Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ...ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multidimensional version of the Zig-Zag process of Ann. Appl. Probab. 27 (2017) 846–882, a continuous-time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible nonreversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme, that is, the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial preprocessing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.
This paper is concerned with the problem of extended dissipativity-based state estimation for discrete-time Markov jump neural networks (NNs), where the variation of the piecewise time-varying ...transition probabilities of Markov chain is subject to a set of switching signals satisfying an average dwell-time property. The communication links between the NNs and the estimator are assumed to be imperfect, where the phenomena of signal quantization and data packet dropouts occur simultaneously. The aim of this paper is to contribute with a Markov switching estimator design method, which ensures that the resulting error system is extended stochastically dissipative, in the simultaneous presences of packet dropouts and signal quantization stemmed from unreliable communication links. Sufficient conditions for the solvability of such a problem are established. Based on the derived conditions, an explicit expression of the desired Markov switching estimator is presented. Finally, two illustrated examples are given to show the effectiveness of the proposed design method.
Successful development of cloud computing paradigm necessitates accurate performance evaluation of cloud data centers. As exact modeling of cloud centers is not feasible due to the nature of cloud ...centers and diversity of user requests, we describe a novel approximate analytical model for performance evaluation of cloud server farms and solve it to obtain accurate estimation of the complete probability distribution of the request response time and other important performance indicators. The model allows cloud operators to determine the relationship between the number of servers and input buffer size, on one side, and the performance indicators such as mean number of tasks in the system, blocking probability, and probability that a task will obtain immediate service, on the other.
This paper addresses the problem of state estimation for a class of discrete-time stochastic complex networks with a constrained and randomly varying coupling and uncertain measurements. The randomly ...varying coupling is governed by a Markov chain, and the capacity constraint is handled by introducing a logarithmic quantizer. The uncertainty of measurements is modeled by a multiplicative noise. An asynchronous estimator is designed to overcome the difficulty that each node cannot access to the coupling information, and an augmented estimation error system is obtained using the Kronecker product. Sufficient conditions are established, which guarantee that the estimation error system is stochastically stable and achieves the strict (Q, S, R)-γ-dissipativity. Then, the estimator gains are derived using the linear matrix inequality method. Finally, a numerical example is provided to illustrate the effectiveness of the proposed new design techniques.
Inference, prediction, and control of complex dynamical systems from time series is important in many areas, including financial markets, power grid management, climate and weather modeling, or ...molecular dynamics. The analysis of such highly nonlinear dynamical systems is facilitated by the fact that we can often find a (generally nonlinear) transformation of the system coordinates to features in which the dynamics can be excellently approximated by a linear Markovian model. Moreover, the large number of system variables often change collectively on large time- and length-scales, facilitating a low-dimensional analysis in feature space. In this paper, we introduce a variational approach for Markov processes (VAMP) that allows us to find optimal feature mappings and optimal Markovian models of the dynamics from given time series data. The key insight is that the best linear model can be obtained from the top singular components of the Koopman operator. This leads to the definition of a family of score functions called VAMP-
r
which can be calculated from data, and can be employed to optimize a Markovian model. In addition, based on the relationship between the variational scores and approximation errors of Koopman operators, we propose a new VAMP-E score, which can be applied to cross-validation for hyper-parameter optimization and model selection in VAMP. VAMP is valid for both reversible and nonreversible processes and for stationary and nonstationary processes or realizations.
•Two balanced systems with protective devices (PDs) are built for the first time.•New triggering policies of PDs based on the system characteristics are proposed.•The protective effects of PDs differ ...when they are in different operation stages.•Markov process imbedding methods are built to evaluate reliability of two models.
In the existing studies, the positive effects of a protective device on the reliability of balanced systems have not been attached importance, and the research on protective devices is limited and insufficient with respect to the triggering and protection mechanisms. Driven by practical engineering applications, this paper establishes two reliability models for balanced systems with multi-state protective devices from the perspectives of dynamic and static balanced concepts, respectively. The rebalanced mechanisms are considered in the proposed dynamic balanced system, and system imbalance is included in the failure criteria of the proposed static balanced system. In both models, new triggering mechanisms of protective devices are proposed based on the respective characteristics of the balanced systems. Moreover, variable protection mechanisms of the devices are proposed by introducing different reduction factors when the protective device operates in different stages. The protective devices reduce the integrated impact of shocks and internal degradation exerted on the units in the new models. The Markov process imbedding method combined with Monte-Carlo simulation is employed to derive the reliability of the proposed models. The applicability of proposed systems and correctness of the method are validated by practical engineering applications.
This article investigates the resilient proportional-integral observer (PIO) problem for Markov switching memristive neural networks (MSMNNs) with randomly occurring sensor saturation within a ...finite-time interval. The Markov switching of memristive neural networks is regulated by a higher level deterministic switching signal, whose transition probabilities are piecewise time-varying and can be depicted by the average dwell-time strategy. Meanwhile, a Bernoulli stochastic process associated with an uncertain packet arriving rate is adopted to describe the randomly occurring sensor saturation. The aim is to design a resilient PIO such that the augmented dynamic has the property of stochastic finite-time boundedness while meeting the desired <inline-formula> <tex-math notation="LaTeX">\mathcal {H}_{\infty } </tex-math></inline-formula> performance index. By applying the Lyapunov method and the average dwell-time scheme, sufficient criteria are established for MSMNNs, and a unified design method is presented for the existence of the PIO. Lastly, the attained theoretical results are validated via a numerical simulation.
The existence of an optimal reverse-waterfilling algorithm to compute the nonanticipative rate distortion function (NRDF) for time-invariant vector-valued Gauss-Markov processes with a ...mean-squared-error distortion has been an open question since the pioneering work of Tatikonda et al. on stochastic linear control over a communication channel, in 2004. In this article, we derive strong structural properties on the time-invariant multidimensional Gauss-Markov processes that allow for an optimization problem that can be computed optimally via a reverse-waterfilling algorithm. Moreover, we propose an elegant optimal iterative scheme that computes this reverse-waterfilling algorithm. We show that the specific scheme operates much faster than any existing algorithmic approach that solves the same problem optimally and is also scalable. Finally, using our new results, we derive for the first time a nontrivial analytical solution of the asymptotic NRDF using a correlated time-invariant 2-D Gauss-Markov process.
Mean-field deterministic epidemic models have been successful in uncovering several important dynamic properties of stochastic epidemic spreading processes over complex networks. In particular, ...individual-based epidemic models isolate the impact of the network topology on spreading dynamics. In this paper, the existing models are generalized to develop a class of models that includes the spreading process in multilayer complex networks. We provide a detailed description of the stochastic process at the agent level where the agents interact through different layers, each represented by a graph. The set of differential equations that describes the time evolution of the state occupancy probabilities has an exponentially growing state-space size in terms of the number of the agents. Based on a mean-field type approximation, we developed a set of nonlinear differential equations that has linearly growing state-space size. We find that the latter system, referred to as the generalized epidemic mean-field (GEMF) model, has a simple structure characterized by the elements of the adjacency matrices of the network layers and the Laplacian matrices of the transition rate graphs. Finally, we present several examples of epidemic models, including spreading of virus and information in computer networks and spreading of multiple pathogens in a host population .
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