62,879 research outputs found
Quantifying uncertainties on excursion sets under a Gaussian random field prior
We focus on the problem of estimating and quantifying uncertainties on the
excursion set of a function under a limited evaluation budget. We adopt a
Bayesian approach where the objective function is assumed to be a realization
of a Gaussian random field. In this setting, the posterior distribution on the
objective function gives rise to a posterior distribution on excursion sets.
Several approaches exist to summarize the distribution of such sets based on
random closed set theory. While the recently proposed Vorob'ev approach
exploits analytical formulae, further notions of variability require Monte
Carlo estimators relying on Gaussian random field conditional simulations. In
the present work we propose a method to choose Monte Carlo simulation points
and obtain quasi-realizations of the conditional field at fine designs through
affine predictors. The points are chosen optimally in the sense that they
minimize the posterior expected distance in measure between the excursion set
and its reconstruction. The proposed method reduces the computational costs due
to Monte Carlo simulations and enables the computation of quasi-realizations on
fine designs in large dimensions. We apply this reconstruction approach to
obtain realizations of an excursion set on a fine grid which allow us to give a
new measure of uncertainty based on the distance transform of the excursion
set. Finally we present a safety engineering test case where the simulation
method is employed to compute a Monte Carlo estimate of a contour line
Using the bootstrap to quantify the authority of an empirical ranking
The bootstrap is a popular and convenient method for quantifying the
authority of an empirical ordering of attributes, for example of a ranking of
the performance of institutions or of the influence of genes on a response
variable. In the first of these examples, the number, , of quantities being
ordered is sometimes only moderate in size; in the second it can be very large,
often much greater than sample size. However, we show that in both types of
problem the conventional bootstrap can produce inconsistency. Moreover, the
standard -out-of- bootstrap estimator of the distribution of an empirical
rank may not converge in the usual sense; the estimator may converge in
distribution, but not in probability. Nevertheless, in many cases the bootstrap
correctly identifies the support of the asymptotic distribution of ranks. In
some contemporary problems, bootstrap prediction intervals for ranks are
particularly long, and in this context, we also quantify the accuracy of
bootstrap methods, showing that the standard bootstrap gets the order of
magnitude of the interval right, but not the constant multiplier of interval
length. The -out-of- bootstrap can improve performance and produce
statistical consistency, but it requires empirical choice of ; we suggest a
tuning solution to this problem. We show that in genomic examples, where it
might be expected that the standard, ``synchronous'' bootstrap will
successfully accommodate nonindependence of vector components, that approach
can produce misleading results. An ``independent component'' bootstrap can
overcome these difficulties, even in cases where components are not strictly
independent.Comment: Published in at http://dx.doi.org/10.1214/09-AOS699 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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