Theory Of Stochastic Processes Research Articles

Spurious self-feedback of mean-field predictions inflates infection curves.

The susceptible-infected-recovered (SIR) model and its variants form the foundation of our understanding of the spread of diseases. Here, each agent can be in one of three states (susceptible, infected, or recovered), and transitions between these states follow a stochastic process. The probability of an agent becoming infected depends on the number of its infected neighbors, hence all agents are correlated. The simplest mean-field theory of the same stochastic process, however, assumes that the agents are statistically independent. This leads to a self-feedback effect in the approximation: when an agent infects its neighbors, this infection may subsequently travel back to the original agent at a later time, leading to a self-infection of the agent which is not present in the underlying stochastic process. We here compute the first-order correction to the mean-field assumption from a systematic expansion, called dynamical TAP theory. This correction, which takes fluctuations up to second order in the interaction strength into account, cancels the self-feedback effect, leading to smaller infection rates. The correction significantly improves predictions compared to mean-field theory. In particular, it captures how sparsity dampens the spread of the disease: this indicates that reducing the number of contacts is more effective than predicted by mean-field models. We further apply the expansion to variants of the SIR model, such as the SIRS model, in which the immunity of an individual to the disease wanes over time. We find that up to the second order, the correction terms in the SIR and SIRS model are equivalent, meaning that fluctuations partially cancel the self-feedback effect even when self-feedback is in principle allowed.

Reliability Evaluation of Multi-State Solar Energy Generating System with Inverters Considering Common Cause Failures

To ensure the efficient functioning of solar energy generation systems, it is crucial to have dependable designs and regular maintenance. However, when these systems or their components operate at multiple working levels, optimizing reliability becomes a complex task for models and analyses. In the context of reliability modeling in solar energy generation systems, researchers often assume that random variables follow an exponential distribution (binary-state representation) as a simplification, although this may not always hold true for real-world engineering systems. In the present paper, a multi-state solar energy generating system with inverters in series configuration is investigated, in which unreliable by-pass changeover switches, common cause failures (CCFs), and multiple repairman vacations are also considered. Furthermore, the arrivals of CCFs and the repair processes of the failed system due to CCFs are governed by different Markovian arrival processes (MAPs), and the lifetimes and repair times of inverters and by-pass changeover switches and the repairman vacation time in the system have different phase-type (PH) distributions. Therefore, the behavior of the system is represented using a Markov process methodology, and reliability measures for the proposed system are derived utilizing aggregated stochastic process theory. Finally, a numerical example and a comparison analysis are presented to demonstrate the findings.

Stochastic process model for interfacial gap of purely normal elastic rough surface contact

In purely normal elastic rough surface contact problems, Persson’s theory of contact shows that the evolution of the probability density function (PDF) of contact pressure with the magnification is governed by a diffusion equation. However, there is no partial differential equation describing the evolution of the PDF of the interfacial gap. In this study, we derive a convection–diffusion equation in terms of the PDF of the interfacial gap based on stochastic process theory, as well as the initial and boundary conditions. A finite difference method is developed to numerically solve the partial differential equation. The predicted PDF of the interfacial gap agrees well with that by Green’s Function Molecular Dynamics (GFMD) and other variants of Persson’s theory of contact at high load ranges. At low load ranges, the obvious deviation between the present work and GFMD is attributed to the overestimated mean interfacial gap and oversimplified magnification-dependent diffusion coefficient used in the present model. As one of its direct application, we show that the present work can effectively solve the adhesive contact problem under the DMT limit. The current study provides an alternative methodology for determining the PDF of the interfacial gap and a unified framework for solving the complementary problem of random contact pressure and random interfacial gap based on stochastic process theory.

Auxiliary model‐based recursive least squares and stochastic gradient algorithms and convergence analysis for feedback nonlinear output‐error systems

SummaryThis paper deals with the problem of the parameter estimation for feedback nonlinear output‐error systems. The auxiliary model‐based recursive least squares algorithm and the auxiliary model‐based stochastic gradient algorithm are derived for parameter estimation. Based on the stochastic process theory, the convergence of the proposed algorithms are proved. The simulation results indicate that the proposed algorithms can estimate the parameters of feedback nonlinear output‐error systems effectively.

Modeling evolutionary power spectral density functions of strong earthquakes via copulas

This paper proposes a new approach for analyzing seismic accelerograms using the evolutionary Power Spectral Density function (EPSD). The accelerogram of an earthquake can be accurately modeled and simulated from its spectrogram, based on the oscillatory stochastic processes theory. To adequately characterize a spectrogram that is consistent with the response spectra, a parametric model with 16 parameters is proposed. This model describes the square of the amplitude spectrum, an envelope of the square of the accelerogram, and a copula that constructs a time–frequency model from the time and frequency marginals. The use of copulas to model a bivariate probability distribution is a common practice in statistics, particularly when the marginal distributions are known. The periodogram can be viewed as an unnormalized probability density function, where the total energy serves as the normalization constant, since the total energy of a seismic motion is always finite. Additionally, a reduced model consisting of only 10 parameters is presented, which may be especially valuable when only shear wave effects are relevant.

