Continuous-time Process Research Articles

Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

The spectral theory of random walks on networks of arbitrary topology can be readily extended to study random walks and Lévy flights subject to resetting on these structures. When a discrete-time process is stochastically brought back from time to time to its starting node, the mean search time needed to reach another node of the network may be significantly decreased. In other cases, however, resetting is detrimental to search. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, we derive a general criterion for finite networks that establishes when there exists a non-zero resetting probability that minimizes the mean first passage time (MFPT) at a target node. Right at optimality, the coefficient of variation of the first passage time is not unity, unlike in continuous time processes with instantaneous resetting, but above 1 and depends on the minimal MFPT. The approach is general and applicable to the study of different discrete-time ergodic Markov processes such as Lévy flights, where the long-range dynamics is introduced in terms of the fractional Laplacian of the graph. We apply these results to the study of optimal transport on rings and Cayley trees.

A real-time data assimilative forecasting system for animal tracking.

Monitoring technologies now provide real‐time animal location information, which opens up the possibility of developing forecasting systems to fuse these data with movement models to predict future trajectories. State‐space modeling approaches are well established for retrospective location estimation and behavioral inference through state and parameter estimation. Here we use a state‐space model within a comprehensive data assimilative framework for probabilistic animal movement forecasting. Real‐time location information is combined with stochastic movement model predictions to provide forecasts of future animal locations and trajectories, as well as estimates of key behavioral parameters. Implementation uses ensemble‐based sequential Monte Carlo methods (a particle filter). We first apply the framework to an idealized case using a nondimensional animal movement model based on a continuous‐time random walk process. A set of numerical forecasting experiments demonstrates the workflow and key features, such as the online estimation of behavioral parameters using state augmentation, the use of potential functions for habitat preference, and the role of observation error and sampling frequency on forecast skill. For a realistic demonstration, we adapt the framework to short‐term forecasting of the endangered southern resident killer whale (SRKW) in the Salish Sea using visual sighting information wherein the potential function reflects historical habitat utilization of SRKW. We successfully estimate whale locations up to 2.5 h in advance with a moderate prediction error (<5 km), providing reasonable lead‐in time to mitigate vessel–whale interactions. It is argued that this forecasting framework can be used to synthesize diverse data types and improve animal movement models and behavioral understanding and has the potential to lead to important advances in movement ecology.

Finitary isomorphisms of renewal point processes and continuous-time regenerative processes

We show that a large class of stationary continuous-time regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with exponential tails is finitarily isomorphic to a Poisson point process. We further give simple necessary and sufficient conditions for a renewal point process to be finitarily isomorphic to a Poisson point process. This improves results and answers several questions of Soo [33] and of Kosloff and Soo [19].

Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures

Stability contractors, based on interval analysis, were introduced in recent work as a tool to verify stability domains for nonlinear dynamic systems. These contractors rely on the property that - in case of provable asymptotic stability - a certain domain in a multi-dimensional state space is mapped into its interior after a certain integration time for continuous-time processes or after a certain number of discretization steps in a discrete-time setting. However, a disadvantage of the use of axis-aligned interval boxes in such computations is the omnipresent wrapping effect. As shown in this contribution, the replacement of classical interval representations by ellipsoidal domain enclosures reduces this undesirable effect. It also helps to find suitable ratios for the edge lengths if interval-based domain representations are investigated. Moreover, ellipsoidal domains naturally represent the possible regions of attraction of asymptotically stable equilibrium points that can be analyzed with the help of quadratic Lyapunov functions, for which stability criteria can be cast into linear matrix inequality (LMI) constraints. For that reason, this paper further presents possible interfaces of ellipsoidal enclosure techniques with LMI approaches. This combination aims at the maximization of those domains that can be proven to be stable for a discrete-time range-only localization algorithm in robotics. There, an Extended Kalman Filter (EKF) is applied to a system for which the dynamics are characterized by a discrete-time integrator disturbance model with additive Gaussian noise. In this scenario, the measurement equations correspond to the distances between the object to be localized and beacons with known positions.

