Complex Stochastic Models Research Articles

Vibration suppression in SDOF systems coupled to a nonlinear energy sink under colored noise

This study analyzes the stochastic vibration suppression and optimization of the single-degree-of-freedom (SDOF) system equipped with the cubic stiffness nonlinear energy sink (NES) under colored noise excitation. Two theoretical methods are proposed: an integrated method (denoted as EL-ELM) that combines evolutionary Lyapunov theory with the equivalent linearization method, and the other is an empirical formula. Using EL-ELM, the coupled nonlinear system is simplified to an equivalent linear stochastic system, allowing for a theoretical analysis of the impact of NES structural parameters on suppression performance and the precise determination of optimal parameter configurations for the best suppression effects. Subsequently, based on the results from the EL-ELM and the response data, an empirical formula has been developed that clearly describes the comprehensive laws governing the optimal NES parameters as they vary with vibration system parameters and stochastic excitation. Through error analysis and comparison of the two methods, it is found that the empirical formula significantly outperforms EL-ELM in terms of accuracy and computational cost, but it is contingent on solid prior knowledge. This study explores the influence of NES structural parameters on the system’s dynamic response and energy, further validating the effectiveness of the proposed methods in identifying optimal structural parameters. The phenomenon of targeted energy transfer (TET) under different NES structural parameters is also explained. The methodologies introduced in this study have strengthened the theory of vibration suppression. Specifically, the empirical formula excels in accuracy and computational efficiency by effectively using prior knowledge. The EL-ELM method, owing to its theoretical insights, is vital for analyzing complex stochastic nonlinear models. Combining these approaches offers guidance for advancing vibration control in theoretical and practical domains.

Dynamical bifurcation of a stochastic Holling-II predator–prey model with infinite distributed delays

This paper serves as a continuation and expansion of the previous achievement by B. Han and D. Jiang in their article (Han and Jiang, 2024). Our initial step involves transforming the more complex stochastic Holling-II model featuring a stronger kernel into a simplified yet degenerate stochastic system composed of five interconnected equations. For the deterministic component of this model, we delve into an extensive examination of the local asymptotic stability of its positive equilibrium state. Subsequently, for the stochastic model, we derive critical conditions that determine the thresholds for exponential extinction and persistence of both predator and prey populations. Importantly, our findings not only encompass scenarios where there are no stochastic disturbances but also shed light on how environmental noise impacts the population dynamics within the predator–prey system. Through the exploration of the homologous Fokker–Planck equations, we present approximate representations characterizing the probability density function of the stochastic predator–prey model. To substantiate these theoretical advancements, several illustrative examples are provided, offering numerical elucidations of our proposed results and emphasizing the profound effects of stochastic noises and some important parameters on the behaviors of the stochastic model.

Bridging the Gap Between Biological Plausibility and Practicality in Neuron Modeling: Challenges and Perspectives

Neuron modeling has been a fundamental approach in computational neuroscience for understanding the dynamics and function of the central nervous system (CNS). Over the past decades, a plethora of mathematical models, ranging from the seminal Hodgkin-Huxley model to more complex multicompartmental and stochastic models, have been developed to capture the intricate behavior of neurons and their interactions. However, despite the significant advances in modeling, there remains a substantial disconnect between the complexity of the models and their practical applicability in day-to-day medical and laboratory settings. This article explores the challenges and limitations of current neuron modeling approaches, particularly focusing on the trade-off between biological realism and computational tractability. I discuss the variability of neuronal properties within and across different regions of the brain, which poses a significant challenge for developing models that can accurately represent the full complexity of the CNS. I also highlight the computational bottlenecks associated with simulating highly detailed, which often require extensive computational resources and specialized software. Considering these challenges, we emphasize the need for a balance between biological plausibility and practicality in neuron modeling. We discuss the role of simplified models, such as the leaky integrate-and-fire (LIF) and Izhikevich models, which sacrifice some biological detail in favor of computational efficiency and ease of use. These models have proven valuable for large-scale simulations of neural networks and have provided important insights into the collective behavior of neuronal populations. Furthermore, I explore the potential of data-driven approaches, such as machine learning and artificial neural networks, in bridging the gap between biological realism and practical applicability. These approaches can learn the dynamics of neuronal systems directly from experimental data, bypassing the need for explicit mathematical models. I present a simple example using linear regression to predict neuronal firing rates based on input features, showcasing the potential of machine learning techniques in capturing the complex relationships between neuronal variables. I also highlight the need for more extensive validation of models against experimental data and the development of standardized benchmarks for assessing model performance. In conclusion, while the quest for biologically plausible neuron models has driven significant advances in our understanding of the CNS, the disconnect between model complexity and practical applicability remains a major challenge. By embracing simplified models, data-driven approaches, and interdisciplinary collaborations, we can work towards the development of more efficient and practical neuron models that can bridge the gap between biological realism and computational feasibility, ultimately facilitating the translation of theoretical insights into clinical and experimental applications.

