Nonlinear Stochastic Model Research Articles

Bayesian Objective Functions for Estimating Parameters in Nonlinear Stochastic Differential Equation Models with Limited Data

An Approximate Bayesian Expectation Maximization (ABEM) methodology and a Laplace Approximation Bayesian (LAB) methodology are developed for estimating parameters in nonlinear stochastic differential equation (SDE) models of chemical processes. These new methodologies are more powerful than previous maximum-likelihood methodologies for SDEs because they enable modelers to account for prior information about unknown parameters and initial conditions. The ABEM methodology is suitable for situations in which the modeler can assume that measurement noise variances are well-known, whereas LAB includes measurement noise variances among the parameters that require estimation. Both techniques estimate the magnitude of stochastic terms included in the differential equations to account for model mismatch and unknown process disturbances. The proposed ABEM and LAB methodologies are illustrated using a nonlinear continuous stirred tank reactor (CSTR) case study, with simulated data sets generated using a variety of s...

Impact of Poor Health of Maize Farmers on Farm Performance in Southwestern Cameroon

The study analyzed the impact of poor health on maize farmers’ performance in the Buea municipality. It made use of primary data collected with the help of a well-structured questionnaire, administered to 60 randomly sampled maize farmers in Buea municipality. The Ordinary Least Squares technique was used to analyze a non-linear stochastic model that captured the relationship between maize output, poor health and other important inputs. It was observed that a 1% increase in working hours of labour would increase output by 0.319% in the study area. Also, a 1% improvement in the health condition of the farmer will increase output by 0.291%. On average, 5,965.1 FCFA was spent by each farmer on health care and this led to a 29.2% loss in income. Given that the labour hours variable had the greatest magnitude, it shows the importance of health in the productivity of these workers, through the quantity and quality of labour. This implies the greater part of poor performance on the farm is a result of poor health, and so an improvement in the health condition will improve maize production significantly. Health should thus be given priority both by the farmer, where possible, and the government in any policy aimed at increasing maize production particularly in the study area.

Predicting Monsoon Intraseasonal Precipitation using a Low-Order Nonlinear Stochastic Model

Abstract The authors assess the predictability of large-scale monsoon intraseasonal oscillations (MISOs) as measured by precipitation. An advanced nonlinear data analysis technique, nonlinear Laplacian spectral analysis (NLSA), is applied to the daily precipitation data, resulting in two spatial modes associated with the MISO. The large-scale MISO patterns are predicted in two steps. First, a physics-constrained low-order nonlinear stochastic model is developed to predict the highly intermittent time series of these two MISO modes. The model involves two observed MISO variables and two hidden variables that characterize the strong intermittency and random oscillations in the MISO time series. It is shown that the precipitation MISO indices can be skillfully predicted from 20 to 50 days in advance. Second, an effective and practical spatiotemporal reconstruction algorithm is designed, which overcomes the fundamental difficulty in most data decomposition techniques with lagged embedding that requires extra information in the future beyond the predicted range of the time series. The predicted spatiotemporal patterns often have comparable skill to the MISO indices. One of the main advantages of the proposed model is that a short (3 year) training period is sufficient to describe the essential characteristics of the MISO and retain skillful predictions. In addition, both model statistics and prediction skill indicate that outgoing longwave radiation is an accurate proxy for precipitation in describing the MISO. Notably, the length of the lagged embedding window used in NLSA is crucial in capturing the main features and assessing the predictability of MISOs.

A subset simulation based approach with modified conditional sampling and estimator for loss exceedance curve computation

A new stochastic simulation-based approach for the evaluation of loss exceedance curve without repeated reliability analyses, and the generation of samples of input random variables and any function of them conditioned on different levels of loss exceedance is proposed for a comprehensive risk and loss analysis, and investigation. The proposed approach involves the modification of the simulation algorithms in the Subset Simulation and the development of new estimators. It allows for a more comprehensive characterization of the probability distribution of the loss including the tail parts due to combinations of scenarios which can lead to extreme and catastrophic consequences. The approach is robust to the number of random variables involved. The effectiveness and efficiency of the proposed method are shown by an illustrative example involving a seismic loss analysis of a multi-story inelastic structure. A stochastic ground motion model coupled with a stochastic nonlinear dynamic model, and probabilistic fragility and loss functions are considered.

