Tilted Euler characteristic densities for central limit random fields, with application to “bubbles”. (English) Zbl 1226.60075

Summary: Local increases in the mean of a random field are detected (conservatively) by thresholding a field of test statistics at a level \(u\) chosen to control the tail probability or \(p\)-value of its maximum. This \(p\)-value is approximated by the expected Euler characteristic (EC) of the excursion set of the test statistic field above \(u\), denoted \(\mathbb E_{\varphi}(A_u)\). Under isotropy, one can use the expansion \(\mathbb E_{\varphi}(A_u)= \sum _{k} \mathcal V _k \rho _k(u)\), where \(\mathcal V _k\) is an intrinsic volume of the parameter space and \(\rho _k\) is an EC density of the field. EC densities are available for a number of processes, mainly those constructed from (multivariate) Gaussian fields via smooth functions. Using saddlepoint methods, we derive an expansion for \(\rho _k(u)\) for fields which are only approximately Gaussian, but for which higher-order cumulants are available. We focus on linear combinations of \(n\) independent non-Gaussian fields, whence a central limit theorem is in force. The threshold \(u\) is allowed to grow with the sample size \(n\), in which case our expression has a smaller relative asymptotic error than the Gaussian EC density. Several illustrative examples including an application to “bubbles” data accompany the theory.


60G60 Random fields
62E20 Asymptotic distribution theory in statistics
62M40 Random fields; image analysis
53A99 Classical differential geometry
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
Full Text: DOI arXiv


