265 research outputs found

    Generalised particle filters

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    The ability to analyse, interpret and make inferences about evolving dynamical systems is of great importance in different areas of the world we live in today. Various examples include the control of engineering systems, data assimilation in meteorology, volatility estimation in financial markets, computer vision and vehicle tracking. In general, the dynamical systems are not directly observable, quite often only partial information, which is deteriorated by the presence noise, is available. This naturally leads us to the area of stochastic filtering, which is defined as the estimation of dynamical systems whose trajectory is modelled by a stochastic process called the signal, given the information accumulated from its partial observation. A massive scientific and computational effort is dedicated to the development of various tools for approximating the solution of the filtering problem. Classical PDE methods can be successful, particularly if the state space has low dimensions (one to three). In higher dimensions (up to ten), a class of numerical methods called particle filters have proved the most successful methods to-date. These methods produce approximations of the posterior distribution of the current state of the signal by using the empirical distribution of a cloud of particles that explore the signal’s state space. In this thesis, we discuss a more general class of numerical methods which involve generalised particles, that is, particles that evolve through spaces larger than the signal’s state space. Such generalised particles include Gaussian mixtures, wavelets, orthonormal polynomials, and finite elements in addition to the classical particle methods. This thesis contains a rigorous analysis of the approximation of the solution of the filtering problem using Gaussian mixtures. In particular we deduce the L2-convergence rate and obtain the central limit theorem for the approximating system. Finally, the filtering model associated to the Navier-Stokes equation will be discussed as an example

    Twentieth conference on stochastic processes and their applications

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    A continuous-time analysis of distributed stochastic gradient

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    We analyze the effect of synchronization on distributed stochastic gradient algorithms. By exploiting an analogy with dynamical models of biological quorum sensing -- where synchronization between agents is induced through communication with a common signal -- we quantify how synchronization can significantly reduce the magnitude of the noise felt by the individual distributed agents and by their spatial mean. This noise reduction is in turn associated with a reduction in the smoothing of the loss function imposed by the stochastic gradient approximation. Through simulations on model non-convex objectives, we demonstrate that coupling can stabilize higher noise levels and improve convergence. We provide a convergence analysis for strongly convex functions by deriving a bound on the expected deviation of the spatial mean of the agents from the global minimizer for an algorithm based on quorum sensing, the same algorithm with momentum, and the Elastic Averaging SGD (EASGD) algorithm. We discuss extensions to new algorithms which allow each agent to broadcast its current measure of success and shape the collective computation accordingly. We supplement our theoretical analysis with numerical experiments on convolutional neural networks trained on the CIFAR-10 dataset, where we note a surprising regularizing property of EASGD even when applied to the non-distributed case. This observation suggests alternative second-order in-time algorithms for non-distributed optimization that are competitive with momentum methods.Comment: 9/14/19 : Final version, accepted for publication in Neural Computation. 4/7/19 : Significant edits: addition of simulations, deep network results, and revisions throughout. 12/28/18: Initial submissio

    Stochastic stability research for complex power systems

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    Bibliography: p. 302-311."November 1980." "Midterm report ... ."U.S. Dept. of Energy Contract ET-76-A-01-2295Tobias A. Trygar

    Applied Dynamics and Geometric Mechanics

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    This one week workshop was organized around several central subjects in applied dynamics and geometric mechanics. The specific organization with afternoons free for discussion led to intense exchanges of ideas. Bridges were forged between researchers representing different fields. Links were established between pure mathematical ideas and applications. The meeting was not restricted to any particular application area. One of the main goals of the meeting, like most others in this series for the past twenty years, has been to facilitate cross fertilization between various areas of mathematics, physics, and engineering. New collaborative projects emerged due to this meeting. The workshop was well attended with participants from Europe, North America, and Asia. Young researchers (doctoral students, postdocs, junior faculty) formed about 30% of the participants

    Maximum likelihood estimation of time series models: the Kalman filter and beyond

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    The purpose of this chapter is to provide a comprehensive treatment of likelihood inference for state space models. These are a class of time series models relating an observable time series to quantities called states, which are characterized by a simple temporal dependence structure, typically a first order Markov process. The states have sometimes substantial interpretation. Key estimation problems in economics concern latent variables, such as the output gap, potential output, the non-accelerating-inflation rate of unemployment, or NAIRU, core inflation, and so forth. Time-varying volatility, which is quintessential to finance, is an important feature also in macroeconomics. In the multivariate framework relevant features can be common to different series, meaning that the driving forces of a particular feature and/or the transmission mechanism are the same. The objective of this chapter is reviewing this algorithm and discussing maximum likelihood inference, starting from the linear Gaussian case and discussing the extensions to a nonlinear and non Gaussian framework
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