Nonlinear Stochastic Model Research Articles

Two-Stage emergency material scheduling based on benders decomposition considering traffic congestion after a disaster

This study establishes a stochastic mixed integer nonlinear programming model for a two-stage emergency multiclass material scheduling problem considering traffic congestion with a Bureau of Public Roads (BPR) function. The generalized Benders decomposition (GBD) algorithm is used to solve the model on the basis of the characteristics of the decomposable structure and nonlinear convex function. To prove the effectiveness of the GBD algorithm, we set up a 30-node medium-scale case and a 50-node large-scale case for experimental demonstration and compared the GBD algorithm with the simple branch and bound (SBB) and discrete and continuous optimization solvers. Results show that for the medium-scale experiment, the gap between the solution results of the GBD algorithm and the ones of SBB or other solvers is less than 0.2%. The solution time of the GBD algorithm is only half of the time of SBB and other solvers. The large-scale case of 50 nodes cannot be solved by the SBB or other solvers, whereas the GBD algorithm can solve it normally, and the results are reasonable. In large-scale cases with more than 50 nodes, the GBD algorithm is more effective and efficient compared with SBB and other solvers. The sensitivity of the correlation coefficient α in the BPR function was analyzed. The actual expected total cost increases with the increase in the value of α. Its solving results are accurate and reasonable, proving that the results are more practical and applicable after adding the BPR function.

Dynamical behavior of a stochastic SICR rumor model incorporating media coverage

Rumor propagation in the new media era poses a huge threat to maintaining the normal order of social life. In this context, we put forth a nonlinear dynamics-based stochastic SICR rumor model that integrates media coverage with science education. First, the existence of a unique global positive solution is obtained. Second, sufficient conditions for extinction are constructed on the spread of rumors based on the Lyapunov function methods and Khasminskii’s theory. Finally, the theoretical analysis is verified through numerical simulations. Additionally, it demonstrates how rumor spreading can be hampered by media coverage.

Asymptotic Properties and Application of GSB Process: A Case Study of the COVID-19 Dynamics in Serbia

This paper describes one of the non-linear (and non-stationary) stochastic models, the GSB (Gaussian, or Generalized, Split-BREAK) process, which is used in the analysis of time series with pronounced and accentuated fluctuations. In the beginning, the stochastic structure of the GSB process and its important distributional and asymptotic properties are given. To that end, a method based on characteristic functions (CFs) was used. Various procedures for the estimation of model parameters, asymptotic properties, and numerical simulations of the obtained estimators are also investigated. Finally, as an illustration of the practical application of the GSB process, an analysis is presented of the dynamics and stochastic distribution of the infected and immunized population in relation to the disease COVID-19 in the territory of the Republic of Serbia.

Towards probabilistic seismic performance of vehicle‐bridge interaction systems: From stochastic dynamic model to fragility analysis

AbstractThis paper aims to assess the seismic fragility of vehicle‐bridge‐interaction (VBI) systems considering the effects of vehicle types, traffic conditions, and road surface qualities. A stochastic nonlinear mechanical model for the earthquake‐VBI system is developed, and the fragility functions for the proposed VBI model are derived by considering the relevant probabilistic seismic demand parameters. On the basis of a typical four‐span continuous prestressed concrete highway bridge in China, a complete numerical model for the VBI system is built considering multiple uncertainties from bridge and vehicle parameters, as well as the road surface qualities. A total of 120 real ground motion records with different combinations of magnitude‐source‐to‐site distance (M‐R) and earthquake intensity characteristics are selected. Meanwhile, 80 scenarios in terms of different combinations of vehicle types, vehicle speeds, and road surface irregularities are defined. In this context, 96,000 nonlinear time‐history analyses are performed, and the developed fragility models are applied to the VBI system at both component and system levels. Results indicate that the fragilities of pier drift, bearing shear strain, and the overall VBI system increase with the increase of the vehicle weight or the decrease of the vehicle speed, while the vertical deck displacement is dominated by the vehicle weight. It is also found that the road surface quality has a negligible effect on both component and system fragilities.

