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The characterization of the subsurface profile properties that are susceptible to liquefaction during earthquakes is of great importance in accident prevention. Field data can be used in a stochastic inverse problem to estimate the material properties using Markov chain Monte Carlo Methods (McMC) that incorporates prior knowledge about the unknown parameters as well as the data available. Here, we numerically study the Random Walk (RW) and Differential Evolution (DE) variations of the Metropolis algorithm in the context of seismic modeling considering the viscoelastic wave equation. The algorithms are tested with data from the 1987 Superstition Hills earthquake recorded at the Wildlife Site. The results show the effectiveness and accuracy of these algorithms, with emphasis to DE which reached convergence earlier compared to RW. [ABSTRACT FROM AUTHOR]

Traffic congestion in urban areas poses several challenges to municipal authorities including pollution, productivity loss, reckless driving, and delays in dealing with emergencies. Smart cities can use modern IoT infrastructure to solve the congestion problem and reduce pollution and delays. In this article, we focus on congestion mapping and mitigation for emergency vehicles in smart cities. We use a novel traffic light control technique to change the flow of cars on lights of interest thereby making way for emergency vehicles. We use a simulation model for a selected area of Manhattan to implement congestion mapping and to help find the fastest path for routing emergency vehicles based on the congestion metrics. The system controls traffic lights to block off the roads feeding into congestion and allows flow away from the congested path. This helps in clearing the preferred route to help emergency vehicles reach the destination faster. We show that the proposed algorithm can map congestion on city roads with accuracy thus helping to improve the response time of the emergency services and saving precious lives. [ABSTRACT FROM AUTHOR]

In a clickstream analysis setting, Mixture Hidden Markov Models (MHMMs) can be used to examine categorical sequences assuming they evolve according to a mixture of latent Markov processes, each related to a different subpopulation. These models involve identifying both the number of subpopulations and hidden states. This study proposes a model selection criterion based on an integrated completed likelihood approach that accounts for the two latent classes in the model. We implemented a Monte Carlo simulation study to compare selection criteria performance. In scenarios characterised by categorical short length sequences, our proposed measure outperforms the most commonly used model selection criteria in identifying components and states. The paper presents a case study on clickstream data collected from the website of a company operating in the hospitality industry and modelled by an MHMM selected by the proposed score. [ABSTRACT FROM AUTHOR]

This paper models games where the strategies are nodes of a graph G (we denote them as G-games) and in presence of coalition structures. The cases of one-shot and repeated games are presented. In the latter situation, coalitions are assumed to move from a strategy to another one under the constraint that they are adjacent in the graph. We introduce novel concepts of pure and mixed equilibria which are comparable with classical Nash and Berge equilibria. A Folk Theorem for G-games of repeated type is presented. [ABSTRACT FROM AUTHOR]

Stochastic models for collections of interacting populations have crucial roles in many scientific fields such as epidemiology, ecology, performance engineering, and queueing theory, to name a few. However, the standard approach to extending an ordinary differential equation model to a Markov chain does not have sufficient flexibility in the mean-variance relationship to match data. To handle that, we develop new approaches using Dirichlet noise to construct collections of independent or dependent noise processes. This permits the modeling of high-frequency variation in transition rates both within and between the populations under study. Our theory is developed in a general framework of time-inhomogeneous Markov processes equipped with a general graphical structure. We demonstrate our approach on a widely analyzed measles dataset, adding Dirichlet noise to a classical Susceptible–Exposed–Infected–Recovered model. Our methodology shows improved statistical fit measured by log-likelihood and provides new insights into the dynamics of this biological system. [ABSTRACT FROM AUTHOR]

