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 $n$-sphere. The $n\to\infty$ 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 $(X(t),t\in M)$, $M\subset {\mathbb{R}}^d$, with regularly varying Levy measure, and are interested in the geometric characteristics of the excursion sets $$A_u=\{t\in M:X(t)>u\}$$ over high levels u. For a large class of such random fields, we compute the $u\to\infty$ asymptotic joint distribution of the numbers of critical points, of various types, of X in $A_u$, conditional on $A_u$ 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