Mathematical tools for analysis, modeling and simulation
of spatial networks on various length scales
Part I
Volker Schmidt
Ulm University, Institute of Stochastics
Blanton Museum of Art, UT Austin
May 20, 2015
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Mathematical tools for spatial networks on various length scales |
Contents
Introduction
Point processes and Palm calculus
Random tessellations
Local simulation of typical Voronoi cells
Cox processes on random tessellations
Multiscale network modeling (Outlook to part II)
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Mathematical tools for spatial networks on various length scales |
Contents
Introduction
Point processes and Palm calculus
Random tessellations
Local simulation of typical Voronoi cells
Cox processes on random tessellations
Multiscale network modeling (Outlook to part II)
Introduction
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Mathematical tools for spatial networks on various length scales |
Motivation
I
Aim: Stochastic modeling of networks for
I
description of networks by only a few parameters
Introduction
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Mathematical tools for spatial networks on various length scales |
Introduction
Motivation
I
Aim: Stochastic modeling of networks for
I
I
description of networks by only a few parameters
simulation of present and future network design scenarios
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Mathematical tools for spatial networks on various length scales |
Introduction
Motivation
I
Aim: Stochastic modeling of networks for
I
I
I
description of networks by only a few parameters
simulation of present and future network design scenarios
control of service quality
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Mathematical tools for spatial networks on various length scales |
Introduction
Motivation
I
Aim: Stochastic modeling of networks for
I
I
I
I
description of networks by only a few parameters
simulation of present and future network design scenarios
control of service quality
cost analysis and risk evaluation
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Mathematical tools for spatial networks on various length scales |
Introduction
Motivation
I
Aim: Stochastic modeling of networks for
I
I
I
I
I
description of networks by only a few parameters
simulation of present and future network design scenarios
control of service quality
cost analysis and risk evaluation
Models necessary both
I
for locations of network components (point processes), and
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Mathematical tools for spatial networks on various length scales |
Introduction
Motivation
I
Aim: Stochastic modeling of networks for
I
I
I
I
I
description of networks by only a few parameters
simulation of present and future network design scenarios
control of service quality
cost analysis and risk evaluation
Models necessary both
I
I
for locations of network components (point processes), and
for systems of communication paths and serving zones
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Mathematical tools for spatial networks on various length scales |
Introduction
Motivation
I
Aim: Stochastic modeling of networks for
I
I
I
I
I
description of networks by only a few parameters
simulation of present and future network design scenarios
control of service quality
cost analysis and risk evaluation
Models necessary both
I
I
for locations of network components (point processes), and
for systems of communication paths and serving zones
(edge sets and cells of random tessellations)
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Mathematical tools for spatial networks on various length scales |
Contents
Introduction
Point processes and Palm calculus
Random tessellations
Local simulation of typical Voronoi cells
Cox processes on random tessellations
Multiscale network modeling (Outlook to part II)
Point processes and Palm calculus
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Counting measures
I
Denote by N the set of locally finite counting measures
ϕ : B(R2 ) → {0, 1, . . . } ∪ {∞}
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Counting measures
I
Denote by N the set of locally finite counting measures
ϕ : B(R2 ) → {0, 1, . . . } ∪ {∞}
I
Let N be the smallest σ-algebra on N s.t. ϕ → ϕ(B) is measurable for
each bounded B ∈ B(R2 )
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Counting measures
I
Denote by N the set of locally finite counting measures
ϕ : B(R2 ) → {0, 1, . . . } ∪ {∞}
I
Let N be the smallest σ-algebra on N s.t. ϕ → ϕ(B) is measurable for
each bounded B ∈ B(R2 )
Examples:
I
Let {x1 , . . . , xn } ⊂ R2 , then
ϕ(B) =
n
X
i=1
δxi (B) = #{xi ∈ B}
for B ∈ B(R2 )
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Counting measures
I
Denote by N the set of locally finite counting measures
ϕ : B(R2 ) → {0, 1, . . . } ∪ {∞}
I
Let N be the smallest σ-algebra on N s.t. ϕ → ϕ(B) is measurable for
each bounded B ∈ B(R2 )
Examples:
I
Let {x1 , . . . , xn } ⊂ R2 , then
ϕ(B) =
n
X
δxi (B) = #{xi ∈ B}
for B ∈ B(R2 )
i=1
I
Let {xi,j = (i, j) : i ∈ Z, j ∈ Z}, then
ϕ(B) =
XX
i∈Z j∈Z
δ(i,j) (B) = #{x ∈ Z2 ∩ B}
for B ∈ B(R2 )
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Point processes
Definition
Let
I
(Ω, A, P) some probability space,
I
X1 , X2 , · · · : Ω 7−→ R2 a sequence of random vectors such that
#{Xn ∈ B} < ∞
for each bounded
B ∈ B(R2 ) .
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Point processes
Definition
Let
I
(Ω, A, P) some probability space,
I
X1 , X2 , · · · : Ω 7−→ R2 a sequence of random vectors such that
B ∈ B(R2 ) .
P∞
Then the mapping X from (Ω, A, P) into (N, N ) defined by X = n=1 δXn is
called a (random) point process.
#{Xn ∈ B} < ∞
for each bounded
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Point processes
Definition
Let
I
(Ω, A, P) some probability space,
I
X1 , X2 , · · · : Ω 7−→ R2 a sequence of random vectors such that
B ∈ B(R2 ) .
P∞
Then the mapping X from (Ω, A, P) into (N, N ) defined by X = n=1 δXn is
called a (random) point process.
#{Xn ∈ B} < ∞
I
Henceforth: Identify X =
P∞
for each bounded
n=1 δXn
and {Xn }, where we write X = {Xn }
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Point processes
Definition
Let
I
(Ω, A, P) some probability space,
I
X1 , X2 , · · · : Ω 7−→ R2 a sequence of random vectors such that
B ∈ B(R2 ) .
P∞
Then the mapping X from (Ω, A, P) into (N, N ) defined by X = n=1 δXn is
called a (random) point process.
#{Xn ∈ B} < ∞
P∞
for each bounded
and {Xn }, where we write X = {Xn }
I
Henceforth: Identify X =
I
In other words: The point
X = {Xn } is identified with the random
Pprocess
∞
counting measure X = n=1 δXn .
n=1 δXn
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Intensity measure, stationarity and Palm distribution
Definition
Let X be a point process, then
I
the intensity measure µ : B(R2 ) → [0, ∞] of X is defined as
µ(B) = EX (B) ,
B ∈ B(R2 ) ,
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Intensity measure, stationarity and Palm distribution
Definition
Let X be a point process, then
I
the intensity measure µ : B(R2 ) → [0, ∞] of X is defined as
µ(B) = EX (B) ,
I
B ∈ B(R2 ) ,
D
X is called stationary if {Xn − x} = {Xn } for each x ∈ R2 .
