Theory Of Stochastic Processes Research Articles

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.

Stochastic Exceptional Points for Noise-Assisted Sensing.

Noise is a fundamental challenge for sensors deployed in daily environments for ambient sensing, health monitoring, and wireless networking. Current strategies for noise mitigation rely primarily on reducing or removing noise. Here, we introduce stochastic exceptional points and show the utility to reverse the detrimental effect of noise. The stochastic process theory illustrates that the stochastic exceptional points manifest as fluctuating sensory thresholds that give rise to stochastic resonance, a counterintuitive phenomenon in which the added noise increases the system's ability to detect weak signals. Demonstrations using a wearable wireless sensor show that the stochastic exceptional points lead to more accurate tracking of a person's vital signs during exercise. Our results may lead to a distinct class of sensors that overcome and are enhanced by ambient noise for applications ranging from healthcare to the internet of things.

Temporal dynamics of trauma memory persistence.

Traumatic events lead to distressing memories, but such memories are made all the worse when they intrude to mind unbidden and recurrently. Intrusive memories and flashbacks after trauma are prominent in several mental disorders, including post-traumatic stress disorder and can persist for years. Critically, the reduction of intrusive memories provides a treatment target. While cognitive and descriptive models for psychological trauma exist, these lack formal quantitative structure and robust empirical validation. Here, using techniques from stochastic process theory, we develop a mechanistically driven, quantitative framework to extend understanding of the temporal dynamic processes of trauma memory. Our approach is to develop a probabilistic description of memory mechanisms to link to the broader goals of trauma treatment. We show how the marginal gains of treatments for intrusive memories can be enhanced as key properties (intervention strength and reminder strength) of the intervention and memory consolidation (probability memories are labile) vary. Parametrizing the framework with empirical data highlights that while emerging interventions to reduce occurrence of intrusive memories can be effective, counterintuitively, weakening multiple reactivation cues may help reduce intrusive memories more than would stronger cues. More broadly, the approach provides a quantitative framework for associating neural mechanisms of memory with broader cognitive processes.

A Hörmander–Fock space

ABSTRACT In a recent paper, we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical -problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander–Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by leading to a special entire function that is used to express the kernel function of the Hörmander–Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner–Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes.

CFD Uncertainty Quantification using stochastic spectral methods—Exemplary application to a buoyancy-driven mixing process

The consideration of uncertainties is of particular importance for nuclear reactor safety, where high safety standards for example ensure the integrity of the containment. By means of Computational Fluid Dynamics (CFD), buoyancy-induced mixing processes, which can take place during a severe reactor accident, are investigated. However, the CFD models contain uncertainties, which have a large impact on the present flow and have to be analyzed. The method development for the subsequent Uncertainty Quantification of a representative reactor test containment is conducted using the Differentially Heated Cavity of aspect ratio 4 with Ra=2×109 as a generic test case from the literature. In this way, methods for the future quantification of uncertainties in large-scale industrial applications are established. Results from single-fidelity models such as Unsteady Reynolds-Averaged Navier–Stokes, Large Eddy Simulation and Direct Numerical Simulation are presented and combined to three-level multifidelity models. Stochastic representations of scalar random responses are constructed by means of Polynomial Chaos Expansions and Karhunen–Loève Expansions are derived for the representation of stochastic processes. A new approach for the description of highly dynamic transient processes called Random Field Composition (RFC) is presented, which proposes stochastic model construction through combination of multiple random fields. The presented multifidelity (MF) models achieve high accuracy in combination with a justifiable computational effort. Therefore, MF modeling serves as a promising approach for large-scale applications. Furthermore, it is found, that RFC allows for the description of highly dynamic processes with a reasonable number of simulation runs, if the complexity of the stochastic process representation can be reduced by partitioning the stochasticity into single random fields.

From Pavlov Conditioning to Hebb Learning.

Hebb's learning traces its origin in Pavlov's classical conditioning; however, while the former has been extensively modeled in the past decades (e.g., by the Hopfield model and countless variations on theme), as for the latter, modeling has remained largely unaddressed so far. Furthermore, a mathematical bridge connecting these two pillars is totally lacking. The main difficulty toward this goal lies in the intrinsically different scales of the information involved: Pavlov's theory is about correlations between concepts that are (dynamically) stored in the synaptic matrix as exemplified by the celebrated experiment starring a dog and a ringing bell; conversely, Hebb's theory is about correlations between pairs of neurons as summarized by the famous statement that neurons that fire together wire together. In this letter, we rely on stochastic process theory to prove that as long as we keep neurons' and synapses' timescales largely split, Pavlov's mechanism spontaneously takes place and ultimately gives rise to synaptic weights that recover the Hebbian kernel.

