Gaussian Kinematic Formula and the integral geometry of random sets

GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityGaussian Kinematic Formula and the integralgeometry of random setsJonathan TaylorStanford UniversityNovember 11, 20111 / 41

GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityOutlineA model for random sets.Some old integral geometry.Gaussian integral geometry.Accuracy of approximation.2 / 41

Random setsGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityGaussian processes: basic building blocksTwice-differentiable Gaussian process (f t ) t∈MM.Satisfying:E{f t } = 0;E{f 2t } = 1.on a manifoldWhy Gaussian?Gaussian processes are specified by mean and covariancefunction.Finite-dimensional distributions are all simple –multivariate Gaussian.3 / 41

Random setsGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityExcursion setsRandom sets we will consider are of the form:f −1 A = {t ∈ M : f t ∈ A}for A ⊂ R. In particular, the geometry of these sets, and how itis determined by correlation function of f .4 / 41

Excursion above 0GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity5 / 41

Excursion above 1GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity6 / 41

Excursion above 1.5GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity7 / 41

Excursion above 2GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity8 / 41

Excursion above 2.5GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity9 / 41

Excursion above 3GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity10 / 41

Excursion above 3.3GaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity11 / 41

Euler characteristicGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityWhat geometric features?We’re interested in integral geometric properties.Specifically, the (expected) Euler characteristicE { χ ( f −1 [u, +∞) )} = E { χ ( M ∩ f −1 [u, +∞) )}EC tells you very littleLet M be a 2-manifold without boundary, thenE { χ ( M ∩ f −1 [0, +∞) )} = 1 2 · χ(M)With boundary:E { χ ( M ∩ f −1 [0, +∞) )} = 1 2 · χ(M) + 12π · |∂M| 12 / 41

Euler characteristicGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityEC is computableOf all quantities in the studies of Gaussian processes, the ECstands out as being explicitly computable in wide generalityE { χ ( M ∩ f −1 [u, +∞) )} =dim(M)∑j=0L j (M)ρ j (u).Where do L j ’s and ρ j ’s come from?EC tells you a lotFor “nice enough” ME { χ ( M ∩ f −1 [u, +∞) )} }u→∞≃ P{sup f t ≥ ut∈M13 / 41

Euler characteristicGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityAccuracyIf the parameter set is locally convex, then}∣{sup P f t ≥ u − E { χ ( M ∩ f −1 [u, +∞) )}∣ ∣∣t∈M(u→∞ = Oexp e −u2 /2·(1+ 1 ))σc 2 (f )Since, ρ j (u) = O exp (e −u2 /2 ), the approximation hasexponential relative accuracy!14 / 41

Integral geometryGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityTube formulaeSuppose f is (the restriction } to M) of an isotropic process˜f , with Var{d˜f /dx i = 1.For small r, the functionals L j (M) are implicitly definedby Steiner-Weyl formula)H k(x ∈ R k : d(x, M) ≤ r =k∑ω k−j r k−j L j (M; M)j=015 / 41

Integral geometry: tubesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityH 3 (Tube([0, a] × [0, b] × [0, c], r))= abc + 2r · (ab + bc + ac) + (πr 2 ) · (a + b + c) + 4πr 3316 / 41

Integral geometry: tubesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity17 / 41

Integral geometry: tubesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityLocal convexity is important!Singularity means implicit definition is invalid, BUTL j (·) ′ s are still well defined . . .They are defined for a large class of sets . . .18 / 41

Gaussian processesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity19 / 41

Integral geometry: curvature measuresGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityH k−1 (exp ν (A ∩ ∂D, rν)) =k∑∫r j−1 · ω jj=1AL k−j (D; dp)20 / 41

Integral geometryGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityKinematic Fundamental FormulaWhere else do we see L j (·)’s?21 / 41

Integral geometryGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityKinematic Fundamental FormulaConsiders the “average” curvature measures of M 1 ∩ gM 2 ,i.e.22 / 41

Integral geometry: KFFGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityThe KFF on R NIsometry group G N of rigid motions of R N ,G N ∼ R N ⋊ O(N)Fix a Haar measure:ν N ({g N ∈ G N : g N x ∈ A}) = H N (A)∫L i (M 1 ∩ g N M 2 ) dν N (g N )G NN−i∑[ ] [ ] −1 i + j N=Li j i+j (M 1 )L N−j (M 2 )j=023 / 41

