×

zbMATH — the first resource for mathematics

Rotation and scale space random fields and the Gaussian kinematic formula. (English) Zbl 1296.60132
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.

MSC:
60G60 Random fields
60G15 Gaussian processes
60D05 Geometric probability and stochastic geometry
62M30 Inference from spatial processes
52A22 Random convex sets and integral geometry (aspects of convex geometry)
60G70 Extreme value theory; extremal stochastic processes
PDF BibTeX Cite
Full Text: DOI Euclid arXiv
References:
[1] Adler, R. J. (2000). On excursion sets, tube formulas and maxima of random fields. Ann. Appl. Probab. 10 1-74. · Zbl 1171.60338
[2] Adler, R. J., Bartz, K. and Kou, S. C. (2011). Estimating thresholding levels for random fields via Euler characteristics. Unpublished manuscript.
[3] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry . Springer, New York. · Zbl 1149.60003
[4] Adler, R. J., Taylor, J. E. and Worsley, K. J. (2013). Applications of Random Fields and Geometry : Foundations and Case Studies . Springer. To appear. Early chapters available at . · Zbl 1149.60003
[5] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118 . Cambridge Univ. Press, Cambridge. · Zbl 1184.15023
[6] Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418-491. · Zbl 0089.38402
[7] Gibson, C. G., Wirthmüller, K., du Plessis, A. A. and Looijenga, E. J. N. (1976). Topological Stability of Smooth Mappings. Lecture Notes in Mathematics 552 . Springer, Berlin. · Zbl 0377.58006
[8] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2007). Numerical Recipes : The Art of Scientific Computing , 3rd ed. Cambridge Univ. Press, Cambridge. · Zbl 1132.65001
[9] Shafie, K., Sigal, B., Siegmund, D. and Worsley, K. J. (2003). Rotation space random fields with an application to fMRI data. Ann. Statist. 31 1732-1771. · Zbl 1043.92019
[10] Siegmund, D. O. and Worsley, K. J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23 608-639. · Zbl 0898.62119
[11] Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362-1396. · Zbl 1083.60031
[12] Taylor, J. E. (2001). Euler characteristics for Gaussian fields on manifolds. Ph.D. thesis, McGill Univ., Montreal. · Zbl 1026.60039
[13] Taylor, J. E. (2006). A Gaussian kinematic formula. Ann. Probab. 34 122-158. · Zbl 1094.60025
[14] Taylor, J. E. and Adler, R. J. (2009). Gaussian processes, kinematic formulae and Poincaré’s limit. Ann. Probab. 37 1459-1482. · Zbl 1172.60006
[15] Taylor, J. E. and Worsley, K. J. (2007). Detecting sparse signals in random fields, with an application to brain mapping. J. Amer. Statist. Assoc. 102 913-928. · Zbl 05564420
[16] Worsley, K. J. (2001). Testing for signals with unknown location and scale in a \(\chi^2\) random field, with an application to fMRI. Adv. in Appl. Probab. 33 773-793. · Zbl 0991.62075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.