Sensitivity analysis of consumption cycles.

We study the special case of a nonlinear stochastic consumption model taking the form of a 2-dimensional, non-invertible map with an additive stochastic component. Applying the concept of the stochastic sensitivity function and the related technique of confidence domains, we establish the conditions under which the system's complex consumption attractor is likely to become observable. It is shown that the level of noise intensities beyond which the complex consumption attractor is likely to be observed depends on the weight given to past consumption in an individual's preference adjustment.

Reference evapotranspiration forecasting based on local meteorological and global climate information screened by partial mutual information

In this study, reference evapotranspiration (ET0) forecasting models are developed for the least economically developed regions subject to meteorological data scarcity. Firstly, the partial mutual information (PMI) capable of capturing the linear and nonlinear dependence is investigated regarding its utility to identify relevant predictors and exclude those that are redundant through the comparison with partial linear correlation. An efficient input selection technique is crucial for decreasing model data requirements. Then, the interconnection between global climate indices and regional ET0 is identified. Relevant climatic indices are introduced as additional predictors to comprise information regarding ET0, which ought to be provided by meteorological data unavailable. The case study in the Jing River and Beiluo River basins, China, reveals that PMI outperforms the partial linear correlation in excluding the redundant information, favouring the yield of smaller predictor sets. The teleconnection analysis identifies the correlation between Nino 1 + 2 and regional ET0, indicating influences of ENSO events on the evapotranspiration process in the study area. Furthermore, introducing Nino 1 + 2 as predictors helps to yield more accurate ET0 forecasts. A model performance comparison also shows that non-linear stochastic models (SVR or RF with input selection through PMI) do not always outperform linear models (MLR with inputs screen by linear correlation). However, the former can offer quite comparable performance depending on smaller predictor sets. Therefore, efforts such as screening model inputs through PMI and incorporating global climatic indices interconnected with ET0 can benefit the development of ET0 forecasting models suitable for data-scarce regions.

Data-Driven Localization Mappings in Filtering the Monsoon–Hadley Multicloud Convective Flows

This paper demonstrates the efficacy of data-driven localization mappings for assimilating satellite-like observations in a dynamical system of intermediate complexity. In particular, a sparse network of synthetic brightness temperature measurements is simulated using an idealized radiative transfer model and assimilated to the monsoon–Hadley multicloud model, a nonlinear stochastic model containing several thousands of model coordinates. A serial ensemble Kalman filter is implemented in which the empirical correlation statistics are improved using localization maps obtained from a supervised learning algorithm. The impact of the localization mappings is assessed in perfect-model observing system simulation experiments (OSSEs) as well as in the presence of model errors resulting from the misspecification of key convective closure parameters. In perfect-model OSSEs, the localization mappings that use adjacent correlations to improve the correlation estimated from small ensemble sizes produce robust accurate analysis estimates. In the presence of model error, the filter skills of the localization maps trained on perfect- and imperfect-model data are comparable.

Study on Nonlinear Vibration Analysis of Gear System with Random Parameters

In order to study the dynamic characteristics of gear nonlinear vibration system and the influence of random parameters, firstly, a nonlinear stochastic vibration analysis model of gear 3-DOF is established based on Newton’s Law. And the random response of gear vibration is simulated by stepwise integration method. Secondly, the influence of stochastic parameters such as meshing damping, tooth side gap and excitation frequency on the dynamic response of gear nonlinear system is analyzed by using the stability analysis method such as bifurcation diagram and Lyapunov exponent method. The analysis shows that the stochastic process can not be neglected, which can cause the random bifurcation and chaos of the system response. This study will provide important reference value for vibration engineering designers.

Gaussian process approximations for fast inference from infectious disease data

We present a flexible framework for deriving and quantifying the accuracy of Gaussian process approximations to non-linear stochastic individual-based models of epidemics. We develop this for the SIR and SEIR models, and we show how it can be used to perform quick maximum likelihood inference for the underlying parameters given population estimates of the number of infecteds or cases at given time points. We also show how the unobserved processes can be inferred at the same time as the underlying parameters.

