Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T09:15:10.180Z Has data issue: false hasContentIssue false
  • English
  • Français

The size of the connected components of excursion sets of χ2, t and F fields

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

J. Cao
J. Cao*
Affiliation:
Bell Laboratories
*
∗ Postal address: Bell Laboratories, Lucent Technologies, 700 Mountain Avenue, Room 2C-260, Murray Hill, NJ 07974-2070, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The distribution of the size of one connected component and the largest connected component of the excursion set is derived for stationary χ2, t and F fields, in the limit of high or low thresholds. This extends previous results for stationary Gaussian fields (Nosko 1969, Adler 1981) and for χ2 fields in one and two dimensions (Aronowich and Adler 1986, 1988). An application of this is to detect regional changes in positron emission tomography (PET) images of blood flow in human brain, using the size of the largest connected component of the excursion set as a test statistic.

Keywords

Excursion setrandom fieldsimage analysis

MSC classification

Primary: 60G15: Gaussian processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 579 - 595
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, R. J. (1981). The Geometry of Random Fields. Wiley, New York.Google Scholar
Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.CrossRefGoogle Scholar
Aronowich, M. and Adler, R. J. (1986). Extrema and level crossings of χ2 processes. Adv. Appl. Prob. 18, 901–920.CrossRefGoogle Scholar
Aronowich, M. and Adler, R. J. (1988). Sample path behaviour of χ2 surfaces at extrema. Adv. Appl. Prob. 20, 719–738.Google Scholar
Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C. and Evans, A. C. (1994). Assessing the significance of focal activations using their spatial extent. Human Brain Mapping, 1, 214–220.CrossRefGoogle ScholarPubMed
Hasofer, A. M. (1978). Upcrossings of random fields. In Proceedings of the conference on spatial patterns and processes, ed. Tweedie, R. L. (Supplement to Adv. Appl. Prob). Applied Probablity Trust, Sheffield, UK, pp. 14–21.Google Scholar
Kac, M. and Slepian, D. (1959). Large excursions of Gaussian processes. Ann. Math. Statist. 30, 1215–1228.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremal and Related Properties of Stationary Processes. Springer, New York.Google Scholar
Lindgren, G. (1972). Local maxima of Gaussian fields. Ark. Math. 10, 195–218.CrossRefGoogle Scholar
Lindgren, G. (1984). Use and structure of Slepian model processes for prediction and detection in crossing and extreme value theory. In Statistical Extremes and Applications, eds. Tiago de Oliveira, J.. Reidel, nopagebreak[5] Dordrecht, pp. 261–284.CrossRefGoogle Scholar
Nosko, V. P. (1969). Local structure of Gaussian random fields in the vicinity of high-level light sources. Soviet Math. Dokl. 10, 1481–1484.Google Scholar
Talbot, J. D., Marrett, S., Evans, A. C., Meyer, E., Bushnell, M. C. and Duncan, G. H. (1991). Multiple representations of pain in human cerebral cortex. Science 251, 1355–1358.CrossRefGoogle ScholarPubMed
Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of χ2, F and t fields. Adv. Appl. Prob. 26, 13–42.CrossRefGoogle Scholar
Worsley, K. J., Evans, A. C., Marrett, S. and Neelin, P. (1992). A three dimensional statistical analysis for CBF activation studies in human brain. J. Cerebral Blood Flow Metab. 12, 900–918.CrossRefGoogle Scholar