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Intensity measure, stationarity and Palm distribution
Definition
Let X be a point process, then
I
the intensity measure µ : B(R2 ) → [0, ∞] of X is defined as
µ(B) = EX (B) ,
B ∈ B(R2 ) ,
D
I
X is called stationary if {Xn − x} = {Xn } for each x ∈ R2 .
I
If X is stationary, then µ(B) = λν2 (B), B ∈ B(R2 ), for some λ > 0, which
is called the intensity of X .
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Intensity measure, stationarity and Palm distribution
Definition
Let X be a point process, then
I
the intensity measure µ : B(R2 ) → [0, ∞] of X is defined as
µ(B) = EX (B) ,
B ∈ B(R2 ) ,
D
I
X is called stationary if {Xn − x} = {Xn } for each x ∈ R2 .
I
If X is stationary, then µ(B) = λν2 (B), B ∈ B(R2 ), for some λ > 0, which
is called the intensity of X .
I
The Palm distribution PXo : N 7→ [0, 1] of a stationary point process X with
intensity λ is defined as
PXo (A) =
1
E#{n : Xn ∈ [0, 1]2 , X ( · + Xn ) ∈ A} ,
λ
A∈N
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Intensity measure, stationarity and Palm distribution
Definition
Let X be a point process, then
I
the intensity measure µ : B(R2 ) → [0, ∞] of X is defined as
µ(B) = EX (B) ,
B ∈ B(R2 ) ,
D
I
X is called stationary if {Xn − x} = {Xn } for each x ∈ R2 .
I
If X is stationary, then µ(B) = λν2 (B), B ∈ B(R2 ), for some λ > 0, which
is called the intensity of X .
I
The Palm distribution PXo : N 7→ [0, 1] of a stationary point process X with
intensity λ is defined as
PXo (A) =
I
1
E#{n : Xn ∈ [0, 1]2 , X ( · + Xn ) ∈ A} ,
λ
A∈N
Note that a point process X o with distribution PXo is called a Palm version
of X .
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Examples: Stationary Poisson processes
For any fixed λ > 0, let X be a point process such that
I X (B) ∼ Poi(λν2 (B)) for each bounded B ∈ B(R2 ), and
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Examples: Stationary Poisson processes
For any fixed λ > 0, let X be a point process such that
I X (B) ∼ Poi(λν2 (B)) for each bounded B ∈ B(R2 ), and
I X (B1 ), . . . , X (Bn ) independent random variables for any pairwise disjoint
B1 , . . . , Bn ∈ B(R2 ).
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Examples: Stationary Poisson processes
For any fixed λ > 0, let X be a point process such that
I X (B) ∼ Poi(λν2 (B)) for each bounded B ∈ B(R2 ), and
I X (B1 ), . . . , X (Bn ) independent random variables for any pairwise disjoint
B1 , . . . , Bn ∈ B(R2 ).
Then X is called a stationary Poisson process with intensity λ.
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Examples: Stationary Poisson processes
For any fixed λ > 0, let X be a point process such that
I X (B) ∼ Poi(λν2 (B)) for each bounded B ∈ B(R2 ), and
I X (B1 ), . . . , X (Bn ) independent random variables for any pairwise disjoint
B1 , . . . , Bn ∈ B(R2 ).
Then X is called a stationary Poisson process with intensity λ.
Realization of a stationary Poisson process
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
General Poisson processes
For any (locally finite) measure µ : B(R2 ) → [0, ∞], let X be a point process
such that
I X (B) ∼ Poi(µ(B)) for each bounded B ∈ B(R2 ), and
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
General Poisson processes
For any (locally finite) measure µ : B(R2 ) → [0, ∞], let X be a point process
such that
I X (B) ∼ Poi(µ(B)) for each bounded B ∈ B(R2 ), and
I X (B1 ), . . . , X (Bn ) independent random variables for any pairwise disjoint
B1 , . . . , Bn ∈ B(R2 ).
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
General Poisson processes
For any (locally finite) measure µ : B(R2 ) → [0, ∞], let X be a point process
such that
I X (B) ∼ Poi(µ(B)) for each bounded B ∈ B(R2 ), and
I X (B1 ), . . . , X (Bn ) independent random variables for any pairwise disjoint
B1 , . . . , Bn ∈ B(R2 ).
Then X is called a Poisson process with intensity measure µ.
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
General Poisson processes
For any (locally finite) measure µ : B(R2 ) → [0, ∞], let X be a point process
such that
I X (B) ∼ Poi(µ(B)) for each bounded B ∈ B(R2 ), and
I X (B1 ), . . . , X (Bn ) independent random variables for any pairwise disjoint
B1 , . . . , Bn ∈ B(R2 ).
Then X is called a Poisson process with intensity measure µ.
Realization of a general (non-stationary) Poisson process
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Poisson-related point processes
I
Poisson cluster processes
I
Poisson hardcore processes
Realizations of a Poisson cluster process (left) and a Poisson hardcore
process (right)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-cluster processes
I
Constructed from Poisson processes (of cluster centers)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-cluster processes
I
Constructed from Poisson processes (of cluster centers)
I
Cluster centers form a stationary Poisson process (with some intensity λ0 )
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-cluster processes
I
Constructed from Poisson processes (of cluster centers)
I
I
Cluster centers form a stationary Poisson process (with some intensity λ0 )
Cluster members form (independent) stationary Poisson processes with
some intensity λ1 , within discs of some radius R around the cluster centers
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-cluster processes
I
Constructed from Poisson processes (of cluster centers)
I
I
I
Cluster centers form a stationary Poisson process (with some intensity λ0 )
Cluster members form (independent) stationary Poisson processes with
some intensity λ1 , within discs of some radius R around the cluster centers
⇒ Spatial interaction between points (mutual attraction)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-cluster processes
I
Constructed from Poisson processes (of cluster centers)
I
I
I
Cluster centers form a stationary Poisson process (with some intensity λ0 )
Cluster members form (independent) stationary Poisson processes with
some intensity λ1 , within discs of some radius R around the cluster centers
⇒ Spatial interaction between points (mutual attraction)
I
Realizations are clustered point patterns (with higher spatial variablility that
in the Poisson case)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-cluster processes
I
Constructed from Poisson processes (of cluster centers)
I
I
I
⇒ Spatial interaction between points (mutual attraction)
I
I
Cluster centers form a stationary Poisson process (with some intensity λ0 )
Cluster members form (independent) stationary Poisson processes with
some intensity λ1 , within discs of some radius R around the cluster centers
Realizations are clustered point patterns (with higher spatial variablility that
in the Poisson case)
Three-parametric model with parameters λ0 , λ1 and R
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-hardcore processes
I
Constructed from Poisson processes (by random deletion of points)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-hardcore processes
I
Constructed from Poisson processes (by random deletion of points)
I
Start from a stationary Poisson process (with some intensity λ)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-hardcore processes
I
Constructed from Poisson processes (by random deletion of points)
I
I
Start from a stationary Poisson process (with some intensity λ)
Cancel those points whose distance to their nearest neighbor is smaller than
some threshold R
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-hardcore processes
I
Constructed from Poisson processes (by random deletion of points)
I
I
I
Start from a stationary Poisson process (with some intensity λ)
Cancel those points whose distance to their nearest neighbor is smaller than
some threshold R
⇒ Spatial interaction between points (mutual repulsion)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-hardcore processes
I
Constructed from Poisson processes (by random deletion of points)
I
I
I
Start from a stationary Poisson process (with some intensity λ)
Cancel those points whose distance to their nearest neighbor is smaller than
some threshold R
⇒ Spatial interaction between points (mutual repulsion)
I
Realizations are regular point patterns (with smaller spatial variablility that in
the Poisson case)
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Matern-hardcore processes
I
Constructed from Poisson processes (by random deletion of points)
I
I
I
⇒ Spatial interaction between points (mutual repulsion)
I
I
Start from a stationary Poisson process (with some intensity λ)
Cancel those points whose distance to their nearest neighbor is smaller than
some threshold R
Realizations are regular point patterns (with smaller spatial variablility that in
the Poisson case)
Two-parametric model with parameters λ and R
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Random measures and Cox processes
Random measures
I
Denote by M the set of all locally finite measures η : B(R2 ) → [0, ∞]
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Random measures and Cox processes
Random measures
I
Denote by M the set of all locally finite measures η : B(R2 ) → [0, ∞]
I
Let M be the smallest σ-algebra on M s.t. η → η(B) is measurable for
each bounded B ∈ B(R2 ).