A Hybrid Improved-Whale-Optimization–Simulated-Annealing Algorithm for Trajectory Planning of Quadruped Robots

Traditional trajectory-planning methods are unable to achieve time optimization, resulting in slow response times to unexpected situations. To address this issue and improve the smoothness of joint trajectories and the movement time of quadruped robots, we propose a trajectory-planning method based on time optimization. This approach improves the whale optimization algorithm with simulated annealing (IWOA-SA) together with adaptive weights to prevent the whale optimization algorithm (WOA) from falling into local optima and to balance its exploration and exploitation abilities. We also use Markov chains of stochastic process theory to analyze the global convergence of the proposed algorithm. The results show that our optimization algorithm has stronger optimization ability and stability when compared to six representative algorithms using six different test function suites in multiple dimensions. Additionally, the proposed optimization algorithm consistently constrains the angular velocity of each joint within the range of kinematic constraints and reduces joint running time by approximately 6.25%, which indicates the effectiveness of this algorithm.

A Note on Hidden Markov Models with Application to Criminal Intelligence

Hidden Markov Models (HMMs) which fall under the class of latent variable models have received widespread attention in many fields of applications. HMMs were initially developed and applied within the context of speech recognition. The theoretical framework underpinning the formalism of HMMs has also evolved over time and has found an exalted place in the theory of stochastic processes. The three problems HMMs are used to resolve were discussed alongside their solutions in this paper. An application to criminal intelligence in unraveling the culprit in a situation involving theft was also carried out and results obtained indicated that the HMMs approach offered a similar result with that of the well-established Dynamic Programming approach.

Random quantum maps and their associated quantum Markov chains

The notion of ‘quantum family of maps’ (QFM) has been defined by Piotr Sołtan as a noncommutative analogue of ‘parameterized family of continuous maps’ between locally compact spaces. A QFM between C*-algebras B, A, is given by a pair $$(C,\phi )$$ where C is a C*-algebra and $$\phi :B\rightarrow A\check{\otimes }C$$ is a $$*$$ -morphism. The main goal of this note, is to introduce the notion of ‘random quantum map’ (RQM), which is a noncommutative analogue of ‘random continuous map’ between compact spaces. We define a RQM between B, A, to be given by a triple $$(C,\phi ,\nu )$$ where $$(C,\phi )$$ is a QFM and $$\nu $$ a state (normalized positive linear functional) on C. Our first application of RQMs takes place in theory of completely positive maps (CPM): RQMs give rise canonically to a class of CPMs which we call implemented CPMs. We consider some partial results about the natural and important problem of characterization of implemented CPMs. For instance, using Stinespring’s Theorem, we show that any CPM from B to A is implemented if A is finite-dimensional. Our second application of RQMs takes place in theory of quantum stochastic processes: We show that iterations of any RQM with $$B=A$$ , gives rise to a quantum Markov chain in a sense introduced by Luigi Accardi.

Nonequilibrium Thermodynamics of DNA Nanopore Unzipping.

Using theory and simulations, we carried out a first systematic characterization of DNA unzipping via nanopore translocation. Starting from partially unzipped states, we found three dynamical regimes depending on the applied force f: (i)heterogeneous DNA retraction and rezipping (f<17 pN), (ii)normal (17 pN<f<60 pN), and (iii)anomalous (f>60 pN) drift-diffusive behavior. We show that the normal drift-diffusion regime can be effectively modeled as a one-dimensional stochastic process in a tilted periodic potential. We use the theory of stochastic processes to recover the potential from nonequilibrium unzipping trajectories and show that it corresponds to the free-energy landscape for single-base-pair unzipping. Applying this general approach to other single-molecule systems with periodic potentials ought to yield detailed free-energy landscapes from out-of-equilibrium trajectories.

Continuous simultaneous measurement of position and momentum of a particle

We formulate a model of a quantum particle continuously monitored by detectors measuring simultaneously its position and momentum. We implement the postulate of wave-function collapse by assuming that upon detection the particle is found in one of the meters' states chosen as a discrete subset of coherent states. The dynamics, as observed by the meters, is thus a random sequence of jumps between coherent states. We generate such trajectories using the quantum Monte Carlo wave-function method. For sparsely distributed detectors, we use methods from renewal theory of stochastic processes to obtain some semianalytic results. In particular, the different regimes of dynamics of the free particle are identified and quantitatively discussed: from stroboscopic motion in the case of low interrogation frequency, to delayed dynamics reminiscent of the Zeno effect if monitoring is frequent. For a semicontinuous spatial distribution of meters the emergence of classical trajectories is shown. Their statistical properties are discussed and compared to other detection schemes in which the operation on the system due to measurement corresponds to ``spatial filtering'' of the wave function.