Random fields of multivariate test statistics, with applications to shape analysis

Annals of Statistics

Volumn 36, Issue 1, 2008, Pages 1-27

  Taylor, J E ^a,b,d   Worsley, K J ^c  

^a STANFORD UNIVERSITY   (United States)

^b Universitéde Montré ^*

^c MCGILL UNIVERSITY   (Canada)

^d UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

Canonical correlation; Differential topology; Euler characteristic; Hotelling's T2; Integral geometry; Morse theory; Roy's maximum root; Scale space

Indexed keywords

EID: 50649110759     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053607000000406     Document Type: Article

References (31)

1. ADLER, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester. MR0611857
2. ADLER, R. J. (2000). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Probab. 10 1-74. MR 1765203
3. ADLER, R. J. and TAYLOR, J. E. (2007). Random Fields and Geometry. Springer, New York.
4. 2 fields. Ann. Statist. 27 925-942. MR 1724036
5. CAO. J. and WORSLEY, K. J. (1999). The geometry of correlation fields with an application to functional connectivity of the brain. Ann. Appl. Probab. 9 1021-1057. MR1727913
6. COLLINS, D. L., HOLMES, C. J., PETERS, T. M. and EVANS, A. C. (1995). Automatic 3-D model-based neuroanatomical segmentation. Human Brain Mapping 3 190-208.
7. FRISTON, K. J., BÜHEL, C., FINK. G. R., MORRIS. J., ROLLS, E. and DOLAN, R. J. (1997). Psychophysiological and modulatory interactions in neuroimaging. NeuroImage 6 218-229.
8. HADWIGER. H. (1957). Vorlesüngen ÜInhalt, Oberfläund Isoperimetrie. Springer, Berlin. MR0102775
9. JOHANSEN. S. and JOHNSTONE, I. M. (1990). Hotelling's theorem on the volume of tubes: Some illustrations in simultaneous inference and data analysis. Ann. Statist. 18652-684. MR1056331
10. KNOWLES, M. and SIEGMUND, D. (1989). On Hotelling's approach to testing for a nonlinear parameter in a regression. Internat. Statist. Rev. 57 205-220.
11. KURIKI, S. and TAKEMURA, A. (2001). Tail probabilities of the maxima of multilinear forms and their applications. Ann. Statist. 29 328-371. MR1863962
12. LIN, Y. and LINDSAY. B. G. (1997). Projections on cones, chi-bar squared distributions, and Weyl's formula. Statist. Probab. Lett. 32 367-376. MR1602207
13. MORSE, M. and CAIRNS. S. S. (1969). Critical Point Theory in Global Analysis and Differential Topology. Academic Press, New York. MR0245046
14. 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. MR 1332585
15. SUN. J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34-71. MR1207215
16. SUN, J. and LOADER. C. R. (1994). Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 1328-1345. MR1311978
17. SUN. J., LOADER. C. R. and MCCORMICK, W. P. (2000). Confidence bands in generalized linear models. Ann. Statist. 28429-460. MR 1790004
18. 2 distribution for smooth or piecewise smooth cone alternatives. Ann. Statist. 25 2368-2387. MR1604465
19. TAKEMURA, A. and KURIKI, S. (2002). On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab. 12 768-796. MR1910648
20. TAYLOR, J. E. (2006). A Gaussian kinematic formula. Ann. Probab. 34 122-158. MR2206344
21. TAYLOR, J. E. and ADLER, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31533-563. MR1964940
22. TAYLOR, J. E., TAKEMURA, A. and ADLER, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362-1396. MR2150192
23. TAYLOR, J. E. and WORSLEY, K. J. (2007). Detecting sparse cone alternatives for Gaussian random fields, with an application to FMRI. Available at http://www.math.mcgill.ca/keith/cone/cone.pdf.
24. 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.
25. TOMAIUOLO, F., WORSLEY, K. J., LERCH, J., DI PAULO, M., CARLESIMO, G. A., BONANNI, R., CALTAGIRONE, C. and PAUS, T. (2005). Changes in white matter in longterm survivors of severe non-missile traumatic brain injury: A computational analysis of magnetic resonance images. J. Neurotrauma 822 76-82.
26. 2. F and t fields. Adv. in Appl. Probab. 26 13-42. MR1260300
27. 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. MR1358902
28. 2 random field, with an application to fMRI. Adv. in Appl. Probab. 33 773-793. MR 1875779
29. WORSLEY, K. J., ANDERMANN, M., KOULIS, T., MACDONALD, D. and EVANS. A. C. (1999). Detecting changes in nonisotropic images. Human Brain Mapping 8 98-101.
30. WORSLEY, K. J. and FRISTON, K. J. (2000). A test for a conjunction. Statist. Probab. Lett. 47 135-140. MR1747100
31. WORSLEY, K. J., TAYLOR, J. E., TOMAIUOLO, F. and LERCH, J. (2004). Unified univariate and multivariate random field theory. NeuroImage 23 S189-S196.