Average decayed dynamics of one-step transformation process with randomly varying return transition rate

The problem of stochastic averaging of the decayed dynamics of the output state population of a one-step transformation process with constant deterministic forward transition rate and randomly varying return transition rate is solved in approximation where random variation is modeled as a dichotomous stochastic process. The form of the obtained solution represented as a product between bimodal sigmoid rise of average population and its unimodal exponential decay is shown to largely be dependent on the stochastic frequency and amplitude parameters. For example, at high stochastic frequency, the behavior of population is reduced to that of a decayed one-step deterministic system. However, for resonance stochastic amplitude at low stochastic frequency, such behavior coincides with that of three-exponential rise-decay kinetics typical rather of a three-step deterministic slowly decaying process. Thus, there is an equivalence between using a more complex deterministic kinetic model and a less complex stochastic kinetic model for describing the decayed dynamics of different irreversible systems.

A detailed sensitivity analysis identifies the key factors influencing the enzymatic saccharification of lignocellulosic biomass

Corn stover is the most abundant form of crop residue that can serve as a source of lignocellulosic biomass in biorefinery approaches, for instance for the production of bioethanol. In such biorefinery processes, the constituent polysaccharide biopolymers are typically broken down into simple monomeric sugars by enzymatic saccharification, for further downstream fermentation into bioethanol. However, the recalcitrance of this material to enzymatic saccharification invokes the need for innovative pre-treatment methods to increase sugar conversion yield. Here, we focus on experimental glucose conversion time-courses for corn stover lignocellulose that has been pre-treated with different acid-catalysed processes and intensities. We identify the key parameters that determine enzymatic saccharification dynamics by performing a Sobol's sensitivity analysis on the comparison between the simulation results from our complex stochastic biophysical model, and the experimental data that we accurately reproduce. We find that the parameters relating to cellulose crystallinity and those associated with the cellobiohydrolase activity are predominantly driving the enzymatic saccharification dynamics. We confirm our computational results using mathematical calculations for a purely cellulosic substrate. On the one hand, having identified that only five parameters drastically influence the saccharification dynamics allows us to reduce the dimensionality of the parameter space (from nineteen to five parameters), which we expect will significantly speed up our fitting algorithm for comparison of experimental and simulated saccharification time-courses. On the other hand, these parameters directly highlight key targets for experimental endeavours in the optimisation of pre-treatment and saccharification conditions. Finally, this systematic and two-fold theoretical study, based on both mathematical and computational approaches, provides experimentalists with key insights that will support them in rationalising their complex experimental results.

