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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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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Gunter Last Distributional properties of Poisson Voronoi tessellations
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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).
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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 .
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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 .
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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)}.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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 .
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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
].
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
(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| .
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Gunter Last Distributional properties of Poisson Voronoi tessellations
Situation under P0Mk
for k = 1, d = 3.
X1
X2X0
0
F1
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Gunter Last Distributional properties of Poisson Voronoi tessellations
Situation under P0Mk
for k = 1, d = 3.
X1
X2X0
Z0
F1
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Gunter Last Distributional properties of Poisson Voronoi tessellations
Situation under P0Mk
for k = 1, d = 3.
X1
X2X0
RR′
R′′0
F1
B(0, R1)
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
(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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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.
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Gunter Last Distributional properties of Poisson Voronoi tessellations
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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