A Stochastic Reduced-Order Model for Statistical Microstructure Descriptors Evolution

Abstract Integrated computational materials engineering (ICME) models have been a crucial building block for modern materials development, relieving heavy reliance on experiments and significantly accelerating the materials design process. However, ICME models are also computationally expensive, particularly with respect to time integration for dynamics, which hinders the ability to study statistical ensembles and thermodynamic properties of large systems for long time scales. To alleviate the computational bottleneck, we propose to model the evolution of statistical microstructure descriptors as a continuous-time stochastic process using a non-linear Langevin equation, where the probability density function (PDF) of the statistical microstructure descriptors, which are also the quantities of interests (QoIs), is modeled by the Fokker–Planck equation. We discuss how to calibrate the drift and diffusion terms of the Fokker–Planck equation from the theoretical and computational perspectives. The calibrated Fokker–Planck equation can be used as a stochastic reduced-order model to simulate the microstructure evolution of statistical microstructure descriptors PDF. Considering statistical microstructure descriptors in the microstructure evolution as QoIs, we demonstrate our proposed methodology in three integrated computational materials engineering (ICME) models: kinetic Monte Carlo, phase field, and molecular dynamics simulations.

Application of agent-based approach for heat conduction processes simulation

Heat transfer processes are the basis of the most technological processes of the energy sector. Thus, the development of modern approaches for computer simulation and visualization of the phenomena of thermal energy transfer in various objects are of great importance and is relevant. Classical models in the form of differential equations of various types describe processes in continuous space and time. So, it is difficult to apply classical models to study nonlinear phenomena, and processes in inhomogeneous media in the presence of discontinuous solutions at the boundaries. In these cases, simplifying assumptions are used, thus, the adequacy of the models is reduced. It is of great interest to apply fundamentally different approaches to describe transfer processes, which include discrete dynamic models. The purpose of this project is to study the possibilities to apply discrete approaches to simulate nonlinear heat transfer processes under conditions of material inhomogeneity and the presence of volume sources of variable power. The paper studies the possibilities to apply the agent-based approach to simulate models of complex systems. This approach allows us to consider a continuum as a set of interacting elements (agents). The behavior of the elements is completely described by local dependencies. At the same time, the laws of functioning of individual elements are accepted as deterministic and they correspond to the fundamental principles of the theory of heat transfer. The possibility to apply a discrete approach for simulating the process of heat transfer by the molecular mechanism has been studied. The general methodology to develop an agent-based deterministic model is described. Its applicability to describe quasi-linear and nonlinear heat conduction processes is considered. The examples of simulation of combustion processes complicated by exothermic and endothermic effects are considered. The advantages and disadvantages of the proposed method are indicated. The results of the study have shown that discrete agent models are a good alternative to classical continuum approaches to study heat transfer processes in inhomogeneous media. The results obtained do not contradict modern approaches to the description of thermal processes. It has also been found that the simulation algorithms used in the agent-based approach are quite universal and easily adapt to changes under the conditions of problem setting. The analysis of the results makes it possible to recommend a discrete agent-based approach to develop simulation models of complex technological processes and systems.

Non-Markovian Inverse Hawkes Processes

Hawkes processes are a class of self-exciting point processes with a clustering effect whose jump rate is determined by its past history. They are generally regarded as continuous-time processes and have been widely applied in a number of fields, such as insurance, finance, queueing, and statistics. The Hawkes model is generally non-Markovian because its future development depends on the timing of past events. However, it can be Markovian under certain circumstances. If the exciting function is an exponential function or a sum of exponential functions, the model can be Markovian with a generator of the model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and self-excitation indicates severity. Inverse Markovian Hawkes processes were introduced by Seol, who studied some asymptotic behaviors. An extended version of inverse Markovian Hawkes processes was also studied by Seol. With this paper, we propose a non-Markovian inverse Hawkes process, which is a more general inverse Hawkes process that features several existing models of self-exciting processes. In particular, we established both the law of large numbers (LLN) and Central limit theorems (CLT) for a newly considered non-Markovian inverse Hawkes process.

A generative model for fBm with deep ReLU neural networks

We provide a large probability bound on the uniform approximation of fractional Brownian motion with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a N-dimensional Gaussian vector, with bounds on the network design (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error of O(N−H) is possible with log⁡(N) hidden layers and O(Nlog⁡N) parameters. Our analysis relies, in the standard Brownian motion case (H=1/2), on the Levy construction and in the general fractional Brownian motion case (H≠1/2), on the Lemarié-Meyer wavelet representation. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes.