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

The pinewood nematode, which causes a disease known as pine wilt, is a serious threat to pine forests all over the world. In this paper, we analyze the dynamics of pine wilt disease and its effects on forest ecosystems using mathematical modeling of pine wilt disease, its numerical simulations, and analytical techniques. Understanding the nematode’s spread, evaluating the efficiency of management efforts, and foreseeing the long-term effects on pine populations are some of our research’s goals. To gain a comprehensive understanding of the dynamics of the disease, we categorize pine trees and vectors systematically into susceptible, exposed, and infected groups. We first presented disease dynamics through a flow diagram which is then transformed to system of ordinary differential equations along with non-negative initial conditions. The deterministic model is then transformed to a stochastic model. Using both deterministic and stochastic modeling approaches, this study explores the complex transmission dynamics of Pine Wilt disease. Basic reproduction number has been calculated via next generation technique and some cases of stability analysis on the basis of the values of reproduction number has been discussed. For the solution of stochastic model Legendre Spectral collocation method has been used as it is known for its high accuracy in approximating solutions to complex stochastic models. It can achieve exponential convergence rates, making it suitable for problems where high precision is required. The results of both deterministic and stochastic models are compared via different figures. We find crucial thresholds for disease transmission and highlight significant factors affecting the spread of diseases through simulations. Our research shows that preventing the severe impacts of pine wilt disease requires early detection and intervention. Additionally, our mathematical model is a useful tool for enhancing disease management plans. This research advances our knowledge of the dynamics of the pine wilt disease and provides useful advice for managing forests and promoting conservation. Our research highlights the need for quick action to protect pine forests from this harmful virus.

Efficient multifidelity likelihood-free Bayesian inference with adaptive computational resource allocation

Likelihood-free Bayesian inference algorithms are popular methods for inferring the parameters of complex stochastic models with intractable likelihoods. These algorithms characteristically rely heavily on repeated model simulations. However, whenever the computational cost of simulation is even moderately expensive, the significant burden incurred by likelihood-free algorithms leaves them infeasible for many practical applications. The multifidelity approach has been introduced in the context of approximate Bayesian computation to reduce the simulation burden of likelihood-free inference without loss of accuracy, by using the information provided by simulating computationally cheap, approximate models in place of the model of interest. In this work we demonstrate that multifidelity techniques can be applied in the general likelihood-free Bayesian inference setting. Analytical results on the optimal allocation of computational resources to simulations at different levels of fidelity are derived, and subsequently implemented practically. We provide an adaptive multifidelity likelihood-free inference algorithm that learns the relationships between models at different fidelities and adapts resource allocation accordingly, and demonstrate that this algorithm produces posterior estimates with near-optimal efficiency.

Fitting the grain orientation distribution of a polycrystalline material conditioned on a Laguerre tessellation

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel–titanium shape memory alloy.

Modelling species invasion using a metapopulation model with variable mortality and stochastic birth-death processes

A species entering a novel landscape must overcome several obstacles that inhibit invasion, which often leads invaders to go extinct while populations are still small and made up of relatively young individuals. Here, we investigate a theoretical model of invader population dynamics to explore the effects of demographic structure on invader success. To do so, we extended the classic Levins metapopulation model by allowing mortality to vary across individual life stages, such that smaller individuals have a higher chance of dying. We also allow size to vary across individuals, using an application of the Gillespie algorithm. Thus, unlike in the classic Levins model, for which successful invasions occur deterministically whenever colonisation exceeds the background mortality rate, invasion success in our model depends on the distribution of sizes found across individuals in the population. Nevertheless, the resulting likelihood of successful invasion can be estimated as a function of just two parameters – the colonisation rate, and a derived index representing the time-averaged mortality rate. This time-averaged mortality can be expressed analytically as a function of individual-level growth rates, the initial size of individuals, and of the impact of disturbances. While demographic stochasticity also contributes to some invasion failures in our model, we find that these effects are rare except when landscapes and initial fraction of occupied sites are both small. Our results demonstrate that invasion success in a complex stochastic model with explicit demographic structure can be predicted using a relatively simple, analytically tractable function. Applications of these results may therefore be particularly useful for studying general patterns of invasion success among sessile organisms with size dependant mortality, such as terrestrial plants.

Multilevel emulation for stochastic computer models with application to large offshore wind farms

Abstract Renewable energy projects, such as large offshore wind farms, are critical to achieving low-emission targets set by governments. Stochastic computer models allow us to explore future scenarios to aid decision making while considering the most relevant uncertainties. Complex stochastic computer models can be prohibitively slow, and thus an emulator may be constructed and deployed to allow for efficient computation. We present a novel heteroscedastic Gaussian Process emulator that exploits cheap approximations to a stochastic offshore wind farm simulator. We also conduct a probabilistic sensitivity analysis to understand the influence of key parameters in the wind farm model, which will help us to plan a probability elicitation in the future.