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Random measures and Cox processes
Random measures
I
Denote by M the set of all locally finite measures η : B(R2 ) → [0, ∞]
I
Let M be the smallest σ-algebra on M s.t. η → η(B) is measurable for
each bounded B ∈ B(R2 ).
I
A mapping Λ from (Ω, A, P) into (M, M) is called a random measure.
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Random measures and Cox processes
Random measures
I
Denote by M the set of all locally finite measures η : B(R2 ) → [0, ∞]
I
Let M be the smallest σ-algebra on M s.t. η → η(B) is measurable for
each bounded B ∈ B(R2 ).
I
A mapping Λ from (Ω, A, P) into (M, M) is called a random measure.
Cox point processes
I
A point process X is called a Cox process with random intensity measure
Λ if
!
n
Y
Λ(Bi )ki −Λ(Bi )
P(X (B1 ) = k1 , . . . , X (Bn ) = kn ) = E
e
,
ki !
i=1
for all pairwise disjoint, bounded B1 , . . . , Bn ∈ B(R2 ) and k1 , . . . , kn ≥ 0.
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Random measures and Cox processes
Random measures
I
Denote by M the set of all locally finite measures η : B(R2 ) → [0, ∞]
I
Let M be the smallest σ-algebra on M s.t. η → η(B) is measurable for
each bounded B ∈ B(R2 ).
I
A mapping Λ from (Ω, A, P) into (M, M) is called a random measure.
Cox point processes
I
A point process X is called a Cox process with random intensity measure
Λ if
!
n
Y
Λ(Bi )ki −Λ(Bi )
P(X (B1 ) = k1 , . . . , X (Bn ) = kn ) = E
e
,
ki !
i=1
for all pairwise disjoint, bounded B1 , . . . , Bn ∈ B(R2 ) and k1 , . . . , kn ≥ 0.
I
Conditioning on Λ = η, a Cox process X is a Poisson process with
intensity measure η.
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
I
I
Let M be a Polish space with Borel σ-algebra B(M)
Examples:
I
M = R and B(M) = B(R),
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
I
I
Let M be a Polish space with Borel σ-algebra B(M)
Examples:
I
I
M = R and B(M) = B(R),
M = P o = family of all convex and compact polytopes in R2 with their centre
of gravity at o
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
I
I
Let M be a Polish space with Borel σ-algebra B(M)
Examples:
I
I
M = R and B(M) = B(R),
M = P o = family of all convex and compact polytopes in R2 with their centre
of gravity at o and the hitting-σ-algebra B(M) = B(F) ∩ P o , where
I
I
F family of closed sets in R2 , and
B(F ) = σ({F ∈ F : F ∩ K 6= ∅}, K compact ).
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
I
I
Let M be a Polish space with Borel σ-algebra B(M)
Examples:
I
I
M = R and B(M) = B(R),
M = P o = family of all convex and compact polytopes in R2 with their centre
of gravity at o and the hitting-σ-algebra B(M) = B(F) ∩ P o , where
I
I
I
F family of closed sets in R2 , and
B(F ) = σ({F ∈ F : F ∩ K 6= ∅}, K compact ).
Let NM be the set of all counting measures
ψ : B ⊗ B(M) → {0, 1, . . . } ∪ {∞} which are locally finite in the first
component, i.e., ψ(B × M) < ∞ for bounded B ∈ B(R2 ),
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
I
I
Let M be a Polish space with Borel σ-algebra B(M)
Examples:
I
I
M = R and B(M) = B(R),
M = P o = family of all convex and compact polytopes in R2 with their centre
of gravity at o and the hitting-σ-algebra B(M) = B(F) ∩ P o , where
I
I
F family of closed sets in R2 , and
B(F ) = σ({F ∈ F : F ∩ K 6= ∅}, K compact ).
I
Let NM be the set of all counting measures
ψ : B ⊗ B(M) → {0, 1, . . . } ∪ {∞} which are locally finite in the first
component, i.e., ψ(B × M) < ∞ for bounded B ∈ B(R2 ),
I
and NM the smallest σ-algebra on NM such that ψ → ψ(B × G) is
measurable for each bounded B ∈ B(R2 ) and G ∈ B(M).
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
Definition
Let
I
(Ω, A, P) some probability space,
I
X1 , X2 , · · · : Ω 7−→ R2 and M1 , M2 , · · · : Ω 7−→ M two sequences of R2 - and
M-valued random variables, respectively, such that
#{Xn ∈ B} < ∞
for each bounded
B ∈ B(R2 ) .
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
Definition
Let
I
(Ω, A, P) some probability space,
I
X1 , X2 , · · · : Ω 7−→ R2 and M1 , M2 , · · · : Ω 7−→ M two sequences of R2 - and
M-valued random variables, respectively, such that
#{Xn ∈ B} < ∞
I
for each bounded
B ∈ B(R2 ) .
P∞
The measurable mapping XM : Ω 7−→ NM defined by XM = n=1 δ(Xn ,Mn )
is called a marked point process and Mn is called the mark (or label) of
Xn .
