Nonlinear Stochastic Model Research Articles

Integrated model of stochastic dynamics for control of a socio-ecological-oriented innovation economy

On the basis of modern mathematical models, methods and information and technologies, for example, an integrated stochastic nonlinear model of man-made processes and objects, suitable for the conditions of systemic crises, were investigated and developed. Considered in this article are some aspects of integration of a lot of domains and sectors of operation of modern complex systems which are functioning and developing in the present-day conditions of instability, crises and nonlinearity. In order to predict the development of the state of an innovation economy, the nonlinear integrated stochastic model of growthing dynamic in the phase space has been investigated and developed. In the article, problem of the optimization of the management of the activity of modern complex systems that develop and function regarding current conditions of the instability are considered. The prospect of further research based on the developed models is to conduct computer experiments and their practical usisng. The development and study of criterias, methods and models for optimal management of man-made objects and the creation of making decision systems based on the proposed integral models in the state space are also promising.

Development of non-linear stochastic fatigue crack growth models for structural grade steel

This paper describes the analysis and quantification of within specimen and between specimen fatigue crack growth variability from tests undertaken on BS 4360: 1990 Grade 50DD (now BS EN 10025: Part 2: 2004, Grade S355K2) steel specimens. A non-linear fatigue crack growth model is presented which draws upon within and between specimen variability, the application of a newly identified and unique intersection point (referred to as the Gurney Point) and the use of reliability updating. This model, which aggregates fatigue crack growth variability over defined increments of growth, provides crisp fatigue life predictions at a sufficiently early stage of fatigue life.

Investigation of Gear Dynamic Characteristics under Stochastic External Excitations

Engine drive gear systems are widely used in lots of mechanical equipment. In engine drive gear systems, the external excitations, including the driving speed and external load, are generally stochastic due to manufacturing error, energy loss, variation of operating environmental, and so on. The reported works usually considered them as deterministic. In this paper, a stochastic nonlinear gear dynamic model considering the stochastic external excitations is presented. Then, the effects of the stochastic driving speed and external load on gear dynamic characteristics are investigated by Monte Carlo (MC) simulation. By the simulation results, some insightful conclusions, which are important for the gear random dynamics, are obtained.

Estimation and identification in batch processes with particle filters

This paper deals with the problem of real-time estimation and identification in batch processes represented by stochastic nonlinear state-space models (SSMs). To address this, the two problems – estimation and identification – are formulated under a common Bayesian filtering framework. In contrast to a continuous process, a batch process is typically operated for a relatively shorter duration, which limits the number of measurements available in a campaign to effectively solve the filtering problem. To overcome the limitations of existing solutions, we propose a two-dimensional dual particle filtering algorithm to solve the nonlinear filtering problem. The proposed algorithm performs filtering both along the time and the batch directions. This allows to combine measurements from multiple short campaigns for improved estimation and identification results. The efficacy of the proposed method is demonstrated on two simulation examples.

Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market

The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model for price dynamics of financial markets. We study this model with relation to concrete stylised facts about financial markets. We investigate the behavior of the long tail distribution of the volatilities and verify the inverse power law behavior which is obeyed for some financial markets. Furthermore, we obtain the behavior of the long range memory and obtain that it follows to a distinct behavior of other stochastic models that are used as models for the finances. Furthermore, we have made an analysis by using Fokker–Planck equation independent on time with the aim of obtaining the cumulative probability distribution of volatilities , however, the probability density found does not exhibit the cubic inverse law.

A Novel Reverse Logistics Network Design Considering Multi-Level Investments for Facility Reconstruction with Environmental Considerations

Reverse logistics is convincingly one of the most efficient solutions to reduce environmental pollution and waste of resources by capturing and recovering the values of the used products. Many studies have been developed for decision-making at tactical, practical, and operational levels of the reverse supply chain. However, many enterprises face a challenge that is how to design the reverse logistics networks into their existing forward logistics networks to account for both economic and environmental sustainability. In this case, it is necessary to design a novel reverse logistics network by reconstructing the facilities based on the existing forward logistics network. Multi-level investments are considered for facility reconstruction because more investment and more advanced remanufacturing technologies need to be applied to reduce the carbon emissions and improve facility capacities. Besides, uncertain elements include the demand for new products and return quantity of used products, making this problem challenging. To handle those uncertain elements, a bi-objective stochastic integer nonlinear programming model is proposed to facilitate this novel reverse logistics network design problem with economic and environmental objectives, where tactical decisions of facility locations, investment level choices, item flows, and vehicle assignments are involved. To show the applicability and computational efficiency of the proposed model, several numerical experiments with sensitivity analysis are provided. Finally, the trade-off between the profit and carbon emissions is presented and the sensitive analysis of changing several key input parameters is also discussed.