Learning minimal representations of stochastic processes with variational autoencoders.

Stochastic processes have found numerous applications in science, as they are broadly used to model a variety of natural phenomena. Due to their intrinsic randomness and uncertainty, they are, however, difficult to characterize. Here, we introduce an unsupervised machine learning approach to determine the minimal set of parameters required to effectively describe the dynamics of a stochastic process. Our method builds upon an extended β-variational autoencoder architecture. By means of simulated data sets corresponding to paradigmatic diffusion models, we showcase its effectiveness in extracting the minimal relevant parameters that accurately describe these dynamics. Furthermore, the method enables the generation of new trajectories that faithfully replicate the expected stochastic behavior. Overall, our approach enables the autonomous discovery of unknown parameters describing stochastic processes, hence enhancing our comprehension of complex phenomena across various fields.

Modeling coupling impacts of self-healing mechanisms and dynamic environments on systems subject to dependent failure processes

Modern smart products are endowed with the capability to autonomously repair damage from random shocks, thereby highlighting the critical necessity to incorporate this self-healing attribute in reliability assessments. This paper investigates periodically inspected self-healing systems that are exposed to dependent competing failure processes within dynamically evolving environments. The impact of environmental changes is manifested in that both the natural degradation process and the self-healing behavior are governed by distinct laws in varying environments. Explicit formulas for the system reliability and availability are analytically derived using the stochastic process theory, and their correctness is verified by proposed simulation algorithms respectively. Meanwhile, the inspection period is optimized by minimizing the long-run average cost rate. Lastly, a practical example of semiconductor lasers is applied to showcase the application of the presented methods. The sensitivity analysis reveals that increasing the self-healing time threshold or the shock arrival rate can improve the system performance while lowering the long-run average cost rate.

Study on joint effects of modal dispersion, mode-dependent loss and noise by unified density-matrix formalism

We generalize the density-matrix formalism to provide a simple and unified framework for the study of modal dispersion (MD), mode-dependent loss (MDL) and noise in mode-division-multiplexing communication systems. Fundamental concepts and formulas in the original formalism are generalized, while new elements and associated equations are introduced and/or derived. Due to the ability of the density operator to characterize partially-coherent fields, the generalized density-matrix formalism is particularly useful for analyzing the field propagation in presence of noise. Combining the unified density-matrix formalism and the stochastic-process theory, we conduct detailed numerical simulation and analysis on the modal properties of a randomly-perturbed 6-mode fiber with loss and noise. We find that although the MDL only slightly affects the statistics of the group delays, it can cause significant non-orthogonality among the principal modes, which might induce extra crosstalk and temporal spreading in the signal pulses. Furthermore, with the coherency of fields in the fiber being reduced by the noise, the MDL can induce substantial fluctuation in the field coherency even though it does not significantly affect the averaged field coherency.

On the structure factor of jammed particle configurations on the one-dimensional lattice

A broad class of blocked or jammed configurations of particles on the one-dimensional lattice can be characterized in terms of local rules involving only the lengths of clusters of particles (occupied sites) and of holes (empty sites). Examples of physical relevance include the metastable states reached by the zero-temperature dynamics of kinetically constrained spin chains, the attractors of totally irreversible processes such as random sequential adsorption, and arrays of Rydberg atoms in the blockade regime. The configurational entropy of ensembles of such blocked configurations has been investigated recently by means of an approach inspired from the theory of stochastic renewal processes. This approach provides a valuable alternative to the more traditional transfer-matrix formalism. We show that the renewal approach is also an efficient tool to investigate a range of observables in uniform ensembles of blocked configurations, besides their configurational entropy. The main emphasis is on their structure factor and correlation function.

New deterministic model for calculating mesh stiffness and damping of rough-surface gears considering elastic–plastic contact and energy-dissipation mechanism

