distributional properties of poisson voronoi tessellationslast/seite/pub/media/prag06...

Gunter Last

Institut fur Stochastik

Universitat Karlsruhe (TH)

Distributional properties of Poisson Voronoi tessellations

Gunter Last

Universitat Karlsruhe (TH)

joint work with Volker Baumstark (Karlsruhe)

Prague Stochastics 2006

Charles University

22.08.2006

Gunter Last Distributional properties of Poisson Voronoi tessellations

1. Voronoi tessellations

Definition:

(i) The space of all point configurations in Rd is defined as

N := {ϕ ⊂ Rd : ϕ is locally finite}.

(ii) Any ϕ ∈ N is identified with a counting measure:

ϕ(B) := card{x ∈ ϕ : x ∈ B}, B ⊂ Rd.

(iii) The σ-field N is the smallest σ-field of subsets of N making the

mappings ϕ 7→ ϕ(B) for all Borel sets B ⊂ Rd measurable.

22.08.2006, Prague Stochastics Slide 2/ 29


Definition: The points of ϕ ∈ N are in general quadratic position if

the following two conditions are satisfied.

(i) Any k ∈ {2, . . . , d+ 2} points of ϕ are in general position.

(ii) No d+ 2 points of ϕ lie on the boundary of some ball.



Definition: Let ϕ ∈ N.

(i) The Voronoi cell C(ϕ, x) of x ∈ ϕ is the set of all sites y ∈ Rdwhose distance from x is smaller or equal than the distances to

all other points of ϕ.

(ii) The Voronoi tessellation based on ϕ is the system

Sd(ϕ) := {C(ϕ, x) : x ∈ ϕ}.

Remark: If the convex hull of ϕ coincides with Rd, then all Voronoi

cells are bounded and Sd(ϕ) is a face-to face tessellation. Moreover,

if the point of ϕ are in general quadratic position, then Sd(ϕ) is also

normal in the sense that any k-face is contained in exactly d−k+ 1

cells.



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Definition: Let C be a convex polytope. Then

C =⋃

k∈{0,...,d}

⋃

C∈Sk(C)

relintF,

where Sk(C) is a finite set of k-dimensional polytopes whose affine

hulls are pairwise not equal. A polytope F ∈ Sk(C) is called a

k-face of C.

Definition: Let ϕ ∈ N and k ∈ {0, . . . , d}. The system of all k-faces

of the Voronoi tessellation Sd(ϕ) is defined by

Sk(ϕ) :=⋃

C∈Sd(ϕ)

Sk(C).



2. Stationary point processes and random measures

Definition:

(i) For any x ∈ Rd the shift θx : N→ N is defined by

θxϕ = ϕ− x.

(ii) A probability measure P on (N,N ) is stationary if

P ◦ θx = P, x ∈ Rd.



Assumption: P is a stationary probability measure on (N,N ).

Definition:

(i) M denotes the space of all locally finite measures on Rd.

(ii) The σ-field M is the smallest σ-field of subsets of M making

the mappings α 7→ α(B) for all Borel sets B ⊂ Rd measurable.

(iii) A random measure M is a measurable mapping from N to M.

(iv) A random measure M is stationary if

M(ϕ,B + x) = M(θxϕ,B), ϕ ∈M, x ∈ Rd, B ∈ Bd.

Remark: The identity N on N is a stationary random measure.



Definition: Let M be a stationary random measure.

(i) The intensity of M is the number

λM := E[M([0, 1]d)].

(ii) If λM is positive and finite, then

P0M (A) :=

1

λME[∫

1{θxN ∈ A, x ∈ [0, 1]d}M(dx)

], A ∈ N ,

is called Palm probability measure of M .



3. Typical faces

Assumption: P is a stationary probability measure on (N,N ) such

that almost all ϕ ∈ N are non-empty and the points of almost all

ϕ ∈ N are in general quadratic position. We consider the (random)

Voronoi tessellation

Sd(N) = {C(N, x) : x ∈ N}

generated by N .



Definition: Let k ∈ {0, . . . , d} and F ∈ Sk. Take some y in the

relative interior of F and assume that the points of N are in gen-

eral position. Then there are exactly d − k + 1 different points

X0, . . . , Xd−k ∈ N (the neighbours of F ) such that the distances

Ry := ‖Xi − y‖ are the same for all i and such that the open ball

with centre y and radius Ry does not contain any point of N . Let

πk(F ) denote the centre of the unique (d − k)-dimensional ball in

the affine hull of the neighbours containing the neighbours on its

boundary. Define the stationary point process of centres of k-faces

by

Nk := {πk(F ) : F ∈ Sk(N)}.