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Marked point processes
Definition
Let
I
(Ω, A, P) some probability space,
I
X1 , X2 , · · · : Ω 7−→ R2 and M1 , M2 , · · · : Ω 7−→ M two sequences of R2 - and
M-valued random variables, respectively, such that
#{Xn ∈ B} < ∞
I
I
for each bounded
B ∈ B(R2 ) .
P∞
The measurable mapping XM : Ω 7−→ NM defined by XM = n=1 δ(Xn ,Mn )
is called a marked point process and Mn is called the mark (or label) of
Xn .
The intensity measure µ : B(R2 ) ⊗ B(M) → [0, ∞] of XM is defined as
µ(B × C) = EXM (B × C) ,
B ∈ B(R2 ) , C ∈ B(M) .
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
Let X = {Xn } be a stationary Poisson process and consider
the Voronoi cell Ξn of Xn :
Ξn
I
=
{x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n}
Then, {(Xn , Mn )}, where Mn = Ξn − Xn , is a (stationary) marked point
process with mark space P o
Realization of a Poisson process {Xn }
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Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
Let X = {Xn } be a stationary Poisson process and consider
the Voronoi cell Ξn of Xn :
Ξn
I
=
{x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n}
Then, {(Xn , Mn )}, where Mn = Ξn − Xn , is a (stationary) marked point
process with mark space P o
Realization of a Poisson-Voronoi tessellation {(Xn , Ξn − Xn )}
Page 19
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
Further (stationary) marked point processes associated with {(Xn , Ξn − Xn )}:
I
{(Xn , ν2 (Ξn ))} with mark space [0, ∞)
I
{(Xn , ν1 (∂Ξn ))} with mark space [0, ∞)
Realization of a Poisson-Voronoi tessellation {(Xn , Ξn − Xn )}
Page 20
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Palm mark distribution and Palm distribution
Definition
I
D
XM is called stationary if {(Xn − x, Mn )} = {(Xn , Mn )} for each x ∈ R2 .
Page 20
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Palm mark distribution and Palm distribution
Definition
D
I
XM is called stationary if {(Xn − x, Mn )} = {(Xn , Mn )} for each x ∈ R2 .
I
If XM is stationary, then
µ(B × C) = λν2 (B) PX∗M (C) ,
B ∈ B(R2 ) ,
Page 20
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Palm mark distribution and Palm distribution
Definition
D
I
XM is called stationary if {(Xn − x, Mn )} = {(Xn , Mn )} for each x ∈ R2 .
I
If XM is stationary, then
µ(B × C) = λν2 (B) PX∗M (C) ,
I
B ∈ B(R2 ) ,
for some λ > 0, which is called the intensity of XM ,
Page 20
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Palm mark distribution and Palm distribution
Definition
D
I
XM is called stationary if {(Xn − x, Mn )} = {(Xn , Mn )} for each x ∈ R2 .
I
If XM is stationary, then
µ(B × C) = λν2 (B) PX∗M (C) ,
I
I
B ∈ B(R2 ) ,
for some λ > 0, which is called the intensity of XM ,
and some probability measure PX∗M on B(M), which is called the Palm mark
distribution of XM .
Page 20
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Palm mark distribution and Palm distribution
Definition
D
I
XM is called stationary if {(Xn − x, Mn )} = {(Xn , Mn )} for each x ∈ R2 .
I
If XM is stationary, then
µ(B × C) = λν2 (B) PX∗M (C) ,
I
I
I
B ∈ B(R2 ) ,
for some λ > 0, which is called the intensity of XM ,
and some probability measure PX∗M on B(M), which is called the Palm mark
distribution of XM .
For any stationary XM with intensity λ ∈ (0, ∞), the Palm distribution PXoM
of XM on NM ⊗ B(M) is defined as
E#{k : Xk ∈ [0, 1]2 , Mk ∈ C, {(Xn − Xk , Mn )} ∈ A}
λ
for A ∈ NM , C ∈ B(M).
PXoM (A × C) =
Page 20
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Palm mark distribution and Palm distribution
Definition
D
I
XM is called stationary if {(Xn − x, Mn )} = {(Xn , Mn )} for each x ∈ R2 .
I
If XM is stationary, then
µ(B × C) = λν2 (B) PX∗M (C) ,
I
I
I
B ∈ B(R2 ) ,
for some λ > 0, which is called the intensity of XM ,
and some probability measure PX∗M on B(M), which is called the Palm mark
distribution of XM .
For any stationary XM with intensity λ ∈ (0, ∞), the Palm distribution PXoM
of XM on NM ⊗ B(M) is defined as
E#{k : Xk ∈ [0, 1]2 , Mk ∈ C, {(Xn − Xk , Mn )} ∈ A}
λ
for A ∈ NM , C ∈ B(M).
PXoM (A × C) =
I
Note that PX∗M (C) = PXoM (NM × C) for any C ∈ B(M).
Page 21
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Typical mark
Definition
Let XM be a stationary marked point process with Palm mark distribution P∗XM .
I
A random variable M ∗ : Ω −→ M distributed according to P∗XM is called the
typical mark of XM .
Page 21
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Typical mark
Definition
Let XM be a stationary marked point process with Palm mark distribution P∗XM .
I
A random variable M ∗ : Ω −→ M distributed according to P∗XM is called the
typical mark of XM .
I
If XM is ergodic, then M ∗ can be regarded as the mark at a point chosen
purely at random out of {Xn },i.e.,
1
r →∞ #{n : Xn ∈ [−r , r ]2 }
Eh(M ∗ ) = lim
X
i: Xi ∈[−r ,r ]2
almost surely for each measurable h : M 7−→ [0, ∞).
h(Mi )
Page 22
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Independent marking
I
Let X = {XN } be a point process and
I
M1 , M2 , · · · : Ω → R i.i.d. random variables with some distribution P,
which are independent of {Xn }.
Page 22
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Independent marking
I
Let X = {XN } be a point process and
I
M1 , M2 , · · · : Ω → R i.i.d. random variables with some distribution P,
which are independent of {Xn }.
I
Then, XM = {(Xn , Mn )} is called an independently marked point process.
Page 22
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Independent marking
I
Let X = {XN } be a point process and
I
M1 , M2 , · · · : Ω → R i.i.d. random variables with some distribution P,
which are independent of {Xn }.
I
Then, XM = {(Xn , Mn )} is called an independently marked point process.
Palm version of independently marked point processes
I
Let X = {Xn } be stationary and X o = {Xno } a Palm version of X (with
distribution PXo ).
Page 22
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Independent marking
I
Let X = {XN } be a point process and
I
M1 , M2 , · · · : Ω → R i.i.d. random variables with some distribution P,
which are independent of {Xn }.
I
Then, XM = {(Xn , Mn )} is called an independently marked point process.