For a long time, meshing parameter calculations and dynamic modeling of gear systems have assumed smooth tooth surfaces, disregarding the impact of actual three-dimensional (3D) microscopic topography. Based on elastic–plastic contact and energy-dissipation mechanisms, this work developed a comprehensive deterministic model that integrates 3D rough surface modeling, reconstruction, and contact analysis, for calculating the time-varying mesh stiffness (TVMS) and mesh damping (TVMD) of deterministic rough-surface gears. Distinct from fractal and statistical models, the deterministic model analyzes deterministic, real 3D rough surfaces rather than simple surfaces defined by a few parameters. Utilizing stochastic process theory, different-topography surfaces are obtained using height and spatial features and reconstructed employing watershed algorithm and ellipsoidal asperity fitting. Subsequently, a deterministic elastic–plastic contact model based on equivalent rough curved-surface contact is constructed, emphasizing local contact states and energy dissipation for calculating contact stiffness (CS) and damping (CD) of gears. Accordingly, a meshing characteristics analysis model is established for calculating TVMS, TVMD, and load sharing ratio (LSR). Experimental validation and comparisons with several statistical and gear contact models demonstrate the effectiveness, reliability, and superiority of the proposed deterministic model. Multi-parameter impact analysis reveals small Sq (meaning low roughness Sa) and Ssk and high load correlate with increased CS and TVMS, while CD and TVMD depend on their interaction and hardness. Overall, by bridging the micro-surface topography and macro-meshing parameters, our model provides essential insights for the design and manufacturing of high-performance gear to reduce vibration and noise.

A stochastic process representation for time warping functions

Time warping function provides a mathematical representation to measure phase variability in functional data. Recent studies have developed various approaches to estimate optimal warping between functions. However, a principled, linear, generative representation on time warping functions is still under-explored. This is highly challenging because the warping functions are non-linear in the conventional L2 space. To address this problem, a new linear warping space is defined and a stochastic process representation is proposed to characterize time warping functions. The key is to define an inner-product structure on the time warping space, followed by a transformation which maps the warping functions into a sub-space of the L2 space. With certain constraints on the warping functions, this transformation is an isometric isomorphism. In the transformed space, the L2 basis in the Hilbert space is adopted for representation, which can be easily utilized to generate time warping functions by using different types of stochastic process. The effectiveness of this representation is demonstrated through its use as a new penalty in the penalized function registration, accompanied by an efficient gradient method to minimize the cost function. The new penalized method is illustrated through simulations that properly characterize nonuniform and correlated constraints in the time domain. Furthermore, this representation is utilized to develop a boxplot for warping functions, which can estimate templates and identify warping outliers. Finally, this representation is applied to a Covid-19 dataset to construct boxplots and identify states with outlying growth patterns.

A Continuous-Time Markov Chain Model for the Spread of COVID-19

Summary Since late 2019 the novel coronavirus, also known as COVID-19, has caused a pandemic that persists. This paper shows how a continuous-time Markov chain model for the spread of COVID-19 can be used to explain, and justify to undergraduate students, strategies now being used in attempts to control the virus. The material in the paper is written at the level of students who are taking an introductory course on the theory and applications of stochastic processes.

Relating Intermittency and Inverse Cascade to Stochastic Entropy in Solar Wind Turbulence

Turbulent energy transfer in nearly collisionless plasmas can be conceptualized as a scale-to-scale Langevin process. Hence, the statistics of magnetic field fluctuations can be embedded in the framework of stochastic process theory. In this work, we investigate the statistical properties of the pristine solar wind as observed by Parker Solar Probe by defining the cascade trajectories of magnetic field increments and by estimating the stochastic entropy variation along them. Through the stochastic entropy, we can identify two regimes where fluctuations exhibit contrasting statistical properties. In the inertial range, the entropy production is associated with an increase of the flatness indicating the occurrence of intermittency. Otherwise, trajectories associated with an entropy consumption exhibit global scale invariance. In the transition region toward ion scales, the phenomenology switches: entropy-consuming trajectories exhibit a sudden flatness increase, associated with the presence of small-scale intermittency, while entropy-producing trajectories display a nearly constant flatness. Results are interpreted in terms of physical processes consistent with an accumulation of energy at ion scales.

Stability analysis of switched stochastic nonlinear functional systems

AbstractThis article is concerned with stochastic finite time stability (SFTS) and stochastic fixed time stability (SFIXTS) of switched stochastic nonlinear functional systems. By using the Lyapunov–Razumikhin approach, stochastic analysis techniques, and stochastic process theory, several novel theorems on SFTS and SFIXTS are established. Concurrently, the estimations of stochastic settling time are given. Different from the previous general results, this article considers the effects of noise, time‐delays, and switches. The SFTS and SFIXTS conditions are established, which are quite different from the traditional exponential stability or almost sure stability. Furthermore, two examples are given to illustrate the effectiveness of the obtained results.

The Weight of a Ten-Pound Bag of Potatoes

Summary The weight of a ten-pound bag of potatoes is almost certainly not exactly ten pounds. Rather, it is a random variable with a minimum of ten pounds. We investigate the distribution of these weights in the context of a renewal theorem from the theory of stochastic processes and provide a straightforward demonstration of this theorem using basic tools from calculus.

