28,075 research outputs found
Rare-event Simulation and Efficient Discretization for the Supremum of Gaussian Random Fields
In this paper, we consider a classic problem concerning the high excursion
probabilities of a Gaussian random field living on a compact set . We
develop efficient computational methods for the tail probabilities and the conditional expectations as
. For each positive, we present Monte Carlo
algorithms that run in \emph{constant} time and compute the interesting
quantities with relative error for arbitrarily large . The
efficiency results are applicable to a large class of H\"older continuous
Gaussian random fields. Besides computations, the proposed change of measure
and its analysis techniques have several theoretical and practical indications
in the asymptotic analysis of extremes of Gaussian random fields
Integration Mechanisms for Heading Perception
Previous studies of heading perception suggest that human observers employ spatiotemporal pooling to accommodate noise in optic flow stimuli. Here, we investigated how spatial and temporal integration mechanisms are used for judgments of heading through a psychophysical experiment involving three different types of noise. Furthermore, we developed two ideal observer models to study the components of the spatial information used by observers when performing the heading task. In the psychophysical experiment, we applied three types of direction noise to optic flow stimuli to differentiate the involvement of spatial and temporal integration mechanisms. The results indicate that temporal integration mechanisms play a role in heading perception, though their contribution is weaker than that of the spatial integration mechanisms. To elucidate how observers process spatial information to extract heading from a noisy optic flow field, we compared psychophysical performance in response to random-walk direction noise with that of two ideal observer models (IOMs). One model relied on 2D screen-projected flow information (2D-IOM), while the other used environmental, i.e., 3D, flow information (3D-IOM). The results suggest that human observers compensate for the loss of information during the 2D retinal projection of the visual scene for modest amounts of noise. This suggests the likelihood of a 3D reconstruction during heading perception, which breaks down under extreme levels of noise
Log-correlated Gaussian fields: an overview
We survey the properties of the log-correlated Gaussian field (LGF), which is
a centered Gaussian random distribution (generalized function) on , defined up to a global additive constant. Its law is determined by the
covariance formula
which holds for mean-zero test functions . The LGF belongs to
the larger family of fractional Gaussian fields obtained by applying fractional
powers of the Laplacian to a white noise on . It takes the
form . By comparison, the Gaussian free field (GFF)
takes the form in any dimension. The LGFs with coincide with the 2D GFF and its restriction to a line. These objects
arise in the study of conformal field theory and SLE, random surfaces, random
matrices, Liouville quantum gravity, and (when ) finance. Higher
dimensional LGFs appear in models of turbulence and early-universe cosmology.
LGFs are closely related to cascade models and Gaussian branching random walks.
We review LGF approximation schemes, restriction properties, Markov properties,
conformal symmetries, and multiplicative chaos applications.Comment: 24 pages, 2 figure
Neighborhood radius estimation in Variable-neighborhood Random Fields
We consider random fields defined by finite-region conditional probabilities
depending on a neighborhood of the region which changes with the boundary
conditions. To predict the symbols within any finite region it is necessary to
inspect a random number of neighborhood symbols which might change according to
the value of them. In analogy to the one dimensional setting we call these
neighborhood symbols the context of the region. This framework is a natural
extension, to d-dimensional fields, of the notion of variable-length Markov
chains introduced by Rissanen (1983) in his classical paper. We define an
algorithm to estimate the radius of the smallest ball containing the context
based on a realization of the field. We prove the consistency of this
estimator. Our proofs are constructive and yield explicit upper bounds for the
probability of wrong estimation of the radius of the context
On ANOVA decompositions of kernels and Gaussian random field paths
The FANOVA (or "Sobol'-Hoeffding") decomposition of multivariate functions
has been used for high-dimensional model representation and global sensitivity
analysis. When the objective function f has no simple analytic form and is
costly to evaluate, a practical limitation is that computing FANOVA terms may
be unaffordable due to numerical integration costs. Several approximate
approaches relying on random field models have been proposed to alleviate these
costs, where f is substituted by a (kriging) predictor or by conditional
simulations. In the present work, we focus on FANOVA decompositions of Gaussian
random field sample paths, and we notably introduce an associated kernel
decomposition (into 2^{2d} terms) called KANOVA. An interpretation in terms of
tensor product projections is obtained, and it is shown that projected kernels
control both the sparsity of Gaussian random field sample paths and the
dependence structure between FANOVA effects. Applications on simulated data
show the relevance of the approach for designing new classes of covariance
kernels dedicated to high-dimensional kriging
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