In the present we provide the definition of the new concept of the General Non-Homogeneous Markov System (G-NHMS) and establish the expected population structure of a NHMS in the various states. These results will be the basis to build on the new concepts and the basic theorems of what follows.We then establish the set of all possible expected relative distributions of the initial number of memberships at time t and all possible expected relative distributions of the expansion memberships at time t. We call this set the general expected relative population structure in the states of a G-NHMS. We then proceed by providing the new definitions of weak ergodicity in a G-NHMS and weak ergodicity with a geometrical rate of convergence. We then prove the Theorem 4 which is a new building block in the theory of G-NHMS. We also prove a similar theorem under the assumption that relative expansion of the population vanishes at infinity.We then provide a generalization of the coupling theorem for populations. We proceed then to study the asymptotic behavior of a G-NHMS when the input policy consists of independent non-homogeneous Poisson variates for each time interval t - 1 , t . It is founded in Theorem 7 that it displays a kind of weak ergodicity behavior, that is, it converges at each step to the row of a stable matrix. This row is independent of the initial distribution and of the asymptotic input policy unlike the results in previous works. Hence it generalizes the result in that works. Finally we illustrate our results numerically for a manpower system with three states. [ABSTRACT FROM AUTHOR]

In this paper, we establish moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in weighted total variation norm or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider. [ABSTRACT FROM AUTHOR]

We consider the propagation of a stochastic SIR-type epidemic in two connected populations: a relatively small local population of interest which is surrounded by a much larger external population. External infectives can temporarily enter the small population and contribute to the spread of the infection inside this population. The rules for entry of infectives into the small population as well as their length of stay are modeled by a general Markov queueing system. Our main objective is to determine the distribution of the total number of infections within both populations. To do this, the approach we propose consists of deriving a family of martingales for the joint epidemic processes and applying classical stopping time or convergence theorems. The study then focuses on several particular cases where the external infection is described by a linear branching process and the entry of external infectives obeys certain specific rules. Some of the results obtained are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is 1 N plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process. [ABSTRACT FROM AUTHOR]

In the Fastest Mixing Markov Chain problem, we are given a graph G = (V , E) and desire the discrete-time Markov chain with smallest mixing time τ subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time τ RW of the lazy random walk on G is characterised by the edge conductance Φ of G via Cheeger's inequality: Φ - 1 ≲ τ RW ≲ Φ - 2 log | V | . Analogously, we characterise the fastest mixing time τ ⋆ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance Ψ of G: Ψ - 1 ≲ τ ⋆ ≲ Ψ - 2 (log | V |) 2 . This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only ε -close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time τ ≲ ε - 1 (diam G) 2 log | V | . Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains. [ABSTRACT FROM AUTHOR]

A copula-based Markov chain model can flexibly capture serial dependence in a time series. However, the computational developments for copula-based Markov models remain insufficient for discrete marginal models compared with continuous ones. In this article, we develop computational methods for a binomial time series under the Clayton and Joe copulas. The methods include the data-generation, parameter estimation, model selection, and goodness-of-fit tests. We implement the methods in our R package Copula.Markov (). We conduct simulations to see the performance of the developed methods. Finally, the proposed method is illustrated by a real dataset. [ABSTRACT FROM AUTHOR]

Controlled discrete time Markov processes are studied first with long run general discounting functional. It is shown that optimal strategies for average reward per unit time problem are also optimal for average generally discounting functional. Then long run risk sensitive reward functional with general discounting is considered. When risk factor is positive then optimal value of such reward functional is dominated by the reward functional corresponding to the long run risk sensitive control. In the case of negative risk factor we get an asymptotical result, which says that optimal average reward per unit time control is nearly optimal for long run risk sensitive reward functional with general discounting, assuming that risk factor is close to 0. For this purpose we show in Appendix upper estimates for large deviations of weighted empirical measures, which are of independent interest. [ABSTRACT FROM AUTHOR]

This paper discusses a generalized SIR epidemic model that incorporates infectives with or without symptoms, allows arbitrary distributions for infectious periods and assumes infection rates depending on the current size of susceptibles. Our interest lies in the joint distribution of the state of the population and the severity of the disease at the end of infection. The approach is based on a formulation of the epidemic as a so-called collective model. First, a set of martingales is constructed which provides the distribution of this final epidemic outcome. Then, its corresponding distributional transform is expressed in terms of a family of pseudo-polynomials of Abel-Gontcharoff type. Some of the results obtained are illustrated numerically. [ABSTRACT FROM AUTHOR]