Applications of nonlinear recursive identification theory in sentencing data analyses

Judicial data have become an important base for empirical legal studies and the construction of smart court systems. Thus, the interpretability and reliability of the computation results become increasingly important. Therefore, we established a new theory of nonlinear recursive identification and applied it to sentencing data analyses. Specifically, based on the criminal law and sentencing guidelines, we established a nonlinear stochastic model for sentencing, to which the corresponding nonlinear recursive identification theory is applied with a specified finite length of data, which is nearly 200000 judgment documents of the crime of intentional injury. Compared with the traditional linear regression model and the least squares algorithm, the computation results given by the nonlinear recursive identification theory are more consistent with the basic principles and rules of sentencing, can reflect the effects and changes of sentencing factors more accurately, and exhibit better prediction ability.

Dynamical analysis for the sustained harvesting of microorganisms using flocculants in a random environment

The use of flocculants to collect/extract microorganisms is of great practical significance for the development of the application of microorganisms. In this paper, a high-dimensional nonlinear stochastic differential equation model is constructed to describe the continuous culture of microorganisms with multiple nutrients and the flocculation process of microorganisms. The study of the dynamics of this model can provide feasible control strategies for the collection/extraction of microorganisms. The main theoretical results are sufficient conditions for the permanence and extinction of the stochastic differential equation model, which are also extensions of some results in the existing literatures. In addition, through numerical simulations, we vividly demonstrate the statistical characteristics of the stochastic differential equation model.

A new nonlinear stochastic volatility model with regime switching stochastic mean reversion and its applications to option pricing

We establish a new nonlinear stochastic volatility model by modeling the volatility long-run mean with a particular stochastic process, whose parameters can jump between different regimes according to a Markov chain. Introducing the regime switching factor is consistent with the empirical evidence of the economic cycle and nonlinear volatility mean-reversion observed from market data. A closed-form solution that can be used to value European options has been successfully obtained based on the characteristic function approach, followed by some numerical examples comparing the models with and without regime switching.

Generalized Covariance Estimator

ABSTRACT We consider a class of semi-parametric dynamic models with iid errors, including the nonlinear mixed causal-noncausal Vector Autoregressive (VAR), Double-Autoregressive (DAR) and stochastic volatility models. To estimate the parameters characterizing the (nonlinear) serial dependence, we introduce a generic Generalized Covariance (GCov) estimator, which minimizes a residual-based multivariate portmanteau statistic. In comparison to the standard methods of moments, the GCov estimator has an interpretable objective function, circumvents the inversion of high-dimensional matrices, and achieves semi-parametric efficiency in one step. We derive the asymptotic properties of the GCov estimator and show its semi-parametric efficiency. We also prove that the associated residual-based portmanteau statistic is asymptotically chi-square distributed. The finite sample performance of the GCov estimator is illustrated in a simulation study. The estimator is then applied to a dynamic model of commodity futures.

Nonlinear Autoregression With Convergent Dynamics on Novel Computational Platforms

Nonlinear stochastic modeling is useful for describing complex engineering systems. Meanwhile, neuromorphic (brain-inspired) computing paradigms are being developed to tackle tasks that are challenging and resource-intensive on digital computers. An emerging scheme is reservoir computing, which exploits nonlinear dynamical systems for temporal information processing. This brief introduces reservoir computers with output feedback as stationary and ergodic infinite-order nonlinear autoregressive models. We highlight the versatility of this approach by employing classical and quantum reservoir computers to model synthetic and real datasets, further exploring their potential for control applications.

Stochastic nonlinear model of the dynamics of actively Q-switched lasers.

