters of an object. To identify the current motion mode of an object at a known time moments, new algorithm based on the use of the Kalman filters bank and sequential ratio probability test is developed. Numerical experiments confirm the efficiency of the obtained results.

Modeling acoustic attenuation of discrete stochastic fractured media

The acoustic response has many important roles in seismic exploration and nondestructive testing. It enables the development of fracture classification and sizing. In this paper, we combined Hudson’s effective medium scheme and finite-difference time-domain modeling method to simulate acoustic wave propagation in fractured media. Fractures are represented by discrete fracture networks, allowing for a state-of-the-art representation of natural fracture networks by a negative Exponential Law length distribution. The propagation of acoustic waves that are emitted by a point source and reflected from a fractured area in a 2D digital rock model are examined numerically with the purpose of developing an acoustic inference of fracture properties. In these fractured models, we vary the number and mean length of fractures to explore the relation between internal structure of rock and acoustic wave field characters. The modeling results indicate that acoustic wave field is more sensitive to the fracture number than to the mean of the fracture length. Moreover, a fracture-dependent attenuation analysis of the reflection records of discrete stochastic fractured models is obtained. The frequency- and time- dependent attenuation profiles feature two parts in frequency, (1) fracture-to-background at lower frequencies and (2) fracture-to-fracture at higher frequencies. Our results indicate that accounting for attenuation effects may not only allow for improving estimation of fracture number, but also provide information about geometrical characteristics of length distribution. Such an approach can be used to estimate nature fracture network properties with given acoustic records.

Organisation-Oriented Coarse Graining and Refinement of Stochastic Reaction Networks.

Chemical organisation theory is a framework developed to simplify the analysis of long-term behaviour of chemical systems. In this work, we build on these ideas to develop novel techniques for formal quantitative analysis of chemical reaction networks, using discrete stochastic models represented as continuous-time Markov chains. We propose methods to identify organisations, and to study quantitative properties regarding movements between these organisations. We then construct and formalise a coarse-grained Markov chain model of hierarchic organisations for a given reaction network, which can be used to approximate the behaviour of the original reaction network. As an application of the coarse-grained model, we predict the behaviour of the reaction network systems over time via the master equation. Experiments show that our predictions can mimic the main pattern of the concrete behaviour in the long run, but the precision varies for different models and reaction rule rates. Finally, we propose an algorithm to selectively refine the coarse-grained models and show experiments demonstrating that the precision of the prediction has been improved.