We consider populations with time-varying growth rates living in sinks. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. We provide a mathematical analysis of this surprising phenomenon, in the context of a deterministic model with periodic variation of growth rates and non-symmetric migration which are assumed to be piecewise continuous. We also consider a stochastic model with random variation of growth rates and migration. This work extends existing results of the literature on the DIG effects obtained for periodic continuous growth rates and time independent symmetric migration. [ABSTRACT FROM AUTHOR]

In this study, we consider disorder detection problems for statistical models with dependent observations defined by Markov chains in a Bayesian setting for a uniform prior distribution when the number of observations is limited by some known quantity. To achieve this, a new optimal sequential detection procedure is constructed using optimal stopping methods. This procedure minimizes the average delay time in the class of sequential procedures with false alarm probabilities that do not exceed a fixed value. The main difference between the proposed detection procedure and the conventional ones is that it is based not on the posterior probabilities but on the weighted Shiryaev-Roberts statistic. This allows for optimal detection in a nonasymptotic sense over any bounded time interval. Thereafter, we applied the constructed procedures to the early detection problem for the beginning of the epidemic spread. We used two epidemic models: the binomial model proposed by Baron, Choudhary, and Yu (2013) and the model based on the Gaussian approximation introduced by Pergamenchtchikov, Tartakovsky, and Spivak (2022). The theoretical results were confirmed through numerical simulations using the Monte Carlo method. [ABSTRACT FROM AUTHOR]

We derive asymptotic formulae in the limit when population size N tends to infinity for mean fixation times (conditional and unconditional) in a population with two types of individuals, A and B, governed by the Moran process. We consider only the case in which the fitness of the two types do not depend on the population frequencies. Our results start with the important cases in which the initial condition is a single individual of any type, but we also consider the initial condition of a fraction x ∈ (0 , 1) of A individuals, where x is kept fixed and the total population size tends to infinity. In the cases covered by Antal and Scheuring (Bull Math Biol 68(8):1923–1944, 2006), i.e. conditional fixation times for a single individual of any type, it will turn out that our formulae are much more accurate than the ones they found. As quoted, our results include other situations not treated by them. An interesting and counterintuitive consequence of our results on mean conditional fixation times is the following. Suppose that a population consists initially of fitter individuals at fraction x and less fit individuals at a fraction 1 - x . If population size N is large enough, then in the average the fixation of the less fit individuals is faster (provided it occurs) than fixation of the fitter individuals, even if x is close to 1, i.e. fitter individuals are the majority. [ABSTRACT FROM AUTHOR]

Bayesian Learning is an inference method designed to tackle exploration-exploitation trade-off as a function of the uncertainty of a given probability model from observations within the Reinforcement Learning (RL) paradigm. It allows the incorporation of prior knowledge, as probabilistic distributions, into the algorithms. Finding the resulting Bayes-optimal policies is notorious problem. We focus our attention on RL of a special kind of ergodic and controllable Markov games. We propose a new framework for computing the near-optimal policies for each agent, where it is assumed that the Markov chains are regular and the inverse of the behavior strategy is well defined. A fundamental result of this paper is the development of a theoretical method that, based on the formulation of a non-linear problem, computes the near-optimal adaptive-behavior strategies and policies of the game under some restrictions that maximize the expected reward. We prove that such behavior strategies and the policies satisfy the Bayesian-Nash equilibrium. Another important result is that the RL process learn a model through the interaction of the agents with the environment, and shows how the proposed method can finitely approximate and estimate the elements of the transition matrices and utilities maintaining an efficient long-term learning performance measure. We develop the algorithm for implementing this model. A numerical empirical example shows how to deploy the estimation process as a function of agent experiences. [ABSTRACT FROM AUTHOR]