In this paper, we present a novel stochastic and spatially lumped multi-mode model to describe the nonlinear dynamics of actively Q-switched lasers and random perturbations due to amplified spontaneous emission. This model will serve as a basis for the design of (nonlinear) control and estimation strategies and thus a high value is set on its computational efficiency. Therefore, a common traveling-wave model is chosen as a starting point and a number of model-order reduction steps are performed. As a result, a set of nonlinear ordinary differential equations for the dynamic behavior of the laser during a switching cycle is obtained. A semi-analytic solution of these differential equations yields expressions for the population inversion after a switching cycle and for the output energy, which are then used to formulate a nonlinear discrete-time model for the pulse-to-pulse dynamics. Simulation studies including models with different levels of complexity and first experimental results demonstrate the feasibility of the proposed approach.

Constrained tracking control of stochastic multivariable nonlinear systems via Gaussian process predictions

To overcome the difficulty of propagating the stochastic uncertainties through a nonlinear model for successful online implementation of the stochastic nonlinear model predictive control (SNMPC) framework, this paper proposes the utilisation of Gaussian processes (GPs) in the context of SNMPC for the stochastic multivariable nonlinear systems to track a given trajectory in the presence of stochastic uncertainties and system constraints. Taking advantage of the GP regression which not only provides predictions, but also uncertainty quantification in the function estimation, the proposed GP-SNMPC architecture exploits the probability distribution of stochastic uncertainties and utilises the predictions provided by the GPs to formulate the model cost and constraint functions, which yields a tractable framework for handling nonlinear constrained control problems with Gaussian parametric uncertainties. By employing a cancellation strategy, the control law consists of two components, that is, tracking control law and disturbance rejection control law. Theoretical results regarding the performance bounds of the closed-loop system are derived. Numerical examples are provided to verify the effectiveness of the proposed GP-SNMPC scheme.

Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics.

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the "curse of dimensionality" related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). We assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001-29 October 2020.

A Nonlinear Stochastic Matrix Model for the Analysis of the Association between International Trade Growth and Environmental Change

This study uses a nonlinear stochastic matrix for an in-depth analysis of the association between international trade growth and environmental change. First, for a class of uncertain stochastic nonlinear time-lag systems, this study investigates its time-lag-related robust stochastic stabilization problem. A memory-free state feedback controller for the system is then obtained to ensure the effectiveness of the system performance. This makes the system more general and the corresponding theoretical results less conservative. Finally, numerical examples are given to further verify the validity and feasibility of the results. The “tariff measure” based on the magnitude of adverse changes in trade policy is not an accurate measure of trade policy uncertainty; the stochastic volatility method measures trade policy uncertainty by including tariff fluctuations that have been communicated in advance in the trade policy uncertainty indicator and fail to identify the different commodities that occur during the current trade friction. The trade policy uncertainty index constructed through textual analysis of newspapers can better reflect the trade policy uncertainty during this trade friction between China and the United States, and the rich time-varying nature can also reflect the alternating changes of tension and détente in economic and trade relations over some time. The analysis of the mechanism reveals that exporters have fewer emission reduction facilities than nonexporters but significantly lower energy consumption intensity for coal and water than nonexporters, reasoning that the mechanism by which exporters are more environmentally friendly than nonexporters lies in their cleaner energy use mix. Following the conclusions of this study, relevant policy recommendations are put forward, specifically the use of more efficient energy sources in the production process and more investment in energy efficiency and emission reduction to combat pollution.

EXTINCTION AND PERSISTENCE OF A HARVESTED PREY–PREDATOR MODEL INCORPORATING GROUP DEFENCE AND DISEASE IN PREY: SPECIAL EMPHASIS ON STOCHASTIC ENVIRONMENT

In this paper, a nonlinear stochastic eco-epidemic model is presented to study the dynamics of harvested prey–predator model with group defence. One of the key roles of forming a group is to defend themselves from the attack of the predators. Since prey population spreads infection through direct contact; we consider two groups of prey population. Using some certain assumptions, a stochastic model is formulated by considering noisy intrinsic growth rate of susceptible pest population, death rate of infected pest and predator populations, respectively. Next, the existence and uniqueness of positive solution are performed using comparison theorem of stochastic differential equations (SDEs). The boundedness and uniform continuity of the system are analyzed. Then, the bionomic equilibrium and optimal harvesting policy are obtained by using stochastic maximum principle. We verify the conditions of extinction and persistence in mean with the numerical results. To verify our consideration with the reality, the effects of harvesting effort and catchability coefficient are observed. In addition, infected and uninfected regions are reported to specify the infection free system.

