In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl–Steiner tube formulae and the Chern–Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued random fields built up from i.i.d. copies of centered, unit-variance smooth Gaussian fields on a manifold M. Specifically, we consider random fields of the form fp=F(y1(p),…,yk(p)) for F∈C2(ℝk;ℝ) and (y1,…,yk) a vector of C2 i.i.d. centered, unit-variance Gaussian fields.
The analogue of the Weyl–Steiner formula for such Gaussian-related fields involves a power series expansion for the Gaussian, rather than Lebesgue, volume of tubes: that is, power series expansions related to the marginal distribution of the field f. The formal expansions of the Gaussian volume of a tube are of independent geometric interest.
As in the classical Weyl–Steiner formulae, the coefficients in these expansions show up in a kinematic formula for the expected Euler characteristic, χ, of the excursion sets M∩f−1[u,+∞)=M∩y−1(F−1[u,+∞)) of the field f.
The motivation for studying the expected Euler characteristic comes from the well-known approximation ![$\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.aop/1140191534&content-type=gif_1)
.
Primary Subjects: 60G15, 60G60, 53A17, 58A05
Secondary Subjects: 60G17, 62M40, 60G70
References
Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
Mathematical Reviews (MathSciNet):
MR611857
Adler, R. J. (2000). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Probab. 10 1--74.
Adler, R. J. and Taylor, J. E. (2006). Random Fields and Geometry. Birkhäuser, Boston. To appear. Preliminary versions of Chapters 1--12 available at http://iew3.technion.ac.il/~radler/grf.pdf.
Airault, H. (1991). Differential calculus on finite codimensional submanifolds of the Wiener space---the divergence operator. J. Funct. Anal. 100 291--316.
Airault, H. and Malliavin, P. (1988). Intégration géométrique sur l'espace de Wiener. Bull. Sci. Math. (2) 112 3--52.
Mathematical Reviews (MathSciNet):
MR942797
Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
Mathematical Reviews (MathSciNet):
MR969362
Bernig, A. and Bröcker, L. (2002). Lipschitz--Killing invariants. Math. Nachr. 245 5--25.
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.
Chern, S.-S. (1944). A simple intrinsic proof of the Gauss--Bonnet formula for closed Riemannian manifolds. Ann. of Math. (2) 45 747--752.
Mathematical Reviews (MathSciNet):
MR11027
Chern, S.-S. (1966). On the kinematic formula in integral geometry. J. Math. Mech. 16 101--118.
Mathematical Reviews (MathSciNet):
MR198406
Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418--491.
Mathematical Reviews (MathSciNet):
MR110078
Gray, A. (1990). Tubes. Addison--Wesley Publishing Company Advanced Book Program, Redwood City, CA.
Klain, D. A. and Rota, G.-C. (1997). Introduction to Geometric Probability. Cambridge Univ. Press.
Schneider, R. (1993). Convex Bodies: The Brunn--Minkowski Theory. Cambridge Univ. Press.
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.
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.
Taylor, J., Takemura, A. and Adler, R. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362--1396.
Taylor, J. E. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31 533--563.
Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461--472.
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.
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.
Worsley, K. J. (1995). Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist. 23 640--669.
Worsley, K. J. (1996). The geometry of random images. Chance 9 27--40.
Worsley, K. J. and Friston, K. J. (2000). A test for a conjunction. Statist. Probab. Lett. 47 135--140.