Univariate Model-Based Deadband Alarm Design for Nonlinear Processes

Alarm design is an essential industrial problem with significant implications for safety and performance. Standard alarm design algorithms are based on the assumption that the data are uncorrelated and stationary. In this Article, we relax these assumptions and develop a novel approach to design alarms for processes modeled as a stochastic nonlinear time-series model. In particular, we develop an algorithm to design deadband alarms by minimizing the false and missed alarm rates. The resulting algorithm is illustrated through extensive simulations on a reactor system.

Balancing a one‐way corridor capacity and safety‐oriented reliability: A stochastic optimization approach for metro train timetabling

AbstractIn urban rail transit systems of large cities, the headway and following distance of successive trains have been compressed as much as possible to enhance the corridor capacity to satisfy extremely high passenger demand during peak hours. To prevent train collisions and ensure the safety of trains, a safe following distance of trains must be maintained. However, this requirement is subject to a series of complex factors, such as the uncertain train braking performance, train communication delay, and driver reaction time. In this paper, we propose a unified mathematical framework to analyze the safety‐oriented reliability of metro train timetables with different corridor capacities, that is, the train traffic density, and determine the most reliable train timetable for metro lines in an uncertain environment. By employing a space‐time network representation in the formulations, the reliability‐based train timetabling problem is formulated as a nonlinear stochastic programming model, in which we use 0‐1 variables to denote the time‐dependent velocity and position of all involved trains. Several reformulation techniques are developed to obtain an equivalent mixed integer programming model with quadratic constraints (MIQCP) that can be solved to optimality by some commercial solvers. To improve the computational efficiency of the MIQCP model, we develop a dual decomposition solution framework that decomposes the primal problem into several sets of subproblems by dualizing the coupling constraints across different samples. An exact dynamic programming combined with search space reduction strategies is also developed to solve the exact optimal solutions of these subproblems. Two sets of numerical experiments, which involve a relatively small‐scale case and a real‐world instance based on the operation data of the Beijing subway Changping Line are implemented to verify the effectiveness of the proposed approaches.

Output feedback stochastic nonlinear model predictive control for batch processes

Batch processes play a vital role in the chemical industry, but are difficult to control due to highly nonlinear behaviour and unsteady state operation. Nonlinear model predictive control (NMPC) is therefore one of the few promising approaches. Batch process models are however often affected by uncertainties, which can lower the performance and cause constraint violations. In this paper we propose a shrinking horizon NMPC algorithm accounting for these uncertainties to optimize a probabilistic objective subject to chance constraints. At each sampling time only noisy output measurements are observed. Polynomial chaos expansions (PCE) are used to express the probability distributions of the uncertainties, which are updated at each sampling time using a PCE state estimator and exploited in the NMPC formulation. The approach considers feedback by using time-invariant linear feedback gains, which alleviates the conservativeness of the approach. The NMPC scheme is verified on a polymerization semi-batch reactor case study.

A joint chance-constrained data envelopment analysis model with random output data

Data envelopment analysis (DEA) is a mathematical programming approach for evaluating the technical efficiency performances of a set of comparable decision-making units that transform multiple inputs into multiple outputs. The conventional DEA models are based on crisp input and output data, but real-world problems often involve random output data. The main purpose of the paper is to propose a joint chance-constrained DEA model for analyzing a real-world situation characterized by random outputs and crisp inputs. After developing the model, we carry out the following: First, we obtain an upper bound of this stochastic non-linear model deterministically by applying a piecewise linear approximation algorithm based on second-order cone programming; Second, we obtain a lower bound with use of a piecewise tangent approximation algorithm, which is also based on second-order cone programming; and then we use a numerical example to demonstrate the applicability of the proposed joint chance-constrained DEA framework.