Estimating disease incidence rates and transition probabilities in elderly patients using multi-state models: a case study in fragility fracture using a Bayesian approach

BackgroundMulti-state models are complex stochastic models which focus on pathways defined by the temporal and sequential occurrence of numerous events of interest. In particular, the so-called illness-death models are especially useful for studying probabilities associated to diseases whose occurrence competes with other possible diseases, health conditions or death. They can be seen as a generalization of the competing risks models, which are widely used to estimate disease-incidences among populations with a high risk of death, such as elderly or cancer patients. The main advantage of the aforementioned illness-death models is that they allow the treatment of scenarios with non-terminal competing events that may occur sequentially, which competing risks models fail to do.MethodsWe propose an illness-death model using Cox proportional hazards models with Weibull baseline hazard functions, and applied the model to a study of recurrent hip fracture. Data came from the PREV2FO cohort and included 34491 patients aged 65 years and older who were discharged alive after a hospitalization due to an osteoporotic hip fracture between 2008-2015. We used a Bayesian approach to approximate the posterior distribution of each parameter of the model, and thus cumulative incidences and transition probabilities. We also compared these results with a competing risks specification.ResultsPosterior transition probabilities showed higher probabilities of death for men and increasing with age. Women were more likely to refracture as well as less likely to die after it. Free-event time was shown to reduce the probability of death. Estimations from the illness-death and the competing risks models were identical for those common transitions although the illness-death model provided additional information from the transition from refracture to death.ConclusionsWe illustrated how multi-state models, in particular illness-death models, may be especially useful when dealing with survival scenarios which include multiple events, with competing diseases or when death is an unavoidable event to consider. Illness-death models via transition probabilities provide additional information of transitions from non-terminal health conditions to absorbing states such as death, what implies a deeper understanding of the real-world problem involved compared to competing risks models.

An Algorithm for Online Stochastic Error Modeling of Inertial Sensors in Urban Cities.

Regardless of whether the global navigation satellite system (GNSS)/inertial navigation system (INS) is integrated or the INS operates independently during GNSS outages, the stochastic error of the inertial sensor has an important impact on the navigation performance. The structure of stochastic error in low-cost inertial sensors is quite complex; therefore, it is difficult to identify and separate errors in the spectral domain using classical stochastic error methods such as the Allan variance (AV) method and power spectral density (PSD) analysis. However, a recently proposed estimation, based on generalized wavelet moment estimation (GMWM), is applied to the stochastic error modeling of inertial sensors, giving significant advantages. Focusing on the online implementation of GMWM and its integration within a general navigation filter, this paper proposes an algorithm for online stochastic error calibration of inertial sensors in urban cities. We further develop the autonomous stochastic error model by constructing a complete stochastic error model and determining model ranking criterion. Then, a detecting module is designed to work together with the autonomous stochastic error model as feedback for the INS/GNSS integration. Finally, two experiments are conducted to compare the positioning performance of this algorithm with other classical methods. The results validate the capability of this algorithm to improve navigation accuracy and achieve the online realization of complex stochastic models.

Unified Moment-Based Modeling of Integrated Stochastic Processes

Dial M for Simulation For years, systems of stochastic differential equations (SDEs) were simulated by discretization, inevitably introducing a bias, which can be difficult to quantify accurately. To circumvent this, some attempts have been made to simulate exactly various models from the SDE solution. These approaches prove capable of producing accurate results. A serious drawback of such an approach, nevertheless, is the implicit need to use extensive numerical methods, which make the entire simulation computationally heavy and quite impracticable. In the paper “Unified moment-based modeling of integrated stochastic processes,” Kyriakou, Brignone, and Fusai present a methodological framework based on M(oments) for the simulation of such models that overcomes earlier limitations. Theoretical results and extensive numerical experiments show that the proposed approach allows accurate simulation of complex stochastic models with low computational effort.