Assumption: For any k ∈ {0, . . . , d} the intensity

λk := E[Nk([0, 1]d)]

is assumed to be finite.

Remark: We have a.s. that N = Nd and hence λd = λ.

Definition: Let k ∈ {0, . . . , d}. Under the Palm probability measure

P0Nk

we denote by Ck ∈ Sk(N) the k-face satisfying π(Ck) = 0. The

distribution

P0Nk

(Ck ∈ ·)is the distribution of the typical k-face of the Voronoi tessellation

based on N .



Definition: For any k ∈ {0, . . . , d} we define the stationary random

measure

Mk :=∑

F∈Sk(N)

Hk(F ∩ ·),

where Hk denotes k-dimensional Hausdorff measure on Rd.

Assumption: For any k ∈ {0, . . . , d} the intensity

µk := E[Mk([0, 1]d)]

is assumed to be finite.

Remark: We have M0 = N0 and hence λ0 = µ0.



Definition: Let k ∈ {0, . . . , d}. Under the Palm probability measure

P0Mk

we denote by Fk ∈ Sk(N) the k-face satisfying 0 ∈ Fk. The

distribution

P0Mk

(Fk ∈ ·)can be interpreted as an area-biased version of the distribution of

the typical k-face.

Proposition: Consider k ∈ {0, . . . , d} and a measurable and shift-

invariant function g : N→ [0,∞). Then

µkE0Mk

[g] = λkE0Nk

[Hk(Ck) · g

],

λkE0Nk

[g] = µkE0Mk

[Hk(Fk)−1 · g

].



4. Mean values for typical faces

Corollary: For any k ∈ {0, . . . , d} we have

µk = λkE0Nk

[Hk(Ck)

],

λk = µkE0Mk

[Hk(Fk)−1

].

In particular

E0Nd

[Hd(Cd)] = λ−1,

E[Hd(Fd)−1] = λ.

Proposition: We have

d∑

j=0

(−1)jλj = 0.



Definition: Let Pd denote the system of all convex polytopes in Rd.For k ∈ {0, . . . , d} we define νk : Pd → N by

νk(F ) := cardSk(F ).

Proposition: Consider the planar case d = 2. Then λ0 = 2λ and

λ1 = 3λ. Moreover,

E0N2

[H2(C2)] =1

λ,

E0N2

[H1(∂C2)] =2µ1

λ,

E0N2

[ν0(C2)] = 6,

E0N1

[H1(C1)] =µ1

3λ.



Theorem: If N is a stationary Poisson process of intensity λ then

the intensities µk are explicitly known. In case d = 2 we have

µ0 = 2λ, µ1 = 2√λ

and in case d = 3 we have

µ0 =24π2

35λ, µ1 =

48π2

35λ, µ2 =

(24π2

35+ 1)λ.

Problem: Assume thatN is a stationary Poisson process. Determine

the distributions

P0Nk

(Ck ∈ ·), k = 0, . . . , d,

and

P0Mk

(Fk ∈ ·), k = 0, . . . , d.



5. The neighbours of a typical vertex

Assumption: N is a stationary Poisson process of intensity λ > 0.

Definition: Consider the probability measure P0N0

.

(i) Almost surely there are exactly d + 1 different points

X0, . . . , Xd ∈ N (lexicographically ordered) such that

{0} = C(N,X0) ∩ · · · ∩ C(N,Xd).

The points X0, . . . , Xd are the neighbours of the origin.

(ii) Let R := |X0| = · · · = |Xd| denote the distance to the neigh-

bours and define the unit vectors

U0 :=X0

R, . . . , Ud :=

Xd

R.



Theorem: Under the probability measure P0N0

the following holds.

(i) The random variables ({x ∈ N : |x| > R}, R) and (U0, . . . , Ud)

are independent.

(ii) Rd is Gamma distributed with shape parameter d and scale pa-

rameter γκd.

(iii) The conditional distribution of {x ∈ N : |x| > R} given R = r

is the distribution of a Poisson process restricted to the comple-

ment of the ball B(0, r).

(iv) {U0, . . . , Ud} has distribution

c−10

∫· · ·∫

1{{u0, . . . , ud} ∈ ·}∆d(u0, . . . , ud−k) S(du0) . . .S(dud)

where ∆d(u0, . . . , ud) is the volume of the simplex spanned by

the vectors u0, . . . , ud, S is the uniform distribution on the unit

sphere Sd−1 and c0 is an explicitly known constant.



6. The length-biased distribution of the neigbours ofa typical face

Assumption: N is a stationary Poisson process of intensity λ > 0.

