187 research outputs found
Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping
A new class of stochastic field models is constructed using nested stochastic
partial differential equations (SPDEs). The model class is computationally
efficient, applicable to data on general smooth manifolds, and includes both
the Gaussian Mat\'{e}rn fields and a wide family of fields with oscillating
covariance functions. Nonstationary covariance models are obtained by spatially
varying the parameters in the SPDEs, and the model parameters are estimated
using direct numerical optimization, which is more efficient than standard
Markov Chain Monte Carlo procedures. The model class is used to estimate daily
ozone maps using a large data set of spatially irregular global total column
ozone data.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS383 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quantifying the uncertainty of contour maps
Contour maps are widely used to display estimates of spatial fields. Instead
of showing the estimated field, a contour map only shows a fixed number of
contour lines for different levels. However, despite the ubiquitous use of
these maps, the uncertainty associated with them has been given a surprisingly
small amount of attention. We derive measures of the statistical uncertainty,
or quality, of contour maps, and use these to decide an appropriate number of
contour lines, that relates to the uncertainty in the estimated spatial field.
For practical use in geostatistics and medical imaging, computational methods
are constructed, that can be applied to Gaussian Markov random fields, and in
particular be used in combination with integrated nested Laplace approximations
for latent Gaussian models. The methods are demonstrated on simulated data and
an application to temperature estimation is presented
Approximation of bayesian Hawkes process models with Inlabru
Hawkes process are very popular mathematical tools for modelling phenomena
exhibiting a \textit{self-exciting} or \textit{self-correcting} behaviour.
Typical examples are earthquakes occurrence, wild-fires, drought,
capture-recapture, crime violence, trade exchange, and social network activity.
The widespread use of Hawkes process in different fields calls for fast,
reproducible, reliable, easy-to-code techniques to implement such models. We
offer a technique to perform approximate Bayesian inference of Hawkes process
parameters based on the use of the R-package \inlabru. The \inlabru R-package,
in turn, relies on the INLA methodology to approximate the posterior of the
parameters. Our Hawkes process approximation is based on a decomposition of the
log-likelihood in three parts, which are linearly approximated separately. The
linear approximation is performed with respect to the mode of the parameters'
posterior distribution, which is determined with an iterative gradient-based
method. The approximation of the posterior parameters is therefore
deterministic, ensuring full reproducibility of the results. The proposed
technique only requires the user to provide the functions to calculate the
different parts of the decomposed likelihood, which are internally linearly
approximated by the R-package \inlabru. We provide a comparison with the
\bayesianETAS R-package which is based on an MCMC method. The two techniques
provide similar results but our approach requires two to ten times less
computational time to converge, depending on the amount of data.Comment: 2o pages, 7 figures, 5 table
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