8,223 research outputs found
Statistical Inference for Partially Observed Markov Processes via the R Package pomp
Partially observed Markov process (POMP) models, also known as hidden Markov
models or state space models, are ubiquitous tools for time series analysis.
The R package pomp provides a very flexible framework for Monte Carlo
statistical investigations using nonlinear, non-Gaussian POMP models. A range
of modern statistical methods for POMP models have been implemented in this
framework including sequential Monte Carlo, iterated filtering, particle Markov
chain Monte Carlo, approximate Bayesian computation, maximum synthetic
likelihood estimation, nonlinear forecasting, and trajectory matching. In this
paper, we demonstrate the application of these methodologies using some simple
toy problems. We also illustrate the specification of more complex POMP models,
using a nonlinear epidemiological model with a discrete population,
seasonality, and extra-demographic stochasticity. We discuss the specification
of user-defined models and the development of additional methods within the
programming environment provided by pomp.Comment: In press at the Journal of Statistical Software. A version of this
paper is provided at the pomp package website: http://kingaa.github.io/pom
Multidimensional Quasi-Monte Carlo Malliavin Greeks
We investigate the use of Malliavin calculus in order to calculate the Greeks
of multidimensional complex path-dependent options by simulation. For this
purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the
multidimensional case. The multidimensional setting shows the convenience of
the Malliavin Calculus approach over different techniques that have been
previously proposed. Indeed, these techniques may be computationally expensive
and do not provide flexibility for variance reduction. In contrast, the
Malliavin approach exhibits a higher flexibility by providing a class of
functions that return the same expected value (the Greek) with different
accuracies. This versatility for variance reduction is not possible without the
use of the generalized integral by part formula of Malliavin Calculus. In the
multidimensional context, we find convenient formulas that permit to improve
the localization technique, introduced in Fourni\'e et al and reduce both the
computational cost and the variance. Moreover, we show that the parameters
employed for variance reduction can be obtained \textit{on the flight} in the
simulation. We illustrate the efficiency of the proposed procedures, coupled
with the enhanced version of Quasi-Monte Carlo simulations as discussed in
Sabino, for the numerical estimation of the Deltas of call, digital Asian-style
and Exotic basket options with a fixed and a floating strike price in a
multidimensional Black-Scholes market.Comment: 22 pages, 6 figure
Convenient Multiple Directions of Stratification
This paper investigates the use of multiple directions of stratification as a
variance reduction technique for Monte Carlo simulations of path-dependent
options driven by Gaussian vectors. The precision of the method depends on the
choice of the directions of stratification and the allocation rule within each
strata. Several choices have been proposed but, even if they provide variance
reduction, their implementation is computationally intensive and not applicable
to realistic payoffs, in particular not to Asian options with barrier.
Moreover, all these previously published methods employ orthogonal directions
for multiple stratification. In this work we investigate the use of algorithms
producing convenient directions, generally non-orthogonal, combining a lower
computational cost with a comparable variance reduction. In addition, we study
the accuracy of optimal allocation in terms of variance reduction compared to
the Latin Hypercube Sampling. We consider the directions obtained by the Linear
Transformation and the Principal Component Analysis. We introduce a new
procedure based on the Linear Approximation of the explained variance of the
payoff using the law of total variance. In addition, we exhibit a novel
algorithm that permits to correctly generate normal vectors stratified along
non-orthogonal directions. Finally, we illustrate the efficiency of these
algorithms in the computation of the price of different path-dependent options
with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross
markets.Comment: 21 pages, 11 table
Multidimensional integration through Markovian sampling under steered function morphing: a physical guise from statistical mechanics
We present a computational strategy for the evaluation of multidimensional
integrals on hyper-rectangles based on Markovian stochastic exploration of the
integration domain while the integrand is being morphed by starting from an
initial appropriate profile. Thanks to an abstract reformulation of Jarzynski's
equality applied in stochastic thermodynamics to evaluate the free-energy
profiles along selected reaction coordinates via non-equilibrium
transformations, it is possible to cast the original integral into the
exponential average of the distribution of the pseudo-work (that we may term
"computational work") involved in doing the function morphing, which is
straightforwardly solved. Several tests illustrate the basic implementation of
the idea, and show its performance in terms of computational time, accuracy and
precision. The formulation for integrand functions with zeros and possible sign
changes is also presented. It will be stressed that our usage of Jarzynski's
equality shares similarities with a practice already known in statistics as
Annealed Importance Sampling (AIS), when applied to computation of the
normalizing constants of distributions. In a sense, here we dress the AIS with
its "physical" counterpart borrowed from statistical mechanics.Comment: 3 figures Supplementary Material (pdf file named "JEMDI_SI.pdf"
Problems in Lattice Gauge Fixing
We review many topics and results about numeric gauge fixing in lattice QCD.Comment: 47 pages, 16 eps figures. Review article sent to IJMP
Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework
Approximate Bayesian Computation (ABC) is a useful class of methods for
Bayesian inference when the likelihood function is computationally intractable.
In practice, the basic ABC algorithm may be inefficient in the presence of
discrepancy between prior and posterior. Therefore, more elaborate methods,
such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be
used. However, the elaboration of a proposal density for MCMC is a sensitive
issue and very difficult in the ABC setting, where the likelihood is
intractable. We discuss an automatic proposal distribution useful for ABC-MCMC
algorithms. This proposal is inspired by the theory of quasi-likelihood (QL)
functions and is obtained by modelling the distribution of the summary
statistics as a function of the parameters. Essentially, given a real-valued
vector of summary statistics, we reparametrize the model by means of a
regression function of the statistics on parameters, obtained by sampling from
the original model in a pilot-run simulation study. The QL theory is well
established for a scalar parameter, and it is shown that when the conditional
variance of the summary statistic is assumed constant, the QL has a closed-form
normal density. This idea of constructing proposal distributions is extended to
non constant variance and to real-valued parameter vectors. The method is
illustrated by several examples and by an application to a real problem in
population genetics.Comment: Published at http://dx.doi.org/10.1214/14-BA921 in the Bayesian
Analysis (http://projecteuclid.org/euclid.ba) by the International Society of
Bayesian Analysis (http://bayesian.org/
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