1,525 research outputs found

### Particle Efficient Importance Sampling

The efficient importance sampling (EIS) method is a general principle for the
numerical evaluation of high-dimensional integrals that uses the sequential
structure of target integrands to build variance minimising importance
samplers. Despite a number of successful applications in high dimensions, it is
well known that importance sampling strategies are subject to an exponential
growth in variance as the dimension of the integration increases. We solve this
problem by recognising that the EIS framework has an offline sequential Monte
Carlo interpretation. The particle EIS method is based on non-standard
resampling weights that take into account the look-ahead construction of the
importance sampler. We apply the method for a range of univariate and bivariate
stochastic volatility specifications. We also develop a new application of the
EIS approach to state space models with Student's t state innovations. Our
results show that the particle EIS method strongly outperforms both the
standard EIS method and particle filters for likelihood evaluation in high
dimensions. Moreover, the ratio between the variances of the particle EIS and
particle filter methods remains stable as the time series dimension increases.
We illustrate the efficiency of the method for Bayesian inference using the
particle marginal Metropolis-Hastings and importance sampling squared
algorithms

### Blister patterns and energy minimization in compressed thin films on compliant substrates

This paper is motivated by the complex blister patterns sometimes seen in
thin elastic films on thick, compliant substrates. These patterns are often
induced by an elastic misfit which compresses the film. Blistering permits the
film to expand locally, reducing the elastic energy of the system. It is
natural to ask: what is the minimum elastic energy achievable by blistering on
a fixed area fraction of the substrate? This is a variational problem involving
both the {\it elastic deformation} of the film and substrate and the {\it
geometry} of the blistered region. It involves three small parameters: the {\it
nondimensionalized thickness} of the film, the {\it compliance ratio} of the
film/substrate pair and the {\it mismatch strain}. In formulating the problem,
we use a small-slope (F\"oppl-von K\'arm\'an) approximation for the elastic
energy of the film, and a local approximation for the elastic energy of the
substrate.
For a 1D version of the problem, we obtain "matching" upper and lower bounds
on the minimum energy, in the sense that both bounds have the same scaling
behavior with respect to the small parameters. For a 2D version of the problem,
our results are less complete. Our upper and lower bounds only "match" in their
scaling with respect to the nondimensionalized thickness, not in the dependence
on the compliance ratio and the mismatch strain. The upper bound considers a 2D
lattice of blisters, and uses ideas from the literature on the folding or
"crumpling" of a confined elastic sheet. Our main 2D result is that in a
certain parameter regime, the elastic energy of this lattice is significantly
lower than that of a few large blisters

### The coarsening of folds in hanging drapes

We consider the elastic energy of a hanging drape -- a thin elastic sheet,
pulled down by the force of gravity, with fine-scale folding at the top that
achieves approximately uniform confinement. This example of energy-driven
pattern formation in a thin elastic sheet is of particular interest because the
length scale of folding varies with height. We focus on how the minimum elastic
energy depends on the physical parameters. As the sheet thickness vanishes, the
limiting energy is due to the gravitational force and is relatively easy to
understand. Our main accomplishment is to identify the "scaling law" of the
correction due to positive thickness. We do this by (i) proving an upper bound,
by considering the energies of several constructions and taking the best; (ii)
proving an ansatz-free lower bound, which agrees with the upper bound up to a
parameter-independent prefactor. The coarsening of folds in hanging drapes has
also been considered in the recent physics literature, using a self-similar
construction whose basic cell has been called a "wrinklon." Our results
complement and extend that work, by showing that self-similar coarsening
achieves the optimal scaling law in a certain parameter regime, and by showing
that other constructions (involving lateral spreading of the sheet) do better
in other regions of parameter space. Our analysis uses a geometrically linear
F\"{o}ppl-von K\'{a}rm\'{a}n model for the elastic energy, and is restricted to
the case when Poisson's ratio is zero.Comment: 34 page

### On approximating copulas by finite mixtures

Copulas are now frequently used to approximate or estimate multivariate
distributions because of their ability to take into account the multivariate
dependence of the variables while controlling the approximation properties of
the marginal densities. Copula based multivariate models can often also be more
parsimonious than fitting a flexible multivariate model, such as a mixture of
normals model, directly to the data. However, to be effective, it is imperative
that the family of copula models considered is sufficiently flexible. Although
finite mixtures of copulas have been used to construct flexible families of
copulas, their approximation properties are not well understood and we show
that natural candidates such as mixtures of elliptical copulas and mixtures of
Archimedean copulas cannot approximate a general copula arbitrarily well. Our
article develops fundamental tools for approximating a general copula
arbitrarily well by a mixture and proposes a family of finite mixtures that can
do so. We illustrate empirically on a financial data set that our approach for
estimating a copula can be much more parsimonious and results in a better fit
than approximating the copula by a mixture of normal copulas.Comment: 26 pages and 1 figure and 2 table

### Efficient Bayesian inference for multivariate factor stochastic volatility models with leverage

This paper discusses the efficient Bayesian estimation of a multivariate
factor stochastic volatility (Factor MSV) model with leverage. We propose a
novel approach to construct the sampling schemes that converges to the
posterior distribution of the latent volatilities and the parameters of
interest of the Factor MSV model based on recent advances in Particle Markov
chain Monte Carlo (PMCMC). As opposed to the approach of Chib et al. (2006} and
Omori et al. (2007}, our approach does not require approximating the joint
distribution of outcome and volatility innovations by a mixture of bivariate
normal distributions. To sample the free elements of the loading matrix we
employ the interweaving method used in Kastner et al. (2017} in the Particle
Metropolis within Gibbs (PMwG) step. The proposed method is illustrated
empirically using a simulated dataset and a sample of daily US stock returns.Comment: 4 figures and 9 table

### On Scalable Particle Markov Chain Monte Carlo

Particle Markov Chain Monte Carlo (PMCMC) is a general approach to carry out
Bayesian inference in non-linear and non-Gaussian state space models. Our
article shows how to scale up PMCMC in terms of the number of observations and
parameters by expressing the target density of the PMCMC in terms of the basic
uniform or standard normal random numbers, instead of the particles, used in
the sequential Monte Carlo algorithm. Parameters that can be drawn efficiently
conditional on the particles are generated by particle Gibbs. All the other
parameters are drawn by conditioning on the basic uniform or standard normal
random variables; e.g. parameters that are highly correlated with the states,
or parameters whose generation is expensive when conditioning on the states.
The performance of this hybrid sampler is investigated empirically by applying
it to univariate and multivariate stochastic volatility models having both a
large number of parameters and a large number of latent states and shows that
it is much more efficient than competing PMCMC methods. We also show that the
proposed hybrid sampler is ergodic

- â€¦