Constrained tracking control of stochastic multivariable nonlinear systems using unscented Kalman filter

To overcome the challenges, including propagation of the stochastic uncertainties through a nonlinear model and full determination of the states over the prediction horizon for successful online implementation of the stochastic nonlinear model predictive control (SNMPC) framework, this paper proposes the utilisation of unscented Kalman filter (UKF) in the context of SNMPC for the stochastic multivariable nonlinear systems to track a given trajectory in the presence of stochastic uncertainties and system constraints. Taking advantage of the UKF which estimates and propagates the entire conditional distribution of the states along the time horizon by taking into account errors due to external disturbances and measurement noises as well as errors introduced due to the imperfect knowledge of the states through the state estimation, the proposed UKF-SNMPC framework utilises the conditional mean and conditional covariance of the states to reformulate the model cost and constraint functions, which yields a tractable framework for handling nonlinear chance-constrained tracking control problems. By employing a cancellation strategy, the control law consists of two components, that is, a tracking control law is designed to follow the desired trajectory and an uncertainty compensation control law designed to accommodate the nonlinearities in the plant. The concept of ‘robust horizon’ is introduced as the terminal constraint to prevent the covariance from the explosion and guarantee the convergence of the UKF-SNMPC framework. Performance bounds of the closed-loop system are theoretically analysed. Comparative studies on numerical examples are carried out to verify the effectiveness of the proposed UKF-SNMPC scheme.

Adaptive trajectory estimation with power limited steering model under perturbation compensation

Trajectory estimation of maneuvering objects is applied in numerous tasks like navigation, path planning and visual tracking. Many previous works get impressive results in the strictly controlled condition with accurate prior statistics and dedicated dynamic model for certain object. But in challenging conditions without dedicated dynamic model and precise prior statistics, the performance of these methods significantly declines. To solve the problem, a stochastic nonlinear model called the power-limited steering model is proposed to describe the motion of non-cooperative object. It is a natural combination of instantaneous power and instantaneous angular velocity, relying on the nonlinearity to achieve the change of states. And the renormalization group is introduced to compensate the nonlinear effect of perturbation in our model. For robust and efficient trajectory estimation, an adaptive trajectory estimation (AdaTE) algorithm is proposed. By updating the statistics and truncation time online, it corrects the estimation error caused by biased prior statistics and observation drift, while reducing the computational complexity lower than O(n). The experiment of trajectory estimation demonstrates the convergence of AdaTE, and the better robust to the biased prior statistics and the observation drift compared with several typical estimation algorithms. Other experiments demonstrate through slight modification, our method can also be applied to local navigation in random obstacle environment, and trajectory optimization in visual tracking.

Conditional Gaussian nonlinear system: A fast preconditioner and a cheap surrogate model for complex nonlinear systems.

Developing suitable approximate models for analyzing and simulating complex nonlinear systems is practically important. This paper aims at exploring the skill of a rich class of nonlinear stochastic models, known as the conditional Gaussian nonlinear system (CGNS), as both a cheap surrogate model and a fast preconditioner for facilitating many computationally challenging tasks. The CGNS preserves the underlying physics to a large extent and can reproduce intermittency, extreme events, and other non-Gaussian features in many complex systems arising from practical applications. Three interrelated topics are studied. First, the closed analytic formulas of solving the conditional statistics provide an efficient and accurate data assimilation scheme. It is shown that the data assimilation skill of a suitable CGNS approximate forecast model outweighs that by applying an ensemble method even to the perfect model with strong nonlinearity, where the latter suffers from filter divergence. Second, the CGNS allows the development of a fast algorithm for simultaneously estimating the parameters and the unobserved variables with uncertainty quantification in the presence of only partial observations. Utilizing an appropriate CGNS as a preconditioner significantly reduces the computational cost in accurately estimating the parameters in the original complex system. Finally, the CGNS advances rapid and statistically accurate algorithms for computing the probability density function and sampling the trajectories of the unobserved state variables. These fast algorithms facilitate the development of an efficient and accurate data-driven method for predicting the linear response of the original system with respect to parameter perturbations based on a suitable CGNS preconditioner.

