Showing papers on "Stochastic process published in 2019"

Advances in Variational Inference

Abstract: Many modern unsupervised or semi-supervised machine learning algorithms rely on Bayesian probabilistic models. These models are usually intractable and thus require approximate inference. Variational inference (VI) lets us approximate a high-dimensional Bayesian posterior with a simpler variational distribution by solving an optimization problem. This approach has been successfully applied to various models and large-scale applications. In this review, we give an overview of recent trends in variational inference. We first introduce standard mean field variational inference, then review recent advances focusing on the following aspects: (a) scalable VI, which includes stochastic approximations, (b) generic VI, which extends the applicability of VI to a large class of otherwise intractable models, such as non-conjugate models, (c) accurate VI, which includes variational models beyond the mean field approximation or with atypical divergences, and (d) amortized VI, which implements the inference over local latent variables with inference networks. Finally, we provide a summary of promising future research directions.

Fuzzy Adaptive Finite-Time Control Design for Nontriangular Stochastic Nonlinear Systems

Abstract: This paper solves the stochastically finite-time control problem for uncertain stochastic nonlinear systems in nontriangular form. The considered controlled plants are different from the previous results of finite-time control systems, which are the multiple-input and multiple-output (MIMO) stochastic systems with the unknown functions consisting of all states, stochastic disturbance, and immeasurable states. Fuzzy logic systems and a state filter are used to model the uncertain systems and estimate the immeasurable states, respectively. Based on the finite-time theory and It $\hat{o}$ differential equation, a novel stochastically finite-time stability theorem is first raised. By combining the novel criterion and backstepping technique, an adaptive fuzzy stochastically finite-time control method is proposed. It is testified that all signals in the closed-loop signals are semiglobal finite-time stable in probability, and the tracking performances are well. Simulation example results further show the effectiveness of the proposed approach.

Applied Stochastic Differential Equations

Abstract: Stochastic differential equations are differential equations whose solutions are stochastic processes. They exhibit appealing mathematical properties that are useful in modeling uncertainties and noisy phenomena in many disciplines. This book is motivated by applications of stochastic differential equations in target tracking and medical technology and, in particular, their use in methodologies such as filtering, smoothing, parameter estimation, and machine learning. It builds an intuitive hands-on understanding of what stochastic differential equations are all about, but also covers the essentials of Ito calculus, the central theorems in the field, and such approximation schemes as stochastic Runge–Kutta. Greater emphasis is given to solution methods than to analysis of theoretical properties of the equations. The book's practical approach assumes only prior understanding of ordinary differential equations. The numerous worked examples and end-of-chapter exercises include application-driven derivations and computational assignments. MATLAB/Octave source code is available for download, promoting hands-on work with the methods.

Multivariate output analysis for Markov chain Monte Carlo

Abstract: Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.

Output-Feedback Control for Stochastic Nonlinear Systems Subject to Input Saturation and Time-Varying Delay

Abstract: This paper is concerned with the problem of global output-feedback control for a class of stochastic nonlinear time-delay systems in the presence of input saturation and unmeasurable states. As a distinctive feature, the growth assumptions imposed on the drift and diffusion terms are proven to be unnecessary, which can be removed through a technical lemma. In addition, by introducing an auxiliary system whose order is the same as the considered system, and using Lyapunov–Krasovskii functional, the adverse effects generated by input saturation and time-varying delay are eliminated. Then, based on state-observer and backstepping recursive design, an output-feedback controller is constructed to render the closed-loop system be globally bounded almost surely. Particularly, instead of converging to an arbitrarily small neighborhood of zero as in related results, the tracking error is ensured to be tuned by the design parameters and an input saturation error in the mean quartic sense. Finally, a stochastic chemical reactor system is established and shown to demonstrate the effectiveness of the proposed scheme.

Reliability constraint stochastic UC by considering the correlation of random variables with Copula theory

Abstract: This essay performs a reliability constraint stochastic model for unit commitment problem by considering generation and transmission constraints with high wind penetration and volatility of load demands. This query is expressed as a MILP that is based on the linear direct current model. The proposed approach models uncertainty of wind generators output power, load demand fluctuations and stochastic elements outage of the system like generators and transmission lines. In this paper, stochastic interdependence between random variables like wind speed and load demand is recognized. To establish the probability distribution of these correlated random variables, Copula theory is applied. Correlation structure between wind speed of different locations and a group of loads existing in the same area is investigated and studied based on historical data. For representing these uncertainties in the stochastic unit commitment problem, possible scenarios are generated with Monte Carlo simulation method. The reliability constraints are utilized in each scenario to evaluate the feasibility of solutions from a reliability point. The introduced stochastic UC is executed on the RTS 96-bus test system. Numerical results demonstrate the advantages of implementing stochastic programming on the UC problem by taking into account the intermittent behavior of wind energy and load inconstancy.

Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems

Abstract: We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.

Comparison of stochastic and machine learning methods for multi-step ahead forecasting of hydrological processes

Abstract: Research within the field of hydrology often focuses on the statistical problem of comparing stochastic to machine learning (ML) forecasting methods. The performed comparisons are based on case studies, while a study providing large-scale results on the subject is missing. Herein, we compare 11 stochastic and 9 ML methods regarding their multi-step ahead forecasting properties by conducting 12 extensive computational experiments based on simulations. Each of these experiments uses 2000 time series generated by linear stationary stochastic processes. We conduct each simulation experiment twice; the first time using time series of 100 values and the second time using time series of 300 values. Additionally, we conduct a real-world experiment using 405 mean annual river discharge time series of 100 values. We quantify the forecasting performance of the methods using 18 metrics. The results indicate that stochastic and ML methods may produce equally useful forecasts.

Neural Jump Stochastic Differential Equations

Abstract: Many time series are effectively generated by a combination of deterministic continuous flows along with discrete jumps sparked by stochastic events. However, we usually do not have the equation of motion describing the flows, or how they are affected by jumps. To this end, we introduce Neural Jump Stochastic Differential Equations that provide a data-driven approach to learn continuous and discrete dynamic behavior, i.e., hybrid systems that both flow and jump. Our approach extends the framework of Neural Ordinary Differential Equations with a stochastic process term that models discrete events. We then model temporal point processes with a piecewise-continuous latent trajectory, where the discontinuities are caused by stochastic events whose conditional intensity depends on the latent state. We demonstrate the predictive capabilities of our model on a range of synthetic and real-world marked point process datasets, including classical point processes (such as Hawkes processes), awards on Stack Overflow, medical records, and earthquake monitoring.

Adaptive Neural-Network-Based Dynamic Surface Control for Stochastic Interconnected Nonlinear Nonstrict-Feedback Systems With Dead Zone

Abstract: In this paper, an adaptive neural-network-based dynamic surface control (DSC) method is proposed for a class of stochastic interconnected nonlinear nonstrict-feedback systems with unmeasurable states and dead zone input First, an appropriate state observer is constructed to estimate the unmeasured state variables of the stochastic interconnected system Then radial basis function neural networks combined with adaptive backstepping technique are applied to model the unknown nonlinear system functions of the stochastic interconnected system; and the DSC method is adopted to ensure the computation burden is greatly reduced Furthermore, the proposed controllers guarantee that the closed-loop stochastic interconnected system is semi-globally bounded stable in probability In the end, two simulation examples are provided to show the effectiveness and practicability of the proposed control scheme

Stabilization of Stochastic Uncertain Complex-Valued Delayed Networks via Aperiodically Intermittent Nonlinear Control

Abstract: This paper concerns the stabilization problem of stochastic uncertain complex-valued delayed networks (SUCVDNs) via aperiodically intermittent nonlinear control. It is worth noting that the intermittent control is aperiodic and nonlinear. As a special case, when each control width tends to zero, intermittent control becomes impulsive control. Hence, the stabilization problem of SUCVDNs via impulsive control is also studied. The main method is the combination of the Lyapunov method and Kirchhoff’s matrix tree theorem in graph theory, which does not require us to solve any linear matrix inequalities. Different from previous results on complex-valued system, we consider the complex-valued system on complex space directly by using complex version Ito’s formula and (conjugate) ${\mathbb {R}}$ derivative. Then we give some sufficient criteria, which have a close relationship with the control interval and the topological structure of the considered network. The main results are applied to a class of stochastic complex-valued coupled oscillators, and numerical examples are also presented to show the effectiveness of the control strategies.

Effects of refractory period on stochastic resetting

Abstract: We consider a stochastic process undergoing resetting after which a random refractory period is imposed. In this period the process is quiescent and remains at the resetting position. Using a first-renewal approach, we compute exactly the stationary position distribution and analyse the emergence of a delta peak at the resetting position. In the case of a power-law distribution for the refractory period we find slow relaxation. We generalise our results to the case when the resetting period and the refractory period are correlated, by computing the Laplace transform of the survival probability of the process and the mean first passage time, i.e., the mean time to completion of a task. We also compute exactly the joint distribution of the active and absorption time to a fixed target.

Fuzzy Peak-to-Peak Filtering for Networked Nonlinear Systems With Multipath Data Packet Dropouts

Abstract: In this paper, the peak-to-peak filtering problem is studied for a class of networked nonlinear systems. The nonlinear physical plant is represented by the Takagi–Sugeno fuzzy system. Assume that the data packet dropout phenomenon occurs when the measurement output signal and the performance output signal of the nonlinear systems are transmitted by the digital communication channel. Two stochastic variables satisfying the Bernoulli random binary distribution are used to model this phenomenon. The motive of this paper is to design a peak-to-peak filter such that the filtering error system is stochastically stable and the prescribed peak-to-peak performance index is guaranteed. The developed theoretical results for designing a peak-to-peak filter are expressed in the form of linear matrix inequalities. Finally, a simulation example is presented to illustrate the validity of theoretical analysis.