Stability analysis of Extended, Cubature and Unscented Kalman Filters for estimating stiff continuous–discrete stochastic systems

This paper studies stiffness and stability properties of Extended Cubature and Unscented Kalman Filters applied to continuous–discrete stochastic systems with stiff dynamic behavior. The main part of these methods relies on numerical integration of Moment Differential Equations (MDEs). Our focus is to understand how the stiffness of MDEs influences performance of the filters. The proposed linear stability analysis shows that the MDEs that havearisen within these methods can enlarge the stiffness of continuous-time stochastic model and, hence, require solvers with advanced stability properties for their effective and accurate treatment. Besides, the proposed nonlinear stability analysis proves that such MDEs may become extremely unstable in simulation intervals. The latter raises the uncertainty of state estimation and results in two interesting implications: (i) the lower-order Extended Kalman Filter outperforms the higher-order Cubature and Unscented Kalman Filters in the accuracy of state estimation of stochastic systems with unstable MDE behavior; (ii) the methods under exploration fail when the stiffness is large enough. Our theoretical consideration is supported by numerical tests with filters based on the MATLAB code ode15s, which is a benchmark solver for stiff ODEs. These are examined on linear and nonlinear stiff continuous-time stochastic models.

Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function

In this paper, we propose a new nonlinear stochastic SIRS epidemic model with standard incidence rate and saturated treatment function. The main purpose of this paper is to investigate the threshold dynamics of the nonlinear stochastic SIRS epidemic model by making use of stochastic inequality techniques. By using Lyapunov methods and Itô’s formula, we first prove the existence and uniqueness of a global positive solution for the corresponding limiting system. Furthermore, we obtain sufficient conditions for the extinction and persistence in mean of the nonlinear stochastic SIRS epidemic model by using the techniques of a series of stochastic inequalities. Finally, we provide some numerical simulations to illustrate the performance of our theoretical findings.

A stochastic scheduling approach to minimise the number of risky jobs and total probability of tardiness

In this paper, single machine stochastic scheduling problem is addressed where jobs have probabilistic processing times and deterministic due dates. In classical scheduling literature, a job is called 'tardy' if it is completed after its due date; otherwise it's called 'early'. However, in probabilistic concept, a job can have a non-zero probability of tardiness. To capture this situation, a stochastic nonlinear mathematical model is developed. The objective is to minimise the expected number of tardy jobs and the total probability of tardiness. Experimentation is performed with datasets having varying number of jobs from 10 to 100. Results are compared with deterministic model. The proposed approach resulted in a significant decrease in the number of risky jobs and the total probability of tardiness. In addition, as the variance of processing times increased, the proposed stochastic approach provided safer schedules in terms of the total probability of tardiness.

Stochastic Thermodynamics: A Dynamical Systems Approach

In this paper, we develop an energy-based, large-scale dynamical system model driven by Markov diffusion processes to present a unified framework for statistical thermodynamics predicated on a stochastic dynamical systems formalism. Specifically, using a stochastic state space formulation, we develop a nonlinear stochastic compartmental dynamical system model characterized by energy conservation laws that is consistent with statistical thermodynamic principles. In particular, we show that the difference between the average supplied system energy and the average stored system energy for our stochastic thermodynamic model is a martingale with respect to the system filtration. In addition, we show that the average stored system energy is equal to the mean energy that can be extracted from the system and the mean energy that can be delivered to the system in order to transfer it from a zero energy level to an arbitrary nonempty subset in the state space over a finite stopping time.

Predictive Control of Discrete Time Stochastic Nonlinear State Space Dynamical Systems: A Particle Nonparametric Approach

This paper presents the main principles of a stochastic nonlinear model predictive control (NMPC) novel approach for imperfectly observed discrete time systems described by time varying nonlinear non-Gaussian state space models with unknown parameters. A convergent particle estimator of the conditional expectation of a chosen cost function on a given receding control horizon is built, leading to an almost sure (a.s.) epi-convergent estimator of the NMPC cost-to-go criterion. The estimator of the expected cost function relies upon simulations and on a recently developed nonparametric convergent particle estimator of a multi-step ahead conditional probability density function (pdf) of the state variables. The theory of stochastic epi-convergence is applied to the estimated cost-to-go criterion to prove the almost sure convergence of the optimal solutions of the approximated NMPC problem to their true counterparts, when both simulation and particle numbers grow to infinity.

