Analysis of Random Fields Using CompRandFld
Adler , Eliran Subag , and Jonathan E. Taylor More by Robert J. Adler Search this author in:.
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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.
Modeling and statistical analysis of non-Gaussian random fields with heavy-tailed distributions
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. Source Ann. Zentralblatt MATH identifier Sign In. Access provided by: anon Sign Out.
Stochastic Geometry, Spatial Statistics and Random Fields
Restriction of a Markov random field on a graph and multiresolution statistical image modeling Abstract: The association of statistical models and multiresolution data analysis in a consistent and tractable mathematical framework remains an intricate theoretical and practical issue. Several consistent approaches have been proposed previously to combine Markov random field MRF models and multiresolution algorithms in image analysis: renormalization group, subsampling of stochastic processes, MRFs defined on trees or pyramids, etc.
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For the simulation or a practical use of these models in statistical estimation, an important issue is the preservation of the local Markovian property of the representation at the different resolution levels. It is shown that this key problem may be studied by considering the restriction of a Markov random field defined on some simple finite nondirected graph to a part of its original site set. Several general properties of the restricted field are derived.
The general form of the distribution of the restriction is given. Sufficient conditions for the new neighborhood structure to be "minimal" are derived. Several consequences of these general results related to various "multiresolution" MRF-based modeling approaches in image analysis are presented.