9,951 research outputs found
Gaussian processes, kinematic formulae and Poincar\'e's limit
We consider vector valued, unit variance Gaussian processes defined over
stratified manifolds and the geometry of their excursion sets. In particular,
we develop an explicit formula for the expectation of all the
Lipschitz--Killing curvatures of these sets. Whereas our motivation is
primarily probabilistic, with statistical applications in the background, this
formula has also an interpretation as a version of the classic kinematic
fundamental formula of integral geometry. All of these aspects are developed in
the paper. Particularly novel is the method of proof, which is based on a an
approximation to the canonical Gaussian process on the -sphere. The
limit, which gives the final result, is handled via recent
extensions of the classic Poincar\'e limit theorem.Comment: Published in at http://dx.doi.org/10.1214/08-AOP439 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Rotation and scale space random fields and the Gaussian kinematic formula
We provide a new approach, along with extensions, to results in two important
papers of Worsley, Siegmund and coworkers closely tied to the statistical
analysis of fMRI (functional magnetic resonance imaging) brain data. These
papers studied approximations for the exceedence probabilities of scale and
rotation space random fields, the latter playing an important role in the
statistical analysis of fMRI data. The techniques used there came either from
the Euler characteristic heuristic or via tube formulae, and to a large extent
were carefully attuned to the specific examples of the paper. This paper treats
the same problem, but via calculations based on the so-called Gaussian
kinematic formula. This allows for extensions of the Worsley-Siegmund results
to a wide class of non-Gaussian cases. In addition, it allows one to obtain
results for rotation space random fields in any dimension via reasonably
straightforward Riemannian geometric calculations. Previously only the
two-dimensional case could be covered, and then only via computer algebra. By
adopting this more structured approach to this particular problem, a solution
path for other, related problems becomes clearer.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1055 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Excursion sets of stable random fields
Studying the geometry generated by Gaussian and Gaussian- related random
fields via their excursion sets is now a well developed and well understood
subject. The purely non-Gaussian scenario has, however, not been studied at
all. In this paper we look at three classes of stable random fields, and obtain
asymptotic formulae for the mean values of various geometric characteristics of
their excursion sets over high levels.
While the formulae are asymptotic, they contain enough information to show
that not only do stable random fields exhibit geometric behaviour very
different from that of Gaussian fields, but they also differ significantly
among themselves.Comment: 35 pages, 1 figur
False discovery rate analysis of brain diffusion direction maps
Diffusion tensor imaging (DTI) is a novel modality of magnetic resonance
imaging that allows noninvasive mapping of the brain's white matter. A
particular map derived from DTI measurements is a map of water principal
diffusion directions, which are proxies for neural fiber directions. We
consider a study in which diffusion direction maps were acquired for two groups
of subjects. The objective of the analysis is to find regions of the brain in
which the corresponding diffusion directions differ between the groups. This is
attained by first computing a test statistic for the difference in direction at
every brain location using a Watson model for directional data. Interesting
locations are subsequently selected with control of the false discovery rate.
More accurate modeling of the null distribution is obtained using an empirical
null density based on the empirical distribution of the test statistics across
the brain. Further, substantial improvements in power are achieved by local
spatial averaging of the test statistic map. Although the focus is on one
particular study and imaging technology, the proposed inference methods can be
applied to other large scale simultaneous hypothesis testing problems with a
continuous underlying spatial structure.Comment: Published in at http://dx.doi.org/10.1214/07-AOAS133 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
High level excursion set geometry for non-Gaussian infinitely divisible random fields
We consider smooth, infinitely divisible random fields ,
, with regularly varying Levy measure, and are
interested in the geometric characteristics of the excursion sets over high levels u. For a large class of such random fields, we
compute the asymptotic joint distribution of the numbers of
critical points, of various types, of X in , conditional on being
nonempty. This allows us, for example, to obtain the asymptotic conditional
distribution of the Euler characteristic of the excursion set. In a significant
departure from the Gaussian situation, the high level excursion sets for these
random fields can have quite a complicated geometry. Whereas in the Gaussian
case nonempty excursion sets are, with high probability, roughly ellipsoidal,
in the more general infinitely divisible setting almost any shape is possible.Comment: Published in at http://dx.doi.org/10.1214/11-AOP738 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Cash in on Genetic Opportunities for Feeder Cattle
Too few cattle feeders pay attention to the breeding background of the animals in their lots. Some useful genetic principles based on recent research are summarized in this article
Some effects of tropical storm Agnes on water quality in the Patuxent River estuary
A post Agnes study emphasizing environmental factors...weekly sampling at eight stations from 28 June to August 30, 1972. Spatial and temporal changes in the distribution of many factors, e.g., salinity, dissolved oxygen (DO), seston, particulate carbon and nitrogen, inorganic and organic fractions of dissolved nitrogen and phosphorus, and chlorophyll a were studied and compared to earlier extensive records. Patterns shown by the present data were compared especially with a local heavy storm that occurred in the Patuxent drainage basin during July 1963.
Some interesting correlations were observed in the data. (PDF has 39 pages.
Some effects of Hurricane Agnes on water quality in the Patuxent River Estuary
A post-Agnes study that emphasized environmental factors was carried out on the Patuxent River estuary with weekly sampling at eight stations from 28 June t o 30 August 1972. Spatial and temporal changes in the distribution of many factors , e.g., salinity , dissolved oxygen, seston, particulate carbon and nitrogen, inorganic and organic fractions of dissolved nitrogen and phosphorus, and chlorophyll a were studied and compared t o extensive earlier records. Patterns shown by the present data were compared especially with a local heavy storm that occurred in the Patuxent drainage basin during July 1969. Estimates were made of the amounts of material contributed via upland drainage. A first approximation indicated that 14.8 x l0 (3) metric tons of seston were contributed t o the head of the estuary between 21 and 24 June. (PDF contains 46 pages
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