17,948 research outputs found
The Extended Parameter Filter
The parameters of temporal models, such as dynamic Bayesian networks, may be
modelled in a Bayesian context as static or atemporal variables that influence
transition probabilities at every time step. Particle filters fail for models
that include such variables, while methods that use Gibbs sampling of parameter
variables may incur a per-sample cost that grows linearly with the length of
the observation sequence. Storvik devised a method for incremental computation
of exact sufficient statistics that, for some cases, reduces the per-sample
cost to a constant. In this paper, we demonstrate a connection between
Storvik's filter and a Kalman filter in parameter space and establish more
general conditions under which Storvik's filter works. Drawing on an analogy to
the extended Kalman filter, we develop and analyze, both theoretically and
experimentally, a Taylor approximation to the parameter posterior that allows
Storvik's method to be applied to a broader class of models. Our experiments on
both synthetic examples and real applications show improvement over existing
methods
Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples
We examine some differential geometric approaches to finding approximate
solutions to the continuous time nonlinear filtering problem. Our primary focus
is a new projection method for the optimal filter infinite dimensional
Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric
and on a family of normal mixtures. We compare this method to earlier
projection methods based on the Hellinger distance/Fisher metric and
exponential families, and we compare the L2 mixture projection filter with a
particle method with the same number of parameters, using the Levy metric. We
prove that for a simple choice of the mixture manifold the L2 mixture
projection filter coincides with a Galerkin method, whereas for more general
mixture manifolds the equivalence does not hold and the L2 mixture filter is
more general. We study particular systems that may illustrate the advantages of
this new filter over other algorithms when comparing outputs with the optimal
filter. We finally consider a specific software design that is suited for a
numerically efficient implementation of this filter and provide numerical
examples.Comment: Updated and expanded version published in the Journal reference
below. Preprint updates: January 2016 (v3) added projection of Zakai Equation
and difference with projection of Kushner-Stratonovich (section 4.1). August
2014 (v2) added Galerkin equivalence proof (Section 5) to the March 2013 (v1)
versio
Chebyshev polynomial filtered subspace iteration in the Discontinuous Galerkin method for large-scale electronic structure calculations
The Discontinuous Galerkin (DG) electronic structure method employs an
adaptive local basis (ALB) set to solve the Kohn-Sham equations of density
functional theory (DFT) in a discontinuous Galerkin framework. The adaptive
local basis is generated on-the-fly to capture the local material physics, and
can systematically attain chemical accuracy with only a few tens of degrees of
freedom per atom. A central issue for large-scale calculations, however, is the
computation of the electron density (and subsequently, ground state properties)
from the discretized Hamiltonian in an efficient and scalable manner. We show
in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can
be used to address this issue and push the envelope in large-scale materials
simulations in a discontinuous Galerkin framework. We describe how the subspace
filtering steps can be performed in an efficient and scalable manner using a
two-dimensional parallelization scheme, thanks to the orthogonality of the DG
basis set and block-sparse structure of the DG Hamiltonian matrix. The
on-the-fly nature of the ALBs requires additional care in carrying out the
subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI
approach in calculations of large-scale two-dimensional graphene sheets and
bulk three-dimensional lithium-ion electrolyte systems. Employing 55,296
computational cores, the time per self-consistent field iteration for a sample
of the bulk 3D electrolyte containing 8,586 atoms is 90 seconds, and the time
for a graphene sheet containing 11,520 atoms is 75 seconds.Comment: Submitted to The Journal of Chemical Physic
On the Spectral Properties of Matrices Associated with Trend Filters
This paper is concerned with the spectral properties of matrices associated
with linear filters for the estimation of the underlying trend of a time
series. The interest lies in the fact that the eigenvectors can be interpreted
as the latent components of any time series that the filter smooths through the
corresponding eigenvalues. A difficulty arises because matrices associated with
trend filters are finite approximations of Toeplitz operators and therefore
very little is known about their eigenstructure, which also depends on the
boundary conditions or, equivalently, on the filters for trend estimation at
the end of the sample. Assuming reflecting boundary conditions, we derive a
time series decomposition in terms of periodic latent components and
corresponding smoothing eigenvalues. This decomposition depends on the local
polynomial regression estimator chosen for the interior. Otherwise, the
eigenvalue distribution is derived with an approximation measured by the size
of the perturbation that different boundary conditions apport to the
eigenvalues of matrices belonging to algebras with known spectral properties,
such as the Circulant or the Cosine. The analytical form of the eigenvectors is
then derived with an approximation that involves the extremes only. A further
topic investigated in the paper concerns a strategy for a filter design in the
time domain. Based on cut-off eigenvalues, new estimators are derived, that are
less variable and almost equally biased as the original estimator, based on all
the eigenvalues. Empirical examples illustrate the effectiveness of the method
Markov cubature rules for polynomial processes
We study discretizations of polynomial processes using finite state Markov
processes satisfying suitable moment matching conditions. The states of these
Markov processes together with their transition probabilities can be
interpreted as Markov cubature rules. The polynomial property allows us to
study such rules using algebraic techniques. Markov cubature rules aid the
tractability of path-dependent tasks such as American option pricing in models
where the underlying factors are polynomial processes.Comment: 29 pages, 6 Figures, 2 Tables; forthcoming in Stochastic Processes
and their Application
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