Dynamical capacity drop in a nonlinear stochastic traffic model

In this work, we show that the inverse-λ shape in the fundamental diagram of traffic flow can be produced dynamically by a simple nonlinear mesoscopic model with stochastic noises. The proposed model is based on the gas-kinetic theory of the traffic system. In our approach, the nonlinearity leads to the coexistence of different traffic states. The scattering of the data is thus attributed to the noise terms introduced in the stochastic differential equations and the transition among the various traffic states. Most importantly, the observed inverse-λ shape and the associated sudden jump of physical quantities arise due to the effect of stochastic noises on the stability of the system. The model parameters are calibrated, and a qualitative agreement is obtained between the data and the numerical simulations.

Optimal investment and premium control in a nonlinear diffusion model

This paper considers the optimal investment and premium control problem in a diffusion approximation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases: (i) maximizing the expected utility of terminal wealth, and (ii) minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.

The p53–Mdm2 regulation relationship under different radiation doses based on the continuous–discrete extended Kalman filter algorithm

The tumour suppressor gene p53 plays a key role in cell response to DNA damage, and the p53–Mdm2 regulation relationship is crucial for the expression of p53. In this paper, based on gene expression time-series data of human leukaemia cells after exposure to ionising radiation, with ionising radiation as input, a nonlinear continuous time-delay dynamic stochastic mathematical model of a p53–Mdm2 network is established using a continuous–discrete extended Kalman filter algorithm. The accuracy of the established model is then validated. Numerical simulation is used to simulate the dynamic regulation in p53–Mdm2 networks for low, medium, and high ionising radiation doses. The results show that the proposed algorithm is convergent and that the error rate of the model is only 1.19%. In addition, the model can simulate accurately the response of the p53–Mdm2 network to different doses of ionising radiation. The methods proposed in this paper can supply the foundation for research on the dynamics of the p53–Mdm2 gene regulation relationship and play a guiding role in research on the p53–Mdm2 response process after DNA damage under different doses of ionising radiation.

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part III extensions and applications to kinetic theory and transport

This third part extends the theory of Generalized Poisson–Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker–Planck–Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac’s program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.

Simulated Pseudo Maximum Likelihood Identification of Nonlinear Models

Nonlinear stochastic parametric models are widely used in various fields. However, for these models, the problem of maximum likelihood identification is very challenging due to the intractability of the likelihood function. Recently, several methods have been developed to approximate the analytically intractable likelihood function and compute either the maximum likelihood or a Bayesian estimator. These methods, albeit asymptotically optimal, are computationally expensive. In this contribution, we present a simulation-based pseudo likelihood estimator for nonlinear stochastic models. It relies only on the first two moments of the model, which are easy to approximate using Monte-Carlo simulations on the model. The resulting estimator is consistent and asymptotically normal. We show that the pseudo maximum likelihood estimator, based on a multivariate normal family, solves a prediction error minimization problem using a parameterized norm and an implicit linear predictor. In the light of this interpretation, we compare with the predictor defined by an ensemble Kalman filter. Although not identical, simulations indicate a close relationship. The performance of the simulated pseudo maximum likelihood method is illustrated in three examples. They include a challenging state-space model of dimension 100 with one output and 2 unknown parameters, as well as an application-motivated model with 5 states, 2 outputs and 5 unknown parameters.

Particle Model Predictive Control: Tractable Stochastic Nonlinear Output-Feedback MPC

We combine conditional state density construction with an extension of the Scenario Approach for stochastic Model Predictive Control to nonlinear systems to yield a novel particle-based formulation of stochastic nonlinear output-feedback Model Predictive Control. Conditional densities given noisy measurement data are propagated via the Particle Filter as an approximate implementation of the Bayesian Filter. This enables a particle-based representation of the conditional state density, or information state, which naturally merges with scenario generation from the current system state. This approach attempts to address the computational tractability questions of general nonlinear stochastic optimal control. The Particle Filter and the Scenario Approach are shown to be fully compatible and–based on the time- and measurement-update stages of the Particle Filter–incorporated into the optimization over future control sequences. A numerical example is presented and examined for the dependence of solution and computational burden on the sampling configurations of the densities, scenario generation and the optimization horizon.