A Stochastic Multi-Objective Model for China’s Provincial Generation-Mix Planning: Considering Variable Renewable and Transmission Capacity

The uncertain output of variable renewables adds significant challenges to the generation of affordable, reliable, and sustainable power sources in a country or region. Therefore, we propose a new stochastic nonlinear multi-objective model to optimize the power generation structure in 31 provinces of China. Considering variable renewable integration, we use Monte Carlo simulation to describe the randomness and uncertainty of renewable power output. The learning curve in the exponential expression is used to describe the nonlinear relationship between generation cost and installed capacity. The optimized results show that China can substitute fossil power with clean power. Renewable power will account for more than 42% of total power in the optimal power generation structure in 2040. In particular, the annual average growth rate of non-hydro renewable generation is expected to be 12.06%, with solar photovoltaic (PV) power growing the most by 17.95%. The share of renewable power exceeds that of thermal power in 14 provinces, and PV power represents the highest proportion at 30.21%. Reducing transmission capacity can promote the development of advantageous power in each region, such as wind power in the Northwest region and PV power in the South region, with the share increasing by 36.33% and 132.59%, respectively.

Nonpharmaceutical Stochastic Optimal Control Strategies to Mitigate the COVID-19 Spread

This article proposes a stochastic nonlinear model predictive controller to support policymakers in determining robust optimal nonpharmaceutical strategies to tackle the COVID-19 pandemic waves. First, a time-varying SIRCQTHE epidemiological model is defined to get predictions on the pandemic dynamics. A stochastic model predictive control problem is then formulated to select the necessary control actions (i.e., restrictions on the mobility for different socioeconomic categories) to minimize the socioeconomic costs. In particular, considering the uncertainty characterizing this decision-making process, we ensure that the capacity of the healthcare system is not violated in accordance with a chance constraint approach. The effectiveness of the presented method in properly supporting the definition of diversified nonpharmaceutical strategies for tackling the COVID-19 spread is tested on the network of Italian regions using real data. The proposed approach can be easily extended to cope with other countries’ characteristics and different levels of the spatial scale. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article is motivated by the emerging need for developing effective methods to support policymakers in mitigating the effects of the COVID-19 pandemic. The proposed feedback control strategy—combining a multiregion epidemiological model with a nonlinear stochastic model predictive control approach—allows the robust identification of the most effective restrictive measures considering the corresponding effects on the healthcare and socioeconomic systems. The proposed framework is a general and flexible method that can be applied to various real scenarios, leveraging mobility data, available from the Google mobility service, to recognize patterns and predict future behaviors of individuals.

Stochastic bi-objective optimization for closed wet cooling tower systems based on a simplified analytical model

A fast and accurate representation of the complex heat and mass transfer processes of closed-wet cooling tower (CWCT) systems for optimal design and management is challenging, especially considering the impact of weather variability. This study develops a systematic approach for the stochastic bi-objective optimization of CWCT systems based on a simplified analytical model. The analytical model is constructed by using the effectiveness-number of transfer unit method, which is embedded in a stochastic non-linear programming model for optimizing two competing objectives, namely the specific area and the coefficient of performance. A decomposition strategy is proposed to convert the original problem into multiple bi-objective optimization sub-problems by sequentially discretizing the design and uncertain spaces. Each bi-objective optimization sub-problem is solved using an augmented e-constraint method. The resulting solutions are finally processed by an efficient multi-criteria decision-making method, namely the Technique for Order Preference by Similarity to Ideal Solution to select the preferred Pareto solution. From an illustrative case study, it is found that the stochastic approach outperforms the deterministic approach since it allows for a certain design margin of heat exchanger area to reduce the impact of the variability in weather and to avoid the risks of abnormal running cases.