We study an energy-constrained random walker on a length-N interval of the one-dimensional integer lattice, with boundary reflection. The walker consumes one unit of energy for every step taken in the interior, and energy is replenished up to a capacity of M on each boundary visit. We establish large N, M distributional asymptotics for the lifetime of the walker, i.e., the first time at which the walker runs out of energy while in the interior. Three phases are exhibited. When M ≪ N 2 (energy is scarce), we show that there is an M-scale limit distribution related to a Darling–Mandelbrot law, while when M ≫ N 2 (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where M / N 2 → ρ ∈ (0 , ∞) , we show that there is an M-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions. [ABSTRACT FROM AUTHOR]

A two-class Processor-Sharing queue with one impatient class is studied. Local exponential decay rates for its stationary distribution (N (∞) , M (∞)) are established in the heavy traffic regime where the arrival rate of impatient customers grows proportionally to a large factor A. This regime is characterized by two time-scales, so that no general Large Deviations result is applicable. In the framework of singular perturbation methods, we instead assume that an asymptotic expansion of the solution of associated Kolmogorov equations exists for large A and derive it in the form P (N (∞) = A x , M (∞) = A y) ∼ g (x , y) 2 π A · e − A H (x , y) , x > 0 , y > 0 , with explicit functions g and H. This result is then applied to the model of mobile networks proposed in [Olivier and Simonian] and accounting for the spatial movement of users. We give further evidence of a unusual growth behavior in heavy traffic in that the stationary mean queue length E (N mob (∞)) and E M mob (∞)) of each customer-class increases proportionally to E (N mob (∞)) ∝ E (M mob (∞)) ∝ log (1 1 − ϱ tot ) with system load ϱ tot tending to 1, instead of the usual 1 / (1 − ϱ tot) growth behavior. [ABSTRACT FROM AUTHOR]

In this article we propose a new online control system aiming to lower the instants in which the production process migrates from being in- to out-of-control state, which generates an increase in the non-conformity rates. As shifts from in- to out-of-control are non-deterministic, a sample of size n is collected, for each m or L units produced, and each element from the sample is imprecisely classified as conform or non-conform (that is, there may be classification errors). If the amount of conform units from the sample is equal or greater than a, the process would not be adjusted and another sample would be collected after m units produced. If the quantity of conform units is inferior to a, the process would be adjusted and another sample would be collected after L units produced, given that L > m. A genetic algorithm is proposed to approximately find the values of a, n, m, and L that minimize all costs involved in the process being controlled. All procedures are illustrated through a detailed numerical example that attests the efficacy and efficiency of the proposed online control system. [ABSTRACT FROM AUTHOR]

We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, which are inspired by the DeGroot-Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as nonlinear Markov chains. In this paper, we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the ℓ1-metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear - both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations.

Because civil engineering structures include stressed structural members and unstressed non-structural members, it isn’t easy to obtain the true parameters of the model when modeling. The article uses a virtual reality fuzzy random modeling to establish a steel bearing force model in civil engineering operations. We determined the relevant bearing capacity parameters based on the corrosion of steel bars in civil engineering. The research found that the numerical simulation method determined in this paper has grasped the essence of stress rate, concrete material properties, and dynamic load effect to a considerable extent.

In this paper, we are concerned with a non-asymptotic analysis of sampling algorithms used in nonconvex optimization. In particular, we obtain non-asymptotic estimates in Wasserstein-1 and Wasserstein-2 distances for a popular class of algorithms called Stochastic Gradient Langevin Dynamics (SGLD). In addition, the aforementioned Wasserstein-2 convergence result can be applied to establish a non-asymptotic error bound for the expected excess risk. Crucially, these results are obtained under a local Lipschitz condition and a local dissipativity condition where we remove the uniform dependence in the data stream. We illustrate the importance of this relaxation by presenting examples from variational inference and from index tracking optimization. [ABSTRACT FROM AUTHOR]

In this paper, we consider constrained discounted stochastic games with a countably generated state space and norm continuous transition probability having a density function. We prove existence of approximate stationary equilibria and stationary weak correlated equilibria. Our results imply the existence of stationary Nash equilibrium in ARAT stochastic games. [ABSTRACT FROM AUTHOR]