[1] Adler, R. J. (1981). The Geometry of Random Fields . Wiley, New York. · Zbl 0478.60059
[2] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry . Springer, New York. · Zbl 1149.60003
[3] Adolphs, R., Gosselin, F., Buchanan, T. W., Tranel, D., Schyns, P. and Damasio, A. R. (2005). A mechanism for impaired fear recognition after amygdala damage. Nature 433 68-72.
[4] Barndorff-Nielsen, O. E. and Cox, D. R. (1989). Asymptotic Techniques for Use in Statistics . Chapman and Hall, London. · Zbl 0672.62024
[5] Bhattacharya, R. N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions . Wiley, New York. · Zbl 0331.41023
[6] Cao, J. and Worsley, K. J. (1999). The detection of local shape changes via the geometry of Hotelling’s t 2 fields. Ann. Statist. 27 925-942. · Zbl 0986.62076
[7] Carbonell, F., Galan, L. and Worsley, K. J. (2005). The geometry of the Wilks’s Lambda random field. Submitted. · Zbl 1432.60054
[8] Catelan, P., Lucchin, F. and Matarrese, S. (1988). Peak number density of non-Gaussian random-fields. Phys. Rev. Lett. 61 267-270.
[9] Chamandy, N. (2007). Inference for asymptotically Gaussian random fields. Ph.D. thesis, McGill University, Montréal.
[10] Chamandy, N. (2007). Thresholding asymptotically Gaussian random fields using the tilted mean EC, with application to lesion density maps. In Proceedings of the 56th Session of the International Statistical Institute .
[11] Chamandy, N., Taylor, J. E. and Worsley, K. J. (2008). Euler characteristic densities for asymptotically Gaussian random fields: A geometric approach. In preparation. · Zbl 1226.60075
[12] Daniels, H. E. (1954). Saddlepoint approximations in statistics. Ann. Math. Statist. 25 631-650. · Zbl 0058.35404
[13] Daniels, H. E. (1980). Exact saddlepoint approximations. Biometrika 67 59-63. JSTOR: · Zbl 0423.62018
[14] Daniels, H. E. (1987). Tail probability approximations. Internat. Statist. Rev. 55 37-48. JSTOR: · Zbl 0614.62016
[15] Field, C. and Ronchetti, E. (1990). Small Sample Asymptotics . IMS, Hayward, CA. · Zbl 0742.62016
[16] Gosselin, F. and Schyns, P. G. (2001). Bubbles, a technique to reveal the use of information in recognition. Vision Research 41 2261-2271.
[17] Hadwiger, H. (1957). Vorlesüngen Über Inhalt , Oberfläche und Isoperimetrie . Springer, Berlin. · Zbl 0078.35703
[18] Hoyle, F., Vogeley, M. S., Gott, J. R., Blanton, M., Tegmark, M., Weinberg, D., Bachall, N., Brinchman, J. and York, D. (2002). Two-dimensional topology of the SDSS. Astrophys. J. 580 663-671.
[19] Loader, C. R. and Sun, J. (1997). Robustness of tube formulae. J. Comput. Graph. Statist. 6 242-250. JSTOR:
[20] Lugannani, R. and Rice, S. O. (1980). Saddle point approximations for the distribution of the sum of independent random variables. Adv. in Appl. Probab. 12 475-490. JSTOR: · Zbl 0425.60042
[21] Matsubara, T. (1994). Analytic expression of the genus in weakly non-Gaussian field induced by gravity. Astrophys. J. 434 L43-L46.
[22] Matsubara, T. and Yokoyama, J. (1996). Genus statistics of the large-scale structure with non-Gaussian density fields. Astrophys. J. 463 409-419.
[23] Nardi, Y., Siegmund, D. and Yakir, B. (2008). The distribution of maxima of approximately Gaussian random fields. Ann. Statist. · Zbl 1148.60029
[24] Padmanabhan, N., Schlegel, D. J., Seljak, U., Makarov, A. et. al. (2006). The clustering of luminous red galaxies in the sloan digital sky survey imaging data. Mon. Not. R. Astron. Soc. Submitted. Available at http://arxiv.org/abs/astro–ph/0605302.
[25] Rabinowitz, D. and Siegmund, D. (1997). The approximate distribution of the maximum of a smoothed Poisson random field. Statist. Sinica 7 167-180. · Zbl 0895.60053
[26] Robinson, J. (1982). Saddlepoint approximations for permutation tests and confidence intervals. J. Roy. Statist. Soc. Ser. B 44 91-101. JSTOR: · Zbl 0487.62016
[27] Salmond, C. H., Ashburner, J., Vargha-Khadem, F., Connelly, A., Gadian, D. G. and Friston, K. J. (2002). Distributional assumptions in Voxel-based morphometry. Neuroimage 17 1027-1030.
[28] Schneider, R. (1993). Convex Bodies : The Brunn-Minkowski Theory . Cambridge Univ. Press. · Zbl 0798.52001
[29] Sun, J., Loader, C. R. and McCormick, W. P. (2000). Confidence bands in generalized linear models. Ann. Statist. 28 429-460. · Zbl 1106.62343
[30] Taylor, J. E. (2003). Gaussian volumes of tubes, Euler characteristic densities and correlated conjunction. In Proceedings of Singular Models and Geometric Methods in Statistics 7-33. IMS, Hayward, CA.
[31] Taylor, J. E. (2006). A Gaussian kinematic formula. Ann. Probab. 34 122-158. · Zbl 1094.60025
[32] Taylor, J. E. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31 533-563. · Zbl 1026.60039
[33] Taylor, J. E., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362-1396. · Zbl 1083.60031
[34] Taylor, J. E. and Worsley, K. J. (2008). Detecting sparse signal in random fields, with an application to brain mapping. J. Amer. Statist. Assoc. 102 913-928. · Zbl 1469.62353
[35] Taylor, J. E. and Worsley, K. J. (2008). Random fields of multivariate test statistics, with applications to shape analysis and fMRI. Ann. Statist. 36 1-27. · Zbl 1144.62083
[36] Taylor, J. E., Worsley, K. J. and Gosselin, F. Maxima of discretely sampled random fields, with an application to “bubbles.” Biometrika 94 1-18. · Zbl 1143.62059
[37] Viviani, R., Beschoner, P., Ehrhard, K., Schmitz, B. and Thöne, J. (2007). Non-normality and transformations of random fields, with an application to voxel-based morphometry. Neuroimage 35 121-130.
[38] Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of \chi 2 , F and t fields. Adv. in Appl. Probab. 26 13-42. JSTOR: · Zbl 0797.60042
[39] 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. JSTOR: · Zbl 0836.60043
[40] Worsley, K. J. (1996). The geometry of random images. Chance 9 27-40.
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.