Skip to main content Accessibility help

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

  • J. Cao (a1)


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.


Corresponding author

Postal address: Bell Laboratories, Lucent Technologies, 700 Mountain Avenue, Room 2C-260, Murray Hill, NJ 07974-2070, USA. Email address:


Hide All
Adler, R. J. (1981). The Geometry of Random Fields. Wiley, New York.
Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
Aronowich, M. and Adler, R. J. (1986). Extrema and level crossings of χ2 processes. Adv. Appl. Prob. 18, 901920.
Aronowich, M. and Adler, R. J. (1988). Sample path behaviour of χ2 surfaces at extrema. Adv. Appl. Prob. 20, 719738.
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, 214220.
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. 1421.
Kac, M. and Slepian, D. (1959). Large excursions of Gaussian processes. Ann. Math. Statist. 30, 12151228.
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremal and Related Properties of Stationary Processes. Springer, New York.
Lindgren, G. (1972). Local maxima of Gaussian fields. Ark. Math. 10, 195218.
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. 261284.
Nosko, V. P. (1969). Local structure of Gaussian random fields in the vicinity of high-level light sources. Soviet Math. Dokl. 10, 14811484.
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, 13551358.
Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of χ2, F and t fields. Adv. Appl. Prob. 26, 1342.
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, 900918.


MSC classification

Related content

Powered by UNSILO

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

  • J. Cao (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.