Palm version of independently marked point processes
I
I
Let X = {Xn } be stationary and X o = {Xno } a Palm version of X (with
distribution PXo ).
If X o is independent of {Mn }, then
I
the distribution of the marked point process XMo = {(Xno , Mn )} is given by
PXoM , i.e., XMo is a Palm version of XM = {(Xn , Mn )},
Page 22
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Independent marking
I
Let X = {XN } be a point process and
I
M1 , M2 , · · · : Ω → R i.i.d. random variables with some distribution P,
which are independent of {Xn }.
I
Then, XM = {(Xn , Mn )} is called an independently marked point process.
Palm version of independently marked point processes
I
I
Let X = {Xn } be stationary and X o = {Xno } a Palm version of X (with
distribution PXo ).
If X o is independent of {Mn }, then
I
I
the distribution of the marked point process XMo = {(Xno , Mn )} is given by
PXoM , i.e., XMo is a Palm version of XM = {(Xn , Mn )},
and the typical mark M ∗ of XM = {(Xn , Mn )} has distribution PX∗M = P.
Page 23
Mathematical tools for spatial networks on various length scales |
Example: Poisson-Voronoi tessellation
I
Let X = {Xn } be a stationary Poisson process.
Point processes and Palm calculus
Page 23
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
Let X = {Xn } be a stationary Poisson process.
I
Consider the Voronoi cells Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n},
Page 23
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
Let X = {Xn } be a stationary Poisson process.
I
Consider the Voronoi cells Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n},
I
and the stationary marked point process XM = {(Xn , Ξn −Xn )}
Realization of the Poisson-Voronoi tessellation {(Xn , Ξn − Xn )}
Page 23
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
Let X = {Xn } be a stationary Poisson process.
I
Consider the Voronoi cells Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n},
I
and the stationary marked point process XM = {(Xn , Ξn −Xn )}
Realization of the Poisson-Voronoi tessellation {(Xn , Ξn − Xn )}
Palm version of the Poisson-Voronoi tessellation {(Xn , Ξn − Xn )}
I
Add the origin X0 = o to the stationary Poisson process X = {Xn }.
Page 24
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
Then, by Slivnyak’s theorem, the point process X o = {Xno }, where
{Xno } = {X0 , X1 , X2 , . . .}, is a Palm version of X = {X1 , X2 , . . .}.
Page 24
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
I
Then, by Slivnyak’s theorem, the point process X o = {Xno }, where
{Xno } = {X0 , X1 , X2 , . . .}, is a Palm version of X = {X1 , X2 , . . .}.
Consider the Voronoi cells Ξon induced by X o , where
Ξon = {x ∈ R2 : |x − Xno | ≤ |x − Xko | ∀k ∈ {0, 1, 2, . . .}, k 6= n} .
Page 24
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
I
I
Then, by Slivnyak’s theorem, the point process X o = {Xno }, where
{Xno } = {X0 , X1 , X2 , . . .}, is a Palm version of X = {X1 , X2 , . . .}.
Consider the Voronoi cells Ξon induced by X o , where
Ξon = {x ∈ R2 : |x − Xno | ≤ |x − Xko | ∀k ∈ {0, 1, 2, . . .}, k 6= n} .
Then, the marked point process XMo = {(Xno , Ξon −Xno )} is a Palm version
of XM = {(Xn , Ξn −Xn )}.
Realization of the Palm version {(Xno , Ξon − Xno )}
Page 24
Mathematical tools for spatial networks on various length scales |
Point processes and Palm calculus
Example: Poisson-Voronoi tessellation
I
I
I
Then, by Slivnyak’s theorem, the point process X o = {Xno }, where
{Xno } = {X0 , X1 , X2 , . . .}, is a Palm version of X = {X1 , X2 , . . .}.
Consider the Voronoi cells Ξon induced by X o , where
Ξon = {x ∈ R2 : |x − Xno | ≤ |x − Xko | ∀k ∈ {0, 1, 2, . . .}, k 6= n} .
Then, the marked point process XMo = {(Xno , Ξon −Xno )} is a Palm version
of XM = {(Xn , Ξn −Xn )}.
Realization of the Palm version {(Xno , Ξon − Xno )}
I
The typical mark Ξ∗ of XM is given by Ξ∗ = Ξo0
Page 25
Mathematical tools for spatial networks on various length scales |
Contents
Introduction
Point processes and Palm calculus
Random tessellations
Local simulation of typical Voronoi cells
Cox processes on random tessellations
Multiscale network modeling (Outlook to part II)
Random tessellations
Page 26
Mathematical tools for spatial networks on various length scales |
Random tessellations
General idea
I
Tessellation
I
countable (locally finite) subdivision of R2
Random tessellations
Page 26
Mathematical tools for spatial networks on various length scales |
Random tessellations
Random tessellations
General idea
I
Tessellation
I
I
countable (locally finite) subdivision of R2
into non-overlapping closed sets (with non-empty interiors), called cells
Page 26
Mathematical tools for spatial networks on various length scales |
Random tessellations
Random tessellations
General idea
I
Tessellation
I
I
I
countable (locally finite) subdivision of R2
into non-overlapping closed sets (with non-empty interiors), called cells
Random tessellation
I
Random marked point process T = {Xn , Mn } with mark space (F, B(F)),
Page 26
Mathematical tools for spatial networks on various length scales |
Random tessellations
Random tessellations
General idea
I
Tessellation
I
I
I
countable (locally finite) subdivision of R2
into non-overlapping closed sets (with non-empty interiors), called cells
Random tessellation
I
I
Random marked point process T = {Xn , Mn } with mark space (F, B(F)),
where F = the family of all closed sets in R2 , and
Page 26
Mathematical tools for spatial networks on various length scales |
Random tessellations
Random tessellations
General idea
I
Tessellation
I
I
I
countable (locally finite) subdivision of R2
into non-overlapping closed sets (with non-empty interiors), called cells
Random tessellation
I
I
I
Random marked point process T = {Xn , Mn } with mark space (F, B(F)),
where F = the family of all closed sets in R2 , and
the hitting-σ-algebra B(F) = σ({F ∈ F : F ∩ K 6= ∅}, K compact ).
Page 26
Mathematical tools for spatial networks on various length scales |
Random tessellations
Random tessellations
General idea
I
Tessellation
I
I
I
countable (locally finite) subdivision of R2
into non-overlapping closed sets (with non-empty interiors), called cells
Random tessellation
I
I
I
Random marked point process T = {Xn , Mn } with mark space (F, B(F)),
where F = the family of all closed sets in R2 , and
the hitting-σ-algebra B(F) = σ({F ∈ F : F ∩ K 6= ∅}, K compact ).