Infinite density and relaxation for Lévy walks in an external potential: Hermite polynomial approach.

Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(τ)=v_{0}/sqrt[τ]. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.

Bandit and covariate processes, with finite or non-denumerable set of arms

We introduce herein a new approach to nonparametric multi-armed bandit theory involving both the bandit and the covariate processes. Following Berry et al. (1997), we assume a non-denumerable set of arms for the bandit process. The approach we develop herein can be readily extended to continuous-time processes by using ε-greedy randomization and arm elimination instead of dynamic allocation indices. It also carries out a stochastic search with O(1) expected time for a nearly optimal arm at covariate values in a given set B before applying ε-greedy randomization and arm elimination. The procedure is shown to attain the asymptotically minimal rates for the regret over B.

Risk-sensitive discounted cost criterion for continuous-time Markov decision processes on a general state space

In this paper, we consider risk-sensitive discounted control problem for continuous-time jump Markov processes taking values in general state space. The transition rates of underlying continuous-time jump Markov processes and the cost rates are allowed to be unbounded. Under certain Lyapunov condition, we establish the existence and uniqueness of the solution to the Hamilton–Jacobi–Bellman equation. Also, we prove the existence of optimal risk-sensitive control in the class of Markov control and completely characterized the optimal control.

Cash flow forecasting using Continuous-Time Stochastic Processes

از آنجا که وضعیت نقدینگی مبنای قضاوت بسیاری از اشخاص درباره موقعیت واحد اقتصادی است، این مطلب مورد توجه گروه‌های ذینفع از جمله اعتباردهندگان و سرمایه‌گذاران قرارگرفته‌است. هدف این پژوهش درک جدیدی از رفتار جریان نقد و پیش­بینی مانده نقد است. جامعه آماری پژوهش حاضر مانده نقد روزانه یکساله ۴۸ شعبه بانک معین است. برای این منظور مدل بهینه از بین 4 مدل؛ براونی هندسی، براونی حسابی، واسیسک و مدل ریشه مربعات اصلاح شده در سه سطح میکروسکوپی، مسسکوپی و ماکروسکوپی بررسی گردیده است و مدل براونی هندسی به عنوان مدل بهینه تایید شده است؛ سپس با استفاده از مدل بهینه مذکور پیش‌بینی مانده نقد در افق‌های زمانی متفاوتی صورت‌گرفته‌است. نتیجه نشان می­دهد که قدرت مدلGBM با افزایش طول افق زمانی پیش­بینی ثابت نمی­ماند و با افزایش افق زمانی پیش­بینی، نتایج متفاوتی دارند صحت پیش­بینی مدل با از معیار MAPEدر افق 14روزه در سطح مناسبی می­باشد.

Closed-loop identification of stable and unstable processes with time-delay

This article concerned with the parametric identification problem of continuous-time process dynamics with unknown time-delay. An improved closed-loop approach is developed to carryout process identification in terms of generalized first and second order plus time-delay (GFOPTD & GSOPTD) process models. These generalized models can include both stable and unstable plant dynamics. The key feature of the proposed method over existing techniques is its effective indirect nature, where the mathematical formulation is developed to identify continuous process dynamics and time-delay parameters, indirectly in terms of discrete parameters. The advantage of the proposed indirect method is that it not only allows to accurately identify process parameters but helps in achieving the single parameter based convergence, which is also proved mathematically. The identification accuracy and robustness of the proposed technique in the presence of measurement noise and modeling uncertainties is corroborated with the help of simulation examples on benchmark problems and by using a practical tank process experiment.

Spatial price equilibrium networks with flow-dependent arc multipliers.

The spatial price equilibrium modeling framework, which emphasizes the importance of transportation costs between markets, has been utilized in agricultural, energy, mineral as well as financial applications. In this paper, we construct static and dynamic spatial price equilibrium networks with flow-dependent arc multipliers, which expand the reach of applications. The static model is formulated and analyzed as a variational inequality problem, whereas the dynamic one is formulated as a projected dynamical system, whose set of stationary points coincides with the set of solutions of the variational inequality. Qualitative results are presented, along with an algorithm, the Euler method, which yields a time-discretization of the continuous-time adjustment processes associated with the product shipments from supply markets to demand markets. The algorithm is implemented and applied to compute solutions to numerical examples with flow-dependent arc multipliers addressing losses and/or gains, inspired by perishable agricultural products, and by financial investments. The results in this paper add to the literature on generalized networks as well as that on commodity trade.