Solution Properties of a 3D Stochastic Euler Fluid Equation

Abstract: We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.

A Distributed Synchronous SGD Algorithm with Global Top-k Sparsification for Low Bandwidth Networks

Abstract: Distributed synchronous stochastic gradient descent (S-SGD) with data parallelism has been widely used in training large-scale deep neural networks (DNNs), but it typically requires very high communication bandwidth between computational workers (e.g., GPUs) to exchange gradients iteratively. Recently, Top-k sparsification techniques have been proposed to reduce the volume of data to be exchanged among workers and thus alleviate the network pressure. Top-k sparsification can zero-out a significant portion of gradients without impacting the model convergence. However, the sparse gradients should be transferred with their indices, and the irregular indices make the sparse gradients aggregation difficult. Current methods that use AllGather to accumulate the sparse gradients have a communication complexity of O(kP), where P is the number of workers, which is inefficient on low bandwidth networks with a large number of workers. We observe that not all top-k gradients from P workers are needed for the model update, and therefore we propose a novel global Top-k (gTop-k) sparsification mechanism to address the difficulty of aggregating sparse gradients. Specifically, we choose global top-k largest absolute values of gradients from P workers, instead of accumulating all local top-k gradients to update the model in each iteration. The gradient aggregation method based on gTop-k sparsification, namely gTopKAllReduce, reduces the communication complexity from O(kP) to O(k log P). Through extensive experiments on different DNNs, we verify that gTop-k S-SGD has nearly consistent convergence performance with S-SGD, and it has only slight degradations on generalization performance. In terms of scaling efficiency, we evaluate gTop-k on a cluster with 32 GPU machines which are interconnected with 1 Gbps Ethernet. The experimental results show that our method achieves 2.7-12× higher scaling efficiency than S-SGD with dense gradients and 1.1-1.7× improvement than the existing Top-k S-SGD.

Telegraphic processes with stochastic resetting.

Abstract: We investigate the effects of resetting mechanisms on random processes that follow the telegrapher's equation instead of the usual diffusion equation. We thus study the consequences of a finite speed of signal propagation, the landmark of telegraphic processes. Likewise diffusion processes where signal propagation is instantaneous, we show that in telegraphic processes, where signal propagation is not instantaneous, random resettings also stabilize the random walk around the resetting position and optimize the mean first-arrival time. We also obtain the exact evolution equations for the probability density of the combined process and study the limiting cases.

Data-Based Distributionally Robust Stochastic Optimal Power Flow—Part I: Methodologies

Abstract: We propose a data-based method to solve a multi-stage stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The framework explicitly combines multi-stage feedback policies with any forecasting method and historical forecast error data. The objective is to determine power scheduling policies for controllable devices in a power network to balance operational cost and conditional value-at-risk of device and network constraint violations. These decisions include both nominal power schedules and reserve policies, which specify planned reactions to forecast errors in order to accommodate fluctuating renewable energy sources. Instead of assuming that the uncertainties across the networks follow prescribed probability distributions, we consider ambiguity sets of distributions centered around a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real unknown data-generating distribution, we formulate a multi-stage distributionally robust OPF problem to compute control policies that are robust to both forecast errors and sampling errors inherent in the dataset. Two specific data-based distributionally robust stochastic OPF problems are proposed for distribution networks and transmission systems.

Convergence Rate Analysis of a Stochastic Trust-Region Method via Supermartingales

Abstract: We propose a novel framework for analyzing convergence rates of stochastic optimization algorithms with adaptive step sizes. This framework is based on analyzing properties of an underlying generic...

Stability and Stabilization of Boolean Networks With Stochastic Delays

Abstract: In this paper, stability and stabilization of Boolean networks with stochastic delays are studied via semi-tensor product of matrices. The stochastic delays, randomly attaining finite values, are modeled by Markov chains. By utilizing an augmented method, the considered Boolean network is first converted into two coupled Markovian switching systems without delays. Then, some stochastic stability results are obtained based on stability results of positive systems. Subsequently, the stabilization of Boolean networks with stochastic delays is further investigated, and an equivalent condition for the existence of feedback controllers is provided in terms of a convex programming problem, which can be easily solved and also conveniently applied to design controller gains. Finally, numerical examples are given to illustrate feasibility of the obtained results.