Examples
I Tessellations with convex cells
I
I
I
I
Voronoi tessellations
Laguerre tessellations (generalization of Voronoi tessellations)
Delaunay tessellations
line tessellations
Page 26
Mathematical tools for spatial networks on various length scales |
Random tessellations
Random tessellations
General idea
I
Tessellation
I
I
I
countable (locally finite) subdivision of R2
into non-overlapping closed sets (with non-empty interiors), called cells
Random tessellation
I
I
I
Random marked point process T = {Xn , Mn } with mark space (F, B(F)),
where F = the family of all closed sets in R2 , and
the hitting-σ-algebra B(F) = σ({F ∈ F : F ∩ K 6= ∅}, K compact ).
Examples
I Tessellations with convex cells
I
I
I
I
I
Voronoi tessellations
Laguerre tessellations (generalization of Voronoi tessellations)
Delaunay tessellations
line tessellations
Tessellations with general (not necessarily convex) cells
I
I
I
aggregate tessellations
generalized Laguerre tessellations
β-skeletons (thinnings of Delaunay tessellations)
Page 27
Mathematical tools for spatial networks on various length scales |
Poisson-Voronoi tessellation
I
Cells are generated by a point process {Xn }
Random tessellations
Page 27
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Voronoi tessellation
I
I
Cells are generated by a point process {Xn }
Cell Ξn of point Xn is given by
Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | for all k 6= n}
Page 27
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Voronoi tessellation
I
I
I
Cells are generated by a point process {Xn }
Cell Ξn of point Xn is given by
Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | for all k 6= n}
If {Xn } stationary Poisson point process
⇒ Poisson-Voronoi tessellation (PVT)
Realization of a PVT {(Xn , Ξn − Xn )}
Page 28
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
I
Let XR = {(Xn , Rn )} a marked point process with non-negative marks Rn .
Page 28
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
I
I
Let XR = {(Xn , Rn )} a marked point process with non-negative marks Rn .
The Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2 − Rn2 ≤ |x − Xk |2 − Rk2 , ∀k 6= n}
Page 28
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
I
I
Let XR = {(Xn , Rn )} a marked point process with non-negative marks Rn .
The Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2 − Rn2 ≤ |x − Xk |2 − Rk2 , ∀k 6= n}
I
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
a Laguerre tessellation induced by XR = {(Xn , Rn )},
Page 28
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
I
I
Let XR = {(Xn , Rn )} a marked point process with non-negative marks Rn .
The Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2 − Rn2 ≤ |x − Xk |2 − Rk2 , ∀k 6= n}
I
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
a Laguerre tessellation induced by XR = {(Xn , Rn )},
which specifies to a Voronoi tessellation if R1 = R2 = . . .
Page 28
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
I
I
Let XR = {(Xn , Rn )} a marked point process with non-negative marks Rn .
The Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2 − Rn2 ≤ |x − Xk |2 − Rk2 , ∀k 6= n}
I
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
I
a Laguerre tessellation induced by XR = {(Xn , Rn )},
which specifies to a Voronoi tessellation if R1 = R2 = . . .
Note that
I
the generating point Xn is not necessarily inside the cell Ξn , and
Page 28
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
I
I
Let XR = {(Xn , Rn )} a marked point process with non-negative marks Rn .
The Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2 − Rn2 ≤ |x − Xk |2 − Rk2 , ∀k 6= n}
I
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
I
a Laguerre tessellation induced by XR = {(Xn , Rn )},
which specifies to a Voronoi tessellation if R1 = R2 = . . .
Note that
I
I
the generating point Xn is not necessarily inside the cell Ξn , and
a point Xn does not necessarily generate a cell (because int(Ξn ) can be
empty)
Page 28
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
I
I
Let XR = {(Xn , Rn )} a marked point process with non-negative marks Rn .
The Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2 − Rn2 ≤ |x − Xk |2 − Rk2 , ∀k 6= n}
I
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
I
Note that
I
I
I
a Laguerre tessellation induced by XR = {(Xn , Rn )},
which specifies to a Voronoi tessellation if R1 = R2 = . . .
the generating point Xn is not necessarily inside the cell Ξn , and
a point Xn does not necessarily generate a cell (because int(Ξn ) can be
empty)
If XR = {(Xn , Rn )} is an independently marked (stationary) Poisson
process, then
T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅}
is called a Poisson-Laguerre tessellation.
Page 29
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Laguerre tessellation
Cutout of Voronoi tessellation (left) and cutout of Laguerre tessellation on the
the same set of seed points (right)
Page 30
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Delaunay tessellation
I
Consider a Voronoi tessellation T = {(Xn , Ξn − Xn )} induced by a
stationary Poisson process {Xn }
Page 31
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Delaunay tessellation
I
Consider a Voronoi tessellation T = {(Xn , Ξn − Xn )} induced by a
stationary Poisson process {Xn }
I
For each vertex Xn0 of T construct the cell Ξ0n as the triangle formed by the
nuclei Xi1 , Xi2 , Xi3 of the three neighboring Voronoi cells.
Page 32
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson-Delaunay tessellation
I
Consider a Voronoi tessellation T = {(Xn , Ξn − Xn )} induced by a
stationary Poisson process {Xn }
I
For each vertex Xn0 of T construct the cell Ξ0n as the triangle formed by the
nuclei Xi1 , Xi2 , Xi3 of the three neighboring Voronoi cells.
I
Then T 0 = {(Xn0 , Ξ0n − Xn0 )} is called a Poisson-Delaunay tessellation.
Page 33
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson line tessellation
Let
I
{Rn } a stationary Poisson process on the real line R
Page 33
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson line tessellation
Let
I
I
{Rn } a stationary Poisson process on the real line R
{Φn } i.i.d. r.v.’s, independent of {Rn }, with Φn ∼ U[0, π), and
Page 33
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson line tessellation
Let
I
I
I
{Rn } a stationary Poisson process on the real line R
{Φn } i.i.d. r.v.’s, independent of {Rn }, with Φn ∼ U[0, π), and
`(Φn , Rn ) = {(x, y ) ∈ R2 : x sin Φn − y cos Φn = Rn } the line with direction
Φn and signed distance Rn to the origin o ∈ R2
Page 33
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson line tessellation
Let
I
I
I
{Rn } a stationary Poisson process on the real line R
{Φn } i.i.d. r.v.’s, independent of {Rn }, with Φn ∼ U[0, π), and
`(Φn , Rn ) = {(x, y ) ∈ R2 : x sin Φn − y cos Φn = Rn } the line with direction
Φn and signed distance Rn to the origin o ∈ R2
Then, {`(Φn , Rn )} is called a Poisson line process, where
Page 33
Mathematical tools for spatial networks on various length scales |
Random tessellations
Poisson line tessellation
Let
I
I
I
{Rn } a stationary Poisson process on the real line R
{Φn } i.i.d. r.v.’s, independent of {Rn }, with Φn ∼ U[0, π), and
`(Φn , Rn ) = {(x, y ) ∈ R2 : x sin Φn − y cos Φn = Rn } the line with direction
Φn and signed distance Rn to the origin o ∈ R2
Then, {`(Φ
S n , Rn )} is called a Poisson line process, where
T (1) = n∈Z `(Φn , Rn ) is the edge set of a Poisson line tessellation (PLT).