On the asymptotic properties of some kernel estimators for continuous-time semi-Markov processes

This paper provides kernel estimators of the main characteristics of a continuous-time semi-Markov process, like conditional and unconditional sojourn times in a state, semi-Markov kernel, etc. The main goal of this paper is to establish asymptotic properties of the semi-Markov kernel estimators and of the sojourn time distribution estimators (conditional and unconditional), as well as of the estimators of the associated Radon-Nikodym derivatives, when the sample size becomes large. The approach is illustrated by considering a three state example with detailed calculus and numerical evaluations.

Generalized regression estimation for continuous time processes with values in functional spaces

We consider two continuous time processes; the first one is valued in a semi-metric space, while the second one is real-valued. In some sense, we extend the results of F. Ferraty and P. Vieu in ``Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination'' (2004), by establishing the convergence, with rates, of the generalized regression function when a real-valued continuous time response is considered. As corollaries, we deduce the convergence of the conditional distribution function as well as conditional quantiles. Note that a parametric rate of convergence in probability is reached while working with a naive kernel.

Zig-Zag Sampling for Discrete Structures and Nonreversible Phylogenetic MCMC

We construct a zig-zag process targeting a posterior distribution defined on a hybrid state space consisting of both discrete and continuous variables. The construction does not require any assumptions on the structure among discrete variables. We demonstrate our method on two examples in genetics based on the Kingman coalescent, showing that the zig-zag process can lead to efficiency gains of up to several orders of magnitude over classical Metropolis–Hastings algorithms, and that it is well suited to parallel computation. Our construction resembles existing techniques for Hamiltonian Monte Carlo on a hybrid state space, which suffers from implementationally and analytically complex boundary crossings when applied to the coalescent. We demonstrate that the continuous-time zig-zag process avoids these complications. Supplementary materials for this article are available online.

Reliability Analysis of Multiperformance Multistate System Considering Performance Conversion Process

A large variety of real engineering systems operate with multiple performance measures that are multistate in nature. These systems are usually modeled as multiperformance multistate systems (MPMSSs). However, existing MPMSS models fail to consider an important aspect, i.e., the performance conversion process. For example, in a combined heat and power (CHP) generating unit, apart from the output heat and electricity, decision-makers are also interested in the unit's capacity to convert gas into electricity and heat. The latter is related to the performance conversion process. This article proposes a framework for the reliability evaluation of performance conversion-based MPMSS. In the proposed MPMSS model, the couplings among different types of performances inside the components are quantified into the multistate performance conversion matrix. The performance conversion structure functions are proposed to derive system performance conversion capability based on the conversion capabilities of the components. Two reliability evaluation methods considering the steady-state performance conversion process and the continuous-time performance conversion process are proposed, respectively. Numerical examples are given to demonstrate the developed methods.

Dynamical Lee-Yang zeros for continuous-time and discrete-time stochastic processes.

We describe classical stochastic processes by using dynamical Lee-Yang zeros. The system is in contact with external leads and the time evolution is described by the two-state classical master equation. The cumulant generating function is written in a factorized form and the current distribution is characterized by the dynamical Lee-Yang zeros. We show that a continuous distribution of zeros is obtained by discretizing the time variable. When the transition probability is a periodically oscillating function of time, the distribution of zeros splits into many parts. We study the geometric property of the current by comparing the result with that of the adiabatic approximation. We also use the Floquet-Magnus expansion in the continuous-time case to study dynamical effects on the current at the fast-driving regime.

Modelling persistent stationary processes in continuous time

This paper presents a novel approach to model continuous time processes that capture two countervailing features of many financial time series: persistency and long term stationarity. The processes introduced by our models exhibit persistent behaviors observed typically in non-stationary time series, but they remain stationary instead of tending towards explosive paths. We provide a relevant statistical theory and its implications on inference and forecasting, presenting both simulation evidence and empirical backing for the existence of, as well as the characteristic behavior for, such a series in real financial time series data.