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 a...

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Bibliographic Details
Published in:The Annals of statistics 2012-12, Vol.40 (6), p.2910-2942
Main Authors: Adler, Robert J., Subag, Eliran, Taylor, Jonathan E.
Format: Article
Language:English
Subjects:
52A22
60D05
60G15
60G60
60G70
62M30
Approximation
Curvature
Density
Eigenvalues
Euler characteristic
fMRI
Gaussian kinematic formula
Heuristic
Heuristics
Kinematics
Lipschitz–Killing curvatures
Magnetic fields
Magnetic resonance imaging
Mathematical vectors
Matrices
NMR
Nuclear magnetic resonance
random fields
Rotation space
scale space
Statistical analysis
Studies
thresholding
Vector fields
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/12-AOS1055