Fractional discrete Temimi–Ansari method with singular and nonsingular operators: applications to electrical circuits

The goal of this article is to present a recently developed numerical approach for solving fractional stochastic differential equations with a singular Caputo kernel and a nonsingular Caputo–Fabrizio and Atangana–Baleanu (ABC) kernel. The proposed method is based on the discrete Temimi–Ansari method, which is combined with three different numerical schemes that are appropriate for the new fractional derivative operators. The proposed technique is used to investigate the effects of Gaussian white-noise and Gaussian colored-noise perturbations on the potential source and resistance in fractional stochastic electrical circuits. The proposed method’s robustness and efficiency were demonstrated by comparing its results to those of the stochastic Runge–Kutta method (SRK). The valuable point in this article is that the resulting numerical scheme is able to combine two powerful methods that can be extended into more complex stochastic models. The comparison of different fractional derivatives using Mathematica 12 software has been obtained and the simulation results demonstrate the merit of the contributed method.

Observer-Based Fuzzy Quantized Control for a Stochastic Third-Order Parabolic PDE System

The work addresses an observer-based fuzzy quantized control for stochastic third-order parabolic partial differential equations (PDEs) using discrete point measurements. For the first time, we contribute in introducing three types of quantizer—logarithmic quantizer, uniform quantizer, and hysteresis quantizer into the controller designs for the stochastic PDE system. The main advantage of the quantized control law lies in reducing the energy that the system spends. The stability analysis is nontrivial due to the complex stochastic PDE model. Different stability results are presented for each case. Sufficient LMI-based conditions are investigated to ensure the internal mean-square exponential stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance of the perturbed actuated system. Consistent simulation results that support the proposed theoretical statements are provided.

Framework for assessing and easing global COVID-19 travel restrictions

During the COVID-19 pandemic, many countries implemented international travel restrictions that aimed to contain viral spread while still allowing necessary cross-border travel for social and economic reasons. The relative effectiveness of these approaches for controlling the pandemic has gone largely unstudied. Here we developed a flexible network meta-population model to compare the effectiveness of international travel policies, with a focus on evaluating the benefit of policy coordination. Because country-level epidemiological parameters are unknown, they need to be estimated from data; we accomplished this using approximate Bayesian computation, given the nature of our complex stochastic disease transmission model. Based on simulation and theoretical insights we find that, under our proposed policy, international airline travel may resume up to 58% of the pre-pandemic level with pandemic control comparable to that of a complete shutdown of all airline travel. Our results demonstrate that global coordination is necessary to allow for maximum travel with minimum effect on viral spread.

Efficient Stochastic Programming in Julia

We present StochasticPrograms.jl, a user-friendly and powerful open-source framework for stochastic programming written in the Julia language. The framework includes both modeling tools and structure-exploiting optimization algorithms. Stochastic programming models can be efficiently formulated using an expressive syntax, and models can be instantiated, inspected, and analyzed interactively. The framework scales seamlessly to distributed environments. Small instances of a model can be run locally to ensure correctness, whereas larger instances are automatically distributed in a memory-efficient way onto supercomputers or clouds and solved using parallel optimization algorithms. These structure-exploiting solvers are based on variations of the classical L-shaped, progressive-hedging, and quasi-gradient algorithms. We provide a concise mathematical background for the various tools and constructs available in the framework along with code listings exemplifying their usage. Both software innovations related to the implementation of the framework and algorithmic innovations related to the structured solvers are highlighted. We conclude by demonstrating strong scaling properties of the distributed algorithms on numerical benchmarks in a multinode setup. Summary of Contribution: This paper presents StochasticPrograms.jl, an open-source framework for stochastic programming implemented in the Julia programming language. The framework includes an expressive syntax for formulating stochastic programming models as well as versatile analysis tools and parallel optimization algorithms. The framework will prove useful to researchers, educators, and industrial users alike. Researchers will benefit from the readily extensible open-source framework, in which they can formulate complex stochastic models or quickly typeset and test novel optimization algorithms. Educators of stochastic programming will benefit from the clean and expressive syntax. Moreover, the framework supports analysis tools and stochastic programming constructs from classical theory and leading textbooks. We strongly believe that the StochasticPrograms.jl framework can reduce the barrier to entry for incoming practitioners of stochastic programming. Industrial practitioners can make use of StochasticPrograms.jl to rapidly formulate complex models, analyze small instances locally, and then run large-scale instances in production. In doing so, they get distributed capabilities for free without changing the code and access to well-tested state-of-the-art implementations of parallel structure-exploiting solvers. As the framework is open-source, anyone from these target audiences can contribute with new functionality to the framework. In conclusion, by providing both an intuitive interface for new users and an extensive development environment for expert users, StochasticPrograms.jl has strong potential to further the field of stochastic programming.