Definition: Consider the probability measure P0Mk

for some fixed

k ∈ {1, . . . , d− 1}.(i) Almost surely there is exactly one k-face Fk ∈ Sk(N) such that

0 ∈ Fk.

(ii) Almost surely there are exactly d − k + 1 different points

X0, . . . , Xd−k ∈ N (lexicographically ordered) such that

Fk = C(N,X0) ∩ · · · ∩ C(N,Xd−k).

The points X0, . . . , Xd−k are the neighbours of Fk.



(iii) Let

R := |X0| = · · · = |Xd−k|denote the distance of the origin from the neighbours.

(iv) There is a unique (d − k)-dimensional ball in the affine hull of

the neighbours containing the neighbours on its boundary. We

let Z denote the centre of this ball.

(v) Let

R′ := |X0 − Z|, R′′ := |Z|.so that

R2 = R′2

+R′′2.

(vi) Define the unit vectors

U0 :=X0 − ZR′

, . . . , Ud−k :=Xd−k − Z

R′, U :=

Z

|Z| .



Situation under P0Mk

for k = 1, d = 3.

X1

X2X0

0

F1




for k = 1, d = 3.

X1

X2X0

Z0

F1




for k = 1, d = 3.

X1

X2X0

RR′

R′′0

F1

B(0, R1)



Theorem: Under the probability measure P0Mk

the following holds.

(i) The random variables ({x ∈ N : |x| > R}, R), R′2/R2, and

(U0, . . . , Ud−k, U) are independent.

(ii) Rd is Gamma distributed with shape parameter d− k+ k/d and

scale parameter γκd.

(iii) The conditional distribution of {x ∈ N : |x| > R} given R = r

is the distribution of a Poisson process restricted to the comple-

ment of the ball B(0, r).

(iv) R′2/R2 has a Beta distribution with parameters d(d− k)/2 and

k/2.



(v) Fix a d − k-dimensional linear subspace L ⊂ Rd. The random

pair ({Uk,0, . . . , Uk,d−k}, Uk) has distribution

c−1k

∫· · ·∫

1{({ϑu0, . . . , ϑud−k}, ϑu)) ∈ ·}

∆d−k(u0, . . . , ud−k)k+1SL(du0) . . .SL(dud−k) SL⊥(du) ν(dϑ),

where ∆d−k(u0, . . . , ud−k) is the (d− k)-dimensional volume of

the simplex spanned by the vectors u0, . . . , ud−k, ν is the uni-

form distribution on the rotation group SOd, ck is an explicitly

known constant, and SL and SL⊥ are the uniform distributions

(normalized Haar measures) on the unit spheres in L and the

orthogonal complement L⊥ of L, respectively.



7. The distribution of the typical edge and its neighbours

Definition: Assume that N is a stationary Poisson process of inten-

sity λ > 0 and consider the Palm probability measure P0N1

.

(i) Let L denote the length of the typical edge C1. Further let Φ1

denote the set of the d unit vectors pointing from π1(C1) to the

neighbours of C1.

(ii) Let α′ and α′′ denote the angles in [0, π] spanned by the edge

C1 and the vectors pointing from the endpoints of C1 to one of

the neighbours of C1.

(iii) Let ξ denote the volume of the union of the two balls centered

at the endpoints of the edge C1 whose radii are giben by the

respective distances from the endpoints to one of the neighbours

of C1.



Theorem: Under P0N1

we have the following assertions:

(i) The random variables (α′, α′′), ξ and Φ1 are independent.

(ii) The random variable ξ has a Gamma distribution with shape

parameter d+ 1 and scale parameter 1.

(iii) The distribution of (cosα′, cosα′′) has an explicitly known and

integral-free density.

(iv) The distribution of Φ1 is the same as the distribution of the cor-

responding unit vectors under PM1 . It has been given in Section

6.

Remark: Under P0N1

the random variables α′, α′′, ξ and Φ determine

the edge C1 and the positions of its neighbours.



References:

Miles (1974). A synopsis of ‘Poisson flats in Euclidean spaces’. In

Stochastic Geometry. ed. E. F. Harding and D. G. Kendall, Wiley,

New York.

Møller (1989). Random Tessellations in Rd. Advances in Applied

Probability 21, 37–73.

Schneider and Weil (2000). Stochastische Geometrie. Teubner,

Stuttgart.

Muche (2005). The Poisson-Voronoi tessellation: relationships for

edges. Advances in Applied Probability 37, 279–296.

Baumstark and Last (2006). Some distributional results for Poisson

Voronoi tessellations. submitted for publication.


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