Back to Gaussian processesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityAnalogy with KFFRecall what we were trying to studyE { χ ( M ∩ f −1 [u, +∞) )}∫= L 0 (M ∩ f (ω) −1 [u, +∞)) P(dω)=Ωdim(M)∑j=0L j (M)ρ j (u)This looks like KFF on R N where g N M 2 is replaced byf −1 [u, +∞) = f −1 D.Can replace f with f = (f 1 , . . . , f j ) . Let’s look at someexamples . . .24 / 41

Gaussian processes: D a coneGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity25 / 41

Gaussian processes: f −1 DGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity26 / 41

Gaussian processes: D a varietyGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity27 / 41

Gaussian processes: f −1 DGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity28 / 41

Gaussian processesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityGaussian Kinematic FormulaLet f = (f 1 , . . . , f k ) be made of IID copies of a GaussianfieldConsider the additive functional on R k that takes a“rejection region”D ↦→ E { χ ( M ∩ f −1 D )} .Questions: how do the L j ’s enter into this functional?How about D?29 / 41

Gaussian processesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityGaussian Kinematic FormulaDefine the functionals M γ kj(·) implicitly by)γ k(y ∈ R k : d(y, D) ≤ r = ∑ j≥0(2π) j/2 r jj!M γ kj(D).Then:E { χ ( M ∩ f −1 D )} = ∑ jL j (M) · M γ kj(D)30 / 41

Integral geometry: curvature measuresGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityQ: How do we compute M γ kj(·)?31 / 41

Integral geometry: curvature measuresGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityA: Integrate power series expansion for density overhypersurface at distance r . . .32 / 41

Example: linear statisticGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity33 / 41

Example: χ 2 statisticGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity34 / 41

ExamplesGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityEC densitiesGaussian EC densitiesρ j (u) = (−1) j d jP {N(0, 1) > u}duj χ 2 kEC densitiesρ j,χ 2k(u) = (−1) j d jdx j P {√χ 2 k > x } ∣ ∣∣∣x=√ u35 / 41

t or F statisticGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity36 / 41

t or F statisticGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversity37 / 41

Ideas behind the proofGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityHidden embeddingA Gaussian process is just a mappingt ↦→ f t ∈ L 2 (Ω, F, P)Our assumptions about mean and variance implies theimage is in the unit sphere in L 2 (Ω, F, P), and it is anembedding.This suggests that the relevant “geometry” to prove thisresult is spherical.Short answer: yes.38 / 41

Proof: spherical KFFGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversitySpherical KFFFor κ ∈ R define∞∑L κ (−κ) n (i + 2n)!(·) =(4π) n L i+2n (·).n!i!n=0For M 1 , M 2 ⊂ S n 1/2(R n )∫L n−1i (M 1 ∩ g n M 2 ) dν n,λ (g n )G nn−1−i∑[ ] [ ] −1 i + j n − 1=L n−1i+j (Mi j1 )L n−1−j(M 2 )j=0where G n = O(n), appropriately normalized.39 / 41

Proof: Poincaré’s limitGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityModel processFor M ⊂ S(R j ) define a R k valued processf n (t, g n ) = π k (n 1/2 g n t)where g n ∈ O(n) is a Haar-distributed random matrix andπ k : S n 1/2(R n ) → R k is projection onto the first kcoordinates.Poincaré’s limit (and generalizations) ensures that theprocess f n = (f1 n,. . . , f k n ) converges in variation to avector of IID zero mean, unit variance Gaussian processesf = (f 1 , . . . , f k ).40 / 41

Proof: connecting Gaussian processes with KFFGaussianKinematicFormula andthe integralgeometry ofrandom setsJonathanTaylorStanfordUniversityExpected EC for model processFor D ⊂ R k∫(M ∩ (f n ) −1 (D) ) dν n,λ (g n )G nL 1 i= n −i/2 ∫n−1−i∑= c nj=0L n−1iG n[ i + ji(n 1/2 M ∩ π −1k D )dν n,λ (g n )] [ ] −1 n − 1L 1j i+j(M)L n−1−j(π −1k D)The set π −1kD is the disjoint union of a warped productand D ∩ S n 1/2(R k ). Need to analyse curvatures of warpedproduct asymptotically.The rest is combinatorics . . . almost.41 / 41