Monte Carlo Evidence on the Estimation Method for Industry Dynamics

Abstract This study presents a structural estimation method for nonlinear stochastic dynamic models of heterogeneous firms. I perform a Monte Carlo experiment to evaluate the performance of the estimators for the AR(1) dynamic panel data subject to sample selection without exogenous regressors. The results suggest a strong need to correct the sample selection and that the proposed structural estimation method works well. These results are important for practical situations where the assumptions of the standard sample selection correction methods are not satisfied.

Linear prediction error methods for stochastic nonlinear models

The estimation problem for stochastic parametric nonlinear dynamical models is recognized to be challenging. The main difficulty is the intractability of the likelihood function and the optimal one-step ahead predictor. In this paper, we present relatively simple prediction error methods based on non-stationary predictors that are linear in the outputs. They can be seen as extensions of the linear prediction error methods for the case where the hypothesized model is stochastic and nonlinear. The resulting estimators are defined by analytically tractable objective functions in several common cases. It is shown that, under certain identifiability and standard regularity conditions, the estimators are consistent and asymptotically normal. We discuss the relationship between the suggested estimators and those based on second-order equivalent models as well as the maximum likelihood method. The paper is concluded with a numerical simulation example as well as a real-data benchmark problem.

Facility location and scale optimisation in closed-loop supply chain

This paper solves a strategic-level decision problem on determining the optimal location of (re)manufacturers and logistics centres with the consideration of facility scales in an integrated closed-loop supply chain network. A two-stage stochastic mixed-integer non-linear programming model is established to minimise the fixed cost and the expected operation costs under uncertain demand and return. We develop an improved tabu search heuristic algorithm to solve the model. We also design a distance-based decision rule to validate the effectiveness of the proposed model. Numerical experiments are conducted to test the performance of the proposed model and the solution method. In addition, sensitivity analysis is provided to investigate the influences of varying inspection locations and recovery rates on the final performance.

Rare event simulation for steady-state probabilities via recurrency cycles.

We develop a new algorithm for the estimation of rare event probabilities associated with the steady-state of a Markov stochastic process with continuous state space Rd and discrete time steps (i.e., a discrete-time Rd-valued Markov chain). The algorithm, which we coin Recurrent Multilevel Splitting (RMS), relies on the Markov chain's underlying recurrent structure, in combination with the Multilevel Splitting method. Extensive simulation experiments are performed, including experiments with a nonlinear stochastic model that has some characteristics of complex climate models. The numerical experiments show that RMS can boost the computational efficiency by several orders of magnitude compared to the Monte Carlo method.

Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations

This paper considers the nonlinear stochastic fractional integro-differential equations (SFIDEs) under the non-Lipschitz conditions, which are general and include many stochastic (fractional) integro-differential equations discussed in literature. An important connection between SFIDEs and stochastic Volterra integral equations (SVIEs) is derived in detail by the Fubini theorem. Using the Euler–Maruyama (EM) approximation, we prove the existence, uniqueness and stability results of the solution to SFIDEs. Moreover, it is shown that the modified EM solution of SFIDEs shares strong first-order sharp convergence. The numerical examples are performed to show the accuracy and effectiveness of the numerical scheme and verify the correctness of our theoretical analysis. • The model covers many nonlinear stochastic (fractional) integro-differential models. • Well-posedness of the models is investigated under the non-Lipschitz condition. • The modified EM approximation of the models admits strong first-order convergence.

Nonlinear stochastic modeling for optimal dispatch of distributed energy resources in active distribution grids including reactive power

This paper deals with energy storage system (ESS) in active distribution networks. The purpose is to install ESSs on the grid to minimize network losses. The problem is expressed as an optimization programming to minimize annualized cost of losses and annualized investment cost of ESSs at the same time. The constraints of the programming are given as security constraints of the network and ESS operational constraints. The network is also equipped with distributed energy resource (DER) and its uncertainty is modeled and dealt by means of stochastic programming. Different DERs including diesel, wind, and solar resources are modeled and studied. The proposed nonlinear mixed integer stochastic programming is solved by particle swarm optimization (PSO). AC power flow is adopted to consider both active and reactive powers in the model. The ESSs are modeled including both active and reactive powers. The introduced planning finds optimal location, capacity, and power for ESSs. Furthermore, the charging-discharging regime for active power of ESSs and injection-absorption pattern for reactive power of ESSs are determined. The introduced methodology is successfully simulated on a typical distribution network. The simulation results confirm that the planned strategy properly installs ESSs on the grid and minimizes network losses. The results demonstrate that the ESSs decrease network losses about 22%. Finally, considering reactive power for ESSs results in about 24% cost reduction.