Consider a cyber-physical system which has the purpose of maintaining the security of some physical asset. A particularly relevant scenario is one in which the system must perform its functions while at the same time enduring malicious attacks for an extended amount of time without energy renewal. We will study a theoretical model analogous to this cyber-physical system with an arbitrary number of nodes. The nodes in this system are able to communicate with one another and have the task of maintaining the security of the system. Initially all nodes comprising this system are in working order, however, due to attacks they can eventually become corrupted and then work against the security of the system. Of particular interest is the calculation of a number of metrics related to reliability of the system including the reliability function and the mean-time to failure. A particular case of this model has been examined by many researchers in the past, mostly using numerical methods. The contribution of this paper is the presentation of explicit expressions for the transient-state transition probabilities, reliability function, and the mean-time to failure along with rigorous proofs for the validity of these expressions. A result such as this is stronger than previous results via numerical methods and is an extension of previous analytical results where additional constraints on the model were imposed. [ABSTRACT FROM AUTHOR]

We investigate graph-based Laplacian semi-supervised learning at low labeling rates (ratios of labeled to total number of data points) and establish a threshold for the learning to be well posed. Laplacian learning uses harmonic extension on a graph to propagate labels. It is known that when the number of labeled data points is finite while the number of unlabeled data points tends to infinity, the Laplacian learning becomes degenerate and the solutions become roughly constant with a spike at each labeled data point. In this work, we allow the number of labeled data points to grow to infinity as the total number of data points grows. We show that for a random geometric graph with length scale ε > 0 , if the labeling rate β ≪ ε 2 , then the solution becomes degenerate and spikes form. On the other hand, if β ≫ ε 2 , then Laplacian learning is well-posed and consistent with a continuum Laplace equation. Furthermore, in the well-posed setting we prove quantitative error estimates of O (ε β - 1 / 2) for the difference between the solutions of the discrete problem and continuum PDE, up to logarithmic factors. We also study p-Laplacian regularization and show the same degeneracy result when β ≪ ε p . The proofs of our well-posedness results use the random walk interpretation of Laplacian learning and PDE arguments, while the proofs of the ill-posedness results use Γ -convergence tools from the calculus of variations. We also present numerical results on synthetic and real data to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper presents an axiomatic approach to finite Markov decision processes where the discount rate is zero. One of the principal difficulties in the no discounting case is that, even if attention is restricted to stationary policies, a strong overtaking optimal policy need not exists. We provide preference foundations for two criteria that do admit optimal policies: 0-discount optimality and average overtaking optimality. As a corollary of our results, we obtain conditions on a decision maker's preferences which ensure that an optimal policy exists. These results have implications for disciplines where dynamic programming problems arise, including automatic control, dynamic games, and economic development. [ABSTRACT FROM AUTHOR]

Our main goal in this paper is to propose a mathematical model in order to use it to study the historical achievements of a particular team in order to stand on the evolutionary behavior of the team and to be able to predict the future results of the team by measuring the team's performance through the total number of victories, losses and draws in official matches approved. Our main idea is to use a Markov model to find the results of matches, so that the outcome of each match is one of the specified cases: win or lose or draw. [ABSTRACT FROM AUTHOR]

A controlled discrete-time stochastic process is proposed as a model for the state of a machine. In the proposed model, normal and random wear of the machine are considered. The random wear of the machine during a time unit can be either a discrete or a continuous random variable. The objective is to find a control policy that maximises the profits generated by the machine over its useful lifetime. In this problem, the optimiser must decide whether to do or not to do the maintenance work on the machine at each time unit. The significance of the paper is that the final time in the optimal control problem is a random variable denoting the first time the machine must be replaced. To obtain the optimal control, one can try to solve the dynamic programming equation, which reduces to a difference or an integral equation, satisfied by the value function. Finally, particular cases are solved exactly, or approximately, and explicitly to demonstrate and validate the proposed model. [ABSTRACT FROM AUTHOR]