Toward Efficient Bayesian Approaches to Inference in Hierarchical Hidden Markov Models for Inferring Animal Behavior

The study of animal behavioral states inferred through hidden Markov models and similar state switching models has seen a significant increase in popularity in recent years. The ability to account for varying levels of behavioral scale has become possible through hierarchical hidden Markov models, but additional levels lead to higher complexity and increased correlation between model components. Maximum likelihood approaches to inference using the EM algorithm and direct optimization of likelihoods are more frequently used, with Bayesian approaches being less favored due to computational demands. Given these demands, it is vital that efficient estimation algorithms are developed when Bayesian methods are preferred. We study the use of various approaches to improve convergence times and mixing in Markov chain Monte Carlo methods applied to hierarchical hidden Markov models, including parallel tempering as an inference facilitation mechanism. The method shows promise for analysing complex stochastic models with high levels of correlation between components, but our results show that it requires careful tuning in order to maximize that potential.

Market segmentation in online platforms

This paper studies ranking policies in a stylized trial-offer marketplace model, in which a single firm offers multiple products and has consumers who express heterogeneous preferences. Consumer trials are influenced by past purchases, the inherent appeal of the products, and the ranking of each product. Consumer purchases conditional on trying the product are dependent on the inherent quality for the given consumer segment. The platform owner needs to devise a ranking policy to display the products to maximize the number of purchases in the long run, and to decide whether to display the number of past purchases. The model proposed attempts to understand the impact of market segmentation in a trial-offer market with position bias and social influence. Under our model, consumer choices are based on a very general choice model known as the mixed multinomial logit model, which embeds product appeal, ranking, and past purchases into the taste parameters. We analyze the long-term dynamics of this highly complex stochastic model and we quantify the expected benefits of market segmentation as well as the value of social influence. When past purchases are displayed, consumer heterogeneity makes buyers try the sub-optimal products, reducing the overall sales rate. We show that consumer heterogeneity makes the ranking problem NP-hard. We then analyze the benefits of market segmentation. We find tight bounds to the expected benefits of offering a distinct ranking to each consumer segment. Finally, we show that the market segmentation strategy always benefits from social influence when the average quality ranking is used. One of the managerial implications is that the firm is better off using an aggregate ranking policy when the variety of consumer preference is limited, but it should perform a market segmentation policy when consumers are highly heterogeneous. We also show that this result is robust to relatively small consumer classification mistakes; when these are large, an aggregate ranking is preferred.

A Semiautomatic Method for History Matching Using Sequential Monte Carlo

The aim of the history matching method is to locate nonimplausible regions of the parameter space of complex deterministic or stochastic models by matching model outputs with data. It does this via a series of waves where at each wave an emulator is fitted to a small number of training samples. An implausibility measure is defined which takes into account the closeness of simulated and observed outputs as well as emulator uncertainty. As the waves progress, the emulator becomes more accurate so that training samples are more concentrated on promising regions of the space and poorer parts of the space are rejected with more confidence. While history matching has proved to be useful, existing implementations are not fully automated, and some ad hoc choices are made during the process, which involves user intervention and is time consuming. This occurs especially when the nonimplausible region becomes small and it is difficult to sample this space uniformly to generate new training points. In this article we develop a sequential Monte Carlo (SMC) algorithm for implementing history matching that is semiautomated. Our novel SMC approach reveals that the history matching method yields a nonimplausible region that can be multimodal, highly irregular, and very difficult to sample uniformly. Our SMC approach offers a much more reliable sampling of the nonimplausible space, which requires additional computation compared to other approaches used in the literature.