Local Lagged Adapted Generalized Method of Moments: An Innovative Estimation and Forecasting Approach and its Applications

Abstract In this work, an attempt is made to apply the Local Lagged Adapted Generalized Method of Moments (LLGMM) to estimate state and parameters in stochastic differential dynamic models. The development of LLGMM is motivated by parameter and state estimation problems in continuous-time nonlinear and non-stationary stochastic dynamic model validation problems in biological, chemical, engineering, energy commodity markets, financial, medical, military, physical sciences and social sciences. The byproducts of this innovative approach (LLGMM) are the balance between model specification and model prescription of continuous-time dynamic process and the development of discrete-time interconnected dynamic model of local sample mean and variance statistic process (DTIDMLSMVSP). Moreover, LLGMM is a dynamic non-parametric method. The DTIDMLSMVSP is an alternative approach to the GARCH(1,1) model, and it provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. Furthermore, applications of LLGMM to energy commodities price, U.S. Treasury Bill interest rate and the U.S.–U.K. foreign exchange rate data strongly exhibit its unique role, scope and performance, in particular, in forecasting and confidence-interval problems in applied statistics.

A dynamic network DEA model for accounting and financial indicators: A case of efficiency in MENA banking

Middle East and North Africa (MENA) countries present a banking industry that is well-known for regulatory and cultural heterogeneity, besides ownership, origin, and type diversity. This paper explores these issues by developing a Dynamic Network DEA model in order to handle the underlying relationships among major accounting and financial indicators. Firstly, a relational model encompassing major profit sheet, balance sheet, and financial health indicators is presented under a dynamic network structure. Subsequently, the dynamic effect of carry-over indicators is incorporated into it so that efficiency scores can be properly computed for these three substructures. The impact of contextual variables related to bank ownership, its type, and whether or not it has undergone a previous merger and acquisition process is tested by means of a stochastic non-linear model solved by differential evolution, which combines bootstrapped Simplex, Tobit, Beta, and Simar and Wilson truncated regression results. The results reveal that bank type, origin, and ownership impact efficiency levels differently in terms of profit sheet, balance sheet, and financial health indicators, although the impact of culture and regulatory barriers seem to prevail at the country level.

Review of Solution and Estimation Methods for Nonlinear Dynamic Stochastic General Equilibrium Models with the Zero Lower Bound

We review solution and estimation methods for nonlinear dynamic stochastic general equilibrium models and their application, with a special focus on the zero lower bound on the nominal interest rate. In a fully nonlinear setting, both the solution and estimation methods involve iterative procedures, and their computational expense grows rapidly with an increase in the dimensionality of state variables and parameters. We describe how the procedures deal with the dimensionality problem.

Non-linear forecasting of the state of a socio-eco-oriented innovative economy in the conditions of systemic crises

The paper deals with the problem of sustainable development and innovative integral modeling and forecasting approach in the management of technogenic objects and processes (TOP) as a system of socio-eco-economic and humanitarian type (SEEH). Based on the use of information and innovation technologies in order to forecast the non-linear dynamics of eco-economic and socio-humanitarian systems, integrated stochastic models of objects and processes were developed and studied, suitable for the conditions of systemic crises. The paper handles the aspect of integration of 4 business and functioning areas of the modern complex systems. It proposes a general conceptual integrated model, generalized synergetic model of dynamics, considering different uncertainty (stochastic and chaotic components). The paper examines the aspects of integration of multiple business areas and sectors of the modern complex systems functioning and developing under the present conditions of non-linearity, instability and crises. An integrated stochastic non-linear phase-space growth dynamics model was developed and studied to forecast the development of the state of an innovative economy. The paper looks into the aspects of activity management of the modern complex systems functioning and developing under the present conditions of instability.