2,953 research outputs found
Particle-kernel estimation of the filter density in state-space models
Sequential Monte Carlo (SMC) methods, also known as particle filters, are
simulation-based recursive algorithms for the approximation of the a posteriori
probability measures generated by state-space dynamical models. At any given
time , a SMC method produces a set of samples over the state space of the
system of interest (often termed "particles") that is used to build a discrete
and random approximation of the posterior probability distribution of the state
variables, conditional on a sequence of available observations. One potential
application of the methodology is the estimation of the densities associated to
the sequence of a posteriori distributions. While practitioners have rather
freely applied such density approximations in the past, the issue has received
less attention from a theoretical perspective. In this paper, we address the
problem of constructing kernel-based estimates of the posterior probability
density function and its derivatives, and obtain asymptotic convergence results
for the estimation errors. In particular, we find convergence rates for the
approximation errors that hold uniformly on the state space and guarantee that
the error vanishes almost surely as the number of particles in the filter
grows. Based on this uniform convergence result, we first show how to build
continuous measures that converge almost surely (with known rate) toward the
posterior measure and then address a few applications. The latter include
maximum a posteriori estimation of the system state using the approximate
derivatives of the posterior density and the approximation of functionals of
it, for example, Shannon's entropy.
This manuscript is identical to the published paper, including a gap in the
proof of Theorem 4.2. The Theorem itself is correct. We provide an {\em
erratum} at the end of this document with a complete proof and a brief
discussion.Comment: IMPORTANT: This manuscript is identical to the published paper,
including a gap in the proof of Theorem 4.2. The Theorem itself is correct.
We provide an erratum at the end of this document. Published at
http://dx.doi.org/10.3150/13-BEJ545 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Two adaptive rejection sampling schemes for probability density functions log-convex tails
Monte Carlo methods are often necessary for the implementation of optimal
Bayesian estimators. A fundamental technique that can be used to generate
samples from virtually any target probability distribution is the so-called
rejection sampling method, which generates candidate samples from a proposal
distribution and then accepts them or not by testing the ratio of the target
and proposal densities. The class of adaptive rejection sampling (ARS)
algorithms is particularly interesting because they can achieve high acceptance
rates. However, the standard ARS method can only be used with log-concave
target densities. For this reason, many generalizations have been proposed.
In this work, we investigate two different adaptive schemes that can be used
to draw exactly from a large family of univariate probability density functions
(pdf's), not necessarily log-concave, possibly multimodal and with tails of
arbitrary concavity. These techniques are adaptive in the sense that every time
a candidate sample is rejected, the acceptance rate is improved. The two
proposed algorithms can work properly when the target pdf is multimodal, with
first and second derivatives analytically intractable, and when the tails are
log-convex in a infinite domain. Therefore, they can be applied in a number of
scenarios in which the other generalizations of the standard ARS fail. Two
illustrative numerical examples are shown
Nudging the particle filter
We investigate a new sampling scheme aimed at improving the performance of
particle filters whenever (a) there is a significant mismatch between the
assumed model dynamics and the actual system, or (b) the posterior probability
tends to concentrate in relatively small regions of the state space. The
proposed scheme pushes some particles towards specific regions where the
likelihood is expected to be high, an operation known as nudging in the
geophysics literature. We re-interpret nudging in a form applicable to any
particle filtering scheme, as it does not involve any changes in the rest of
the algorithm. Since the particles are modified, but the importance weights do
not account for this modification, the use of nudging leads to additional bias
in the resulting estimators. However, we prove analytically that nudged
particle filters can still attain asymptotic convergence with the same error
rates as conventional particle methods. Simple analysis also yields an
alternative interpretation of the nudging operation that explains its
robustness to model errors. Finally, we show numerical results that illustrate
the improvements that can be attained using the proposed scheme. In particular,
we present nonlinear tracking examples with synthetic data and a model
inference example using real-world financial data
Adapting the Number of Particles in Sequential Monte Carlo Methods through an Online Scheme for Convergence Assessment
Particle filters are broadly used to approximate posterior distributions of
hidden states in state-space models by means of sets of weighted particles.
While the convergence of the filter is guaranteed when the number of particles
tends to infinity, the quality of the approximation is usually unknown but
strongly dependent on the number of particles. In this paper, we propose a
novel method for assessing the convergence of particle filters online manner,
as well as a simple scheme for the online adaptation of the number of particles
based on the convergence assessment. The method is based on a sequential
comparison between the actual observations and their predictive probability
distributions approximated by the filter. We provide a rigorous theoretical
analysis of the proposed methodology and, as an example of its practical use,
we present simulations of a simple algorithm for the dynamic and online
adaption of the number of particles during the operation of a particle filter
on a stochastic version of the Lorenz system
Relation between growth dynamics and the spatial distribution of intrinsic defects in self-assembled colloidal crystal films
Herein we establish a clear relation between the parameters that govern the growth dynamics and the structural quality of colloidal crystal films. We report an optical analysis of the spatial distribution of intrinsic defects in colloidal crystal films and correlate our results with a theoretical model describing the growth dynamics of such lattices. We find that the amount of defects fluctuates periodically and decreases along the growth direction of the lattice. We demonstrate that these spatial variations are a direct consequence of the temporal oscillations of the crystal film formation velocity, which are inherent to the colloidal particle deposition process.Ministerio de Ciencia y Educación MAT2004-0302
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