Modelling of wind and photovoltaic power output considering dynamic spatio-temporal correlation

The simulation technology of wind and solar power output can provide data support for the planning of new energy stations and the optimization and scheduling of power systems. In order to solve the problem that the existing output models can not accurately describe the dynamic spatio-temporal dependence between wind and solar output, a dynamic spatiotemporal correlation model of wind and solar output based on Markov stochastic process and Copula function theory is proposed. Firstly, wind power and photovoltaic output are regarded as a stochastic process, and the time autocorrelation models of wind power and photovoltaic output are constructed based on a one-dimensional Markov chain and hybrid Copula function. Next, two one-dimensional Markov chains were coupled to form a two-dimensional Markov chain, and a spatial correlation model between wind and solar output was constructed using the dynamic SJC Copula function. Take the measured data of adjacent wind farms and photovoltaic power stations in Hami, Xinjiang as an example for simulation. The simulation results show that the proposed model can effectively reflect the spatio-temporal correlation of the original data and reflect the dynamic changes in the correlation between wind and solar energy.

UAV Path Planning Based on Deep Reinforcement Learning

Abstract Path planning is one of the very important aspects of UAV navigation control, which refers to the UAV searching for an optimal or near-optimal route from the starting point to the end point according to the performance indexes such as time, distance, et al. The path planning problem has a long history and has more abundant algorithms. The path planning problem has a long history and a rich set of algorithms, but most of the current algorithms require a known environment, however, in most cases, the environment model is difficult to describe and obtain, and the algorithms perform less satisfactorily. To address the above problems, this paper proposes a UAV path planning method based on deep reinforcement learning algorithm. Based on the OpenAI-GYM architecture, a 3D map environment model is constructed, with the map grid as the state set and 26 actions as the action set, which does not need an environment model and relies on its own interaction with the environment to complete the path planning task. The algorithm is based on stochastic process theory, modeling the path planning problem as a Markov Decision Process (MDP), fitting the UAV path planning decision function and state-action function, and designing the DQN algorithm model according to the state space, action space and network structure. The algorithm enables the intelligences to carry out strategy iteration efficiently. Through simulation, the DQN algorithm is verified to avoid obstacles and complete the path planning task in only about 160 rounds, which validates the effectiveness of the proposed path planning algorithm.

Stochastic and corpuscular theory of (polarized) light

Both the corpuscular theory of light and the theory of stochastic processes are well known in the literature. However, they are not systematically used together for description of optical phenomena. There are optical phenomena, such as the well-known three-polarizer experiment or other phenomena related to polarization of light, which have never been quantitatively and qualitatively explained using the concept of quantum of light (photon). The situation changed in 2022 when stochastic memoryless and independent (IM) process formulated within the framework of the theory of stochastic processes was introduced. It is suitable for determination of probability (density) functions characterizing interaction (transmission or reflection) of individual photons with optical elements on the basis of experimental data. The process has memoryless (Markov) property, and it is assumed that the interactions of individual photons with an optical system are independent. Formulae needed for analysis of data in the context of polarization of light are derived. An example analysis of the three-polarizer experiment is performed and numerical result of the probability (density) functions are determined. These original results were missing in the literature. The results imply that the possibilities of the corpuscular theory of light to describe optical phenomena can be significantly extended with the help of stochastic IM process and the theory of stochastic processes in general.

A Bayesian Augmented-Learning framework for spectral uncertainty quantification of incomplete records of stochastic processes

A novel Bayesian Augmented-Learning framework, quantifying the uncertainty of spectral representations of stochastic processes in the presence of missing data, is developed. The approach combines additional information (prior domain knowledge) of the physical processes with real, yet incomplete, observations. Bayesian deep learning models are trained to learn the underlying stochastic process, probabilistically capturing temporal dynamics, from the physics-based pre-simulated data. An ensemble of time domain reconstructions are provided through recurrent computations using the learned Bayesian models. Models are characterized by the posterior distribution of model parameters, whereby uncertainties over learned models, reconstructions and spectral representations are all quantified. In particular, three recurrent neural network architectures, (namely long short-term memory, or LSTM, LSTM-Autoencoder, LSTM-Autoencoder with teacher forcing mechanism), which are implemented in a Bayesian framework through stochastic variational inference, are investigated and compared under many missing data scenarios. An example from stochastic dynamics pertaining to the characterization of earthquake-induced stochastic excitations even when the source load data records are incomplete is used to illustrate the framework. Results highlight the superiority of the proposed approach, which adopts additional information, and the versatility of outputting many forms of results in a probabilistic manner.

Finite-Time Ruin Probabilities of Bidimensional Risk Models with Correlated Brownian Motions

The present work concerns the finite-time ruin probabilities for several bidimensional risk models with constant interest force and correlated Brownian motions. Under the condition that the two Brownian motions {B1(t),t≥0} and {B2(t),t≥0} are correlated, we establish new results for the finite-time ruin probabilities. Our research enriches the development of the ruin theory with heavy tails in unidimensional risk models and the dependence theory of stochastic processes.