Fixed-Time Stochastic Synchronization of Complex Networks via Continuous Control

Abstract: This paper investigates the fixed-time synchronization (FDTS) of complex networks with stochastic perturbations. A new control scheme is designed to realize the synchronization goal. Moreover, the designed controller without sign function is continuous, which means the chattering phenomenon in some previous results can be avoided. By constructing Lyapunov functionals, using the properties of the Weiner process as well as applying a designed comparison system, several FDTS criteria are obtained. Synchronization criteria of this paper are very general and can be utilized in directed and undirected weighted networks. Numerical simulations are given to illustrate the theoretical results.

Affine Volterra processes

Abstract: We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier--Laplace functional in terms of the solution of an associated system of deterministic integral equations, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and simplify recent results in the literature on rough volatility models in finance.

First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate.

Abstract: First passage in a stochastic process may be influenced by the presence of an external confining potential, as well as "stochastic resetting" in which the process is repeatedly reset back to its initial position. Here, we study the interplay between these two strategies, for a diffusing particle in a one-dimensional trapping potential V(x), being randomly reset at a constant rate r. Stochastic resetting has been of great interest as it is known to provide an "optimal rate" (r_{*}) at which the mean first passage time is a minimum. On the other hand, an attractive potential also assists in the first capture process. Interestingly, we find that for a sufficiently strong external potential, the advantageous optimal resetting rate vanishes (i.e., r_{*}→0). We derive a condition for this optimal resetting rate vanishing transition, which is continuous. We study this problem for various functional forms of V(x), some analytically, and the rest numerically. We find that the optimal rate r_{*} vanishes with a deviation from the critical strength of the potential as a power law with an exponent β which appears to be universal.

A Framework for Time-Consistent, Risk-Sensitive Model Predictive Control: Theory and Algorithms

Abstract: In this paper, we present a framework for risk-sensitive model predictive control (MPC) of linear systems affected by stochastic multiplicative uncertainty. Our key innovation is to consider a time-consistent, dynamic risk evaluation of the cumulative cost as the objective function to be minimized. This framework is axiomatically justified in terms of time-consistency of risk assessments, is amenable to dynamic optimization, and is unifying in the sense that it captures a full range of risk preferences from risk neutral (i.e., expectation) to worst case. Within this framework, we propose and analyze an online risk-sensitive MPC algorithm that is provably stabilizing. Furthermore, by exploiting the dual representation of time-consistent, dynamic risk measures, we cast the computation of the MPC control law as a convex optimization problem amenable to real-time implementation. Simulation results are presented and discussed.

Scaled Brownian motion with renewal resetting.

Abstract: We investigate an intermittent stochastic process in which the diffusive motion with time-dependent diffusion coefficient $D(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}\ensuremath{-}1}$ with $\ensuremath{\alpha}g0$ (scaled Brownian motion) is stochastically reset to its initial position, and starts anew. In the present work we discuss the situation in which the memory on the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. The situation when the resetting of the coordinate does not affect the diffusion coefficient's time dependence is considered in the other work of this series [A. S. Bodrova et al., Phys. Rev. E 100, 012119 (2019)]. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different. In addition we discuss the first-passage properties of the scaled Brownian motion with renewal resetting and consider the dependence of the efficiency of search on the parameters of the process.

Duality-Free Decomposition Based Data-Driven Stochastic Security-Constrained Unit Commitment

Abstract: To incorporate the superiority of both stochastic and robust approaches, a data-driven stochastic optimization is employed to solve the security-constrained unit commitment model. This approach makes the most use of the historical data to generate a set of possible probability distributions for wind power outputs and then it optimizes the unit commitment under the worst-case probability distribution. However, this model suffers from huge computational burden, as a large number of scenarios are considered. To tackle this issue, a duality-free decomposition method is proposed in this paper. This approach does not require doing duality, which can save a large set of dual variables and constraints, and therefore reduces the computational burden. In addition, the inner max-min problem has a special mathematical structure, where the scenarios have the similar constraint. Thus, the max-min problem can be decomposed into independent subproblems to be solved in parallel, which further improves the computational efficiency. A numerical study on an IEEE 118-bus system with practical data of a wind power system has demonstrated the effectiveness of the proposal.

Particle filtering-based recursive identification for controlled auto-regressive systems with quantised output

Abstract: Recursive prediction error method is one of the main tools for analysis of controlled auto-regressive systems with quantised output. In this study, a recursive identification algorithm is proposed based on the auxiliary model principle by modifying the standard stochastic gradient algorithm. To improve the convergence performance of the algorithm, a particle filtering technique, which approximates the posterior probability density function with a weighted set of discrete random sampling points is utilised to correct the linear output estimates. It can exclude those invalid particles according to their corresponding weights. The performance of the particle filtering technique-based algorithm is much better than that of the auxiliary model-based one. Finally, results are verified by examples from simulation and engineering.