16,168 research outputs found
Particle approximations of the score and observed information matrix for parameter estimation in state space models with linear computational cost
Poyiadjis et al. (2011) show how particle methods can be used to estimate
both the score and the observed information matrix for state space models.
These methods either suffer from a computational cost that is quadratic in the
number of particles, or produce estimates whose variance increases
quadratically with the amount of data. This paper introduces an alternative
approach for estimating these terms at a computational cost that is linear in
the number of particles. The method is derived using a combination of kernel
density estimation, to avoid the particle degeneracy that causes the
quadratically increasing variance, and Rao-Blackwellisation. Crucially, we show
the method is robust to the choice of bandwidth within the kernel density
estimation, as it has good asymptotic properties regardless of this choice. Our
estimates of the score and observed information matrix can be used within both
online and batch procedures for estimating parameters for state space models.
Empirical results show improved parameter estimates compared to existing
methods at a significantly reduced computational cost. Supplementary materials
including code are available.Comment: Accepted to Journal of Computational and Graphical Statistic
Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models
We propose a new class of filtering and smoothing methods for inference in
high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models.
The main idea is to combine the ensemble Kalman filter and smoother, developed
in the geophysics literature, with state-space algorithms from the statistics
literature. Our algorithms address a variety of estimation scenarios, including
on-line and off-line state and parameter estimation. We take a Bayesian
perspective, for which the goal is to generate samples from the joint posterior
distribution of states and parameters. The key benefit of our approach is the
use of ensemble Kalman methods for dimension reduction, which allows inference
for high-dimensional state vectors. We compare our methods to existing ones,
including ensemble Kalman filters, particle filters, and particle MCMC. Using a
real data example of cloud motion and data simulated under a number of
nonlinear and non-Gaussian scenarios, we show that our approaches outperform
these existing methods
Sequential Monte Carlo pricing of American-style options under stochastic volatility models
We introduce a new method to price American-style options on underlying
investments governed by stochastic volatility (SV) models. The method does not
require the volatility process to be observed. Instead, it exploits the fact
that the optimal decision functions in the corresponding dynamic programming
problem can be expressed as functions of conditional distributions of
volatility, given observed data. By constructing statistics summarizing
information about these conditional distributions, one can obtain high quality
approximate solutions. Although the required conditional distributions are in
general intractable, they can be arbitrarily precisely approximated using
sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes,
is potentially heavy computational demand. We present two variants of the
algorithm, one closely related to the well-known least-squares Monte Carlo
algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001)
113-147], and the other solving the same problem using a "brute force" gridding
approach. We estimate an illustrative SV model using Markov chain Monte Carlo
(MCMC) methods for three equities. We also demonstrate the use of our algorithm
by estimating the posterior distribution of the market price of volatility risk
for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sequential Monte Carlo Methods for System Identification
One of the key challenges in identifying nonlinear and possibly non-Gaussian
state space models (SSMs) is the intractability of estimating the system state.
Sequential Monte Carlo (SMC) methods, such as the particle filter (introduced
more than two decades ago), provide numerical solutions to the nonlinear state
estimation problems arising in SSMs. When combined with additional
identification techniques, these algorithms provide solid solutions to the
nonlinear system identification problem. We describe two general strategies for
creating such combinations and discuss why SMC is a natural tool for
implementing these strategies.Comment: In proceedings of the 17th IFAC Symposium on System Identification
(SYSID). Added cover pag
Bayesian subset simulation
We consider the problem of estimating a probability of failure ,
defined as the volume of the excursion set of a function above a given threshold, under a given
probability measure on . In this article, we combine the popular
subset simulation algorithm (Au and Beck, Probab. Eng. Mech. 2001) and our
sequential Bayesian approach for the estimation of a probability of failure
(Bect, Ginsbourger, Li, Picheny and Vazquez, Stat. Comput. 2012). This makes it
possible to estimate when the number of evaluations of is very
limited and is very small. The resulting algorithm is called Bayesian
subset simulation (BSS). A key idea, as in the subset simulation algorithm, is
to estimate the probabilities of a sequence of excursion sets of above
intermediate thresholds, using a sequential Monte Carlo (SMC) approach. A
Gaussian process prior on is used to define the sequence of densities
targeted by the SMC algorithm, and drive the selection of evaluation points of
to estimate the intermediate probabilities. Adaptive procedures are
proposed to determine the intermediate thresholds and the number of evaluations
to be carried out at each stage of the algorithm. Numerical experiments
illustrate that BSS achieves significant savings in the number of function
evaluations with respect to other Monte Carlo approaches
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