265 research outputs found
Generalised particle filters
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
A continuous-time analysis of distributed stochastic gradient
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
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
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
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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