Realization of a Poisson line tessellation
Page 34
Mathematical tools for spatial networks on various length scales |
Random tessellations
Tessellations with general (not necessarily convex) cells
I
Aggregate Voronoi tessellations
Construction principle (left) and cutout of an aggregate tessellation (right)
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
the Rn are non-negative r.v.’s, and
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
I
the Rn are non-negative r.v.’s, and
the An are positive definite random 2 × 2-matrices.
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
I
I
the Rn are non-negative r.v.’s, and
the An are positive definite random 2 × 2-matrices.
The generalized Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2An − Rn2 ≤ |x − Xk |2An − Rk2 , ∀k 6= n} ,
where |x|A =
√
x > Ax for all x ∈ R2 .
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
I
I
the Rn are non-negative r.v.’s, and
the An are positive definite random 2 × 2-matrices.
The generalized Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2An − Rn2 ≤ |x − Xk |2An − Rk2 , ∀k 6= n} ,
where |x|A =
I
√
x > Ax for all x ∈ R2 .
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
a generalized Laguerre tessellation induced by XR = {(Xn , [Rn , An ])},
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
I
I
the Rn are non-negative r.v.’s, and
the An are positive definite random 2 × 2-matrices.
The generalized Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2An − Rn2 ≤ |x − Xk |2An − Rk2 , ∀k 6= n} ,
where |x|A =
I
√
x > Ax for all x ∈ R2 .
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
a generalized Laguerre tessellation induced by XR = {(Xn , [Rn , An ])},
which specifies to a Laguerre tessellation if A1 = A2 = . . . = I and to a
Voronoi tessellation if A1 = A2 = . . . = I and R1 = R2 = . . .
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
I
I
the Rn are non-negative r.v.’s, and
the An are positive definite random 2 × 2-matrices.
The generalized Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2An − Rn2 ≤ |x − Xk |2An − Rk2 , ∀k 6= n} ,
where |x|A =
I
x > Ax for all x ∈ R2 .
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
I
√
a generalized Laguerre tessellation induced by XR = {(Xn , [Rn , An ])},
which specifies to a Laguerre tessellation if A1 = A2 = . . . = I and to a
Voronoi tessellation if A1 = A2 = . . . = I and R1 = R2 = . . .
Note that
I
the generating point Xn is not necessarily inside the cell Ξn , and
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
I
I
the Rn are non-negative r.v.’s, and
the An are positive definite random 2 × 2-matrices.
The generalized Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2An − Rn2 ≤ |x − Xk |2An − Rk2 , ∀k 6= n} ,
where |x|A =
I
x > Ax for all x ∈ R2 .
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
I
√
a generalized Laguerre tessellation induced by XR = {(Xn , [Rn , An ])},
which specifies to a Laguerre tessellation if A1 = A2 = . . . = I and to a
Voronoi tessellation if A1 = A2 = . . . = I and R1 = R2 = . . .
Note that
I
I
the generating point Xn is not necessarily inside the cell Ξn , and
a point Xn does not necessarily generate a cell (because int(Ξn ) can be
empty)
Page 35
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
I
Let XR = {(Xn , [Rn , An ])} be a marked point process, where
I
I
I
the Rn are non-negative r.v.’s, and
the An are positive definite random 2 × 2-matrices.
The generalized Laguerre cell Ξn of Xn is given by
Ξn = {x ∈ R2 : |x − Xn |2An − Rn2 ≤ |x − Xk |2An − Rk2 , ∀k 6= n} ,
where |x|A =
I
x > Ax for all x ∈ R2 .
Then, T = {(Xn , Ξn − Xn ) such that int(Ξn ) 6= ∅} is called
I
I
I
√
a generalized Laguerre tessellation induced by XR = {(Xn , [Rn , An ])},
which specifies to a Laguerre tessellation if A1 = A2 = . . . = I and to a
Voronoi tessellation if A1 = A2 = . . . = I and R1 = R2 = . . .
Note that
I
I
I
the generating point Xn is not necessarily inside the cell Ξn , and
a point Xn does not necessarily generate a cell (because int(Ξn ) can be
empty)
the cells Ξn are not necessarily convex.
Page 36
Mathematical tools for spatial networks on various length scales |
Random tessellations
Generalized Laguerre tessellations
Seed points Xn , radii Rn , ellipse-representation of matrices An (left),
and cutout of generalized Laguerre tessellation (right)
Page 37
Mathematical tools for spatial networks on various length scales |
β-skeletons
I
Let β ∈ [1, 2] any fixed number.
Random tessellations
Page 37
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number.
For x, y ∈ R2 consider the weighted means
(1)
mxy =
β
β
x + (1 − ) y ,
2
2
(2)
mxy = (1 −
β
β
)x+ y,
2
2
Page 37
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number.
For x, y ∈ R2 consider the weighted means
β
β
x + (1 − ) y ,
2
2
and the intersection of two balls
(1)
mxy =
(1)
(1)
(2)
mxy = (1 −
(2)
β
β
)x+ y,
2
2
(2)
Aβ (x, y ) = B(mxy , |mxy − y |) ∩ B(mxy , |mxy − x|) .
Page 37
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number.
For x, y ∈ R2 consider the weighted means
β
β
x + (1 − ) y ,
2
2
and the intersection of two balls
(1)
mxy =
(1)
(2)
mxy = (1 −
(1)
(2)
β
β
)x+ y,
2
2
(2)
Aβ (x, y ) = B(mxy , |mxy − y |) ∩ B(mxy , |mxy − x|) .
y
x
Illustration of the intersection Aβ (x, y ) of the two balls:
Page 37
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number.
For x, y ∈ R2 consider the weighted means
β
β
x + (1 − ) y ,
2
2
and the intersection of two balls
(1)
mxy =
(1)
(2)
mxy = (1 −
(1)
(2)
β
β
)x+ y,
2
2
(2)
Aβ (x, y ) = B(mxy , |mxy − y |) ∩ B(mxy , |mxy − x|) .
y
x
Illustration of the intersection Aβ (x, y ) of the two balls:
for β = 1 (dotted), β = 1.5 (dashed) and β = 2 (solid)
Page 38
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
Let β ∈ [1, 2] any fixed number and X = {Xn } a point process in R2 .
Page 38
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number and X = {Xn } a point process in R2 .
Then, the edge set
[
G(β, X ) =
[x, y ]
x,y ∈X : X ∩ Aβ (x,y )=∅
is called a β-skeleton induced by X = {Xn }.
