Euler characteristics for Gaussian fields on manifolds

Annals of Probability

Volumn 31, Issue 2, 2003, Pages 533-563

  Taylor, Jonathan E ^a   Adler, Robert J ^b  

^a STANFORD UNIVERSITY   (United States)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Euler characteristic; Excursions; Gaussian processes; Manifolds; Random fields; Riemannian geometry

Indexed keywords

EID: 0038336161     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1048516527     Document Type: Article

References (23)

1. ADLER, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
2. ADLER, R. J. (1990). An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA.
3. ADLER, R. J. (2000). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Probab. 10 1-74.
4. BOOTHBY, W. M. (1986). An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, New York.
5. CAO, J. and WORSLEY, K. (1999). The geometry of correlation fields with an application to functional connectivity of the brain. Ann. Appl. Probab. 9 1021-1057.
6. 2 fields. Ann. Statist. 27 925-942.
7. CHEEGER, J., MÜLLER, W. and SCHRADER, R. (1984). On the curvature of piecewise flat spaces. Comm. Math. Phys. 92 405-454.
8. CHERN, S. S., ed. (1989). Global Differential Geometry. Amer. Math. Assoc., Washington, DC.
9. FEDERER, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418-491.
10. GRAY, A. (1990). Tubes. Addison-Wesley, Redwood City, CA.
11. KLAIN, D. A. and ROTA, G.-C. (1997). Introduction to Geometric Probability. Cambridge Univ. Press.
12. KLINGENBERG, W. (1982). Riemannian Geometry. de Guyter, Berlin.
13. MILNOR, J. (1963). Morse Theory. Princeton Univ. Press.
14. MORSE, M. and CAIRNS, S. (1969). Critical Point Theory in Global Analysis and Differential Topology. An Introduction. Academic Press, New York.
15. SANTALÓ, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.
16. SHAFIE, K., SIGAL, B., SIEGMUND, D. and WORSLEY, K.J. (2001). Rotation space random fields with an application to fMRI data. Unpublished manuscript.
17. 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.
18. TAKEMURA, A. and KURIKI, S. (2002). Maximum of Gaussian field on piecewise smooth domain: Equivalence of tube method and Euler characteristic method. Ann. Appl. Probab. 12 768-796.
19. TAYLOR, J. (2001). Euler characteristics for Gaussian fields on manifolds. Ph.D. thesis, McGill Univ.
20. WEYL, H. (1939). On the volume of tubes. Amer. J. Math. 61 461-472.
21. 2, F and t fields. Adv. in Appl. Probab. 26 13-42.
22. WORSLEY, K. J. (1995). Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. in Appl. Probab. 27 943-959.
23. YAGLOM, A. M. (1961). Second-order homogeneous random fields. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 2 593-622. Univ. California Press, Berkeley.