Page 38
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number and X = {Xn } a point process in R2 .
Then, the edge set
[
G(β, X ) =
[x, y ]
x,y ∈X : X ∩ Aβ (x,y )=∅
is called a β-skeleton induced by X = {Xn }.
Examples of β-skeletons for β = 1, β = 1.5 and β = 2 (left to right)
Page 38
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number and X = {Xn } a point process in R2 .
Then, the edge set
[
G(β, X ) =
[x, y ]
x,y ∈X : X ∩ Aβ (x,y )=∅
is called a β-skeleton induced by X = {Xn }.
Examples of β-skeletons for β = 1, β = 1.5 and β = 2 (left to right)
I
Note that the edge set G(β, I) is monotonously decreasing in β, and
Page 38
Mathematical tools for spatial networks on various length scales |
Random tessellations
β-skeletons
I
I
Let β ∈ [1, 2] any fixed number and X = {Xn } a point process in R2 .
Then, the edge set
[
G(β, X ) =
[x, y ]
x,y ∈X : X ∩ Aβ (x,y )=∅
is called a β-skeleton induced by X = {Xn }.
Examples of β-skeletons for β = 1, β = 1.5 and β = 2 (left to right)
I
I
Note that the edge set G(β, I) is monotonously decreasing in β, and
for β = 1, β-skeletons specify to the edge sets of Delaunay tessellations.
Page 39
Mathematical tools for spatial networks on various length scales |
Contents
Introduction
Point processes and Palm calculus
Random tessellations
Local simulation of typical Voronoi cells
Cox processes on random tessellations
Multiscale network modeling (Outlook to part II)
Local simulation of typical Voronoi cells
Page 40
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical Poisson-Voronoi cell
General idea
I
Consider a stationary Poisson process X with some intensity λ > 0.
Page 40
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical Poisson-Voronoi cell
General idea
I
Consider a stationary Poisson process X with some intensity λ > 0.
I
Use Slivnyak’s theorem, which says that the Palm version X 0 of X is
given by
X 0 = X ∪ {o}
.
Page 40
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical Poisson-Voronoi cell
General idea
I
Consider a stationary Poisson process X with some intensity λ > 0.
I
Use Slivnyak’s theorem, which says that the Palm version X 0 of X is
given by
X 0 = X ∪ {o}
.
I
Simulate n points X1 , X2 , . . . , Xn of X radially and
Page 40
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical Poisson-Voronoi cell
General idea
I
Consider a stationary Poisson process X with some intensity λ > 0.
I
Use Slivnyak’s theorem, which says that the Palm version X 0 of X is
given by
X 0 = X ∪ {o}
.
I
Simulate n points X1 , X2 , . . . , Xn of X radially and
I
compute the zero cell of the Voronoi tessellation corresponding to
{X1 , X2 , . . . , Xn } = X ∪ {o}.
Page 40
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical Poisson-Voronoi cell
General idea
I
Consider a stationary Poisson process X with some intensity λ > 0.
I
Use Slivnyak’s theorem, which says that the Palm version X 0 of X is
given by
X 0 = X ∪ {o}
.
I
Simulate n points X1 , X2 , . . . , Xn of X radially and
I
compute the zero cell of the Voronoi tessellation corresponding to
{X1 , X2 , . . . , Xn } = X ∪ {o}.
I
Use a suitable stopping rule to reduce runtime.
Page 41
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
Theorem
Let
I
λ > 0 be an arbitrary, but fixed number,
I
Y1 , Y2 , . . . i.i.d. Exp(1)–distributed,
qP
n
Yk
Rn =
k =1 πλ for n = 1, 2, . . . ,
I
I
U1 , U2 , . . . i.i.d. U[0, 2π)-distributed and
I
Xn = (Rn cos Un , Rn sin Un ) for n = 1, 2, . . . .
Page 41
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
Theorem
Let
I
λ > 0 be an arbitrary, but fixed number,
I
Y1 , Y2 , . . . i.i.d. Exp(1)–distributed,
qP
n
Yk
Rn =
k =1 πλ for n = 1, 2, . . . ,
I
I
U1 , U2 , . . . i.i.d. U[0, 2π)-distributed and
I
Xn = (Rn cos Un , Rn sin Un ) for n = 1, 2, . . . .
Then {Xn } is a stationary Poisson process in R2 with intensity λ.
Page 41
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
Theorem
Let
I
λ > 0 be an arbitrary, but fixed number,
I
Y1 , Y2 , . . . i.i.d. Exp(1)–distributed,
qP
n
Yk
Rn =
k =1 πλ for n = 1, 2, . . . ,
I
I
U1 , U2 , . . . i.i.d. U[0, 2π)-distributed and
I
Xn = (Rn cos Un , Rn sin Un ) for n = 1, 2, . . . .
Then {Xn } is a stationary Poisson process in R2 with intensity λ.
Proof Idea: Show that X (B) ∼ Poi(λν2 (B)) and X (B1 ), . . . , X (Bn ) are
independent for B1 , . . . , Bn ∈ B with Bi ∩ Bj = ∅ for i 6= j, e.g.,
Page 41
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
Theorem
Let
I
λ > 0 be an arbitrary, but fixed number,
I
Y1 , Y2 , . . . i.i.d. Exp(1)–distributed,
qP
n
Yk
Rn =
k =1 πλ for n = 1, 2, . . . ,
I
I
U1 , U2 , . . . i.i.d. U[0, 2π)-distributed and
I
Xn = (Rn cos Un , Rn sin Un ) for n = 1, 2, . . . .
Then {Xn } is a stationary Poisson process in R2 with intensity λ.
Proof Idea: Show that X (B) ∼ Poi(λν2 (B)) and X (B1 ), . . . , X (Bn ) are
independent for B1 , . . . , Bn ∈ B with BP
i ∩ Bj = ∅ for i 6= j, e.g.,
n
X (B(o, r )) = #{n : Rn ≤ r } = #{n : k =1 Yk ≤ λπr 2 } ∼ Poi(λπr 2 ).
Page 42
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
I
Algorithm:
I
I
Simulate Yn ∼ Exp(1), Un ∼ U[0, 2π) independent of
Y1 , . . . , Yn−1 , U1 , . . . , Un−1
qP
n
Construct Xn = (Rn cos Un , Rn sin Un ) with Rn =
k =1 Yk /(πλ)
Page 43
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
I
Algorithm:
I
I
Simulate Yn ∼ Exp(1), Un ∼ U[0, 2π) independent of
Y1 , . . . , Yn−1 , U1 , . . . , Un−1
qP
n
Construct Xn = (Rn cos Un , Rn sin Un ) with Rn =
k =1 Yk /(πλ)
Page 44
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
I
Algorithm:
I
I
Simulate Yn ∼ Exp(1), Un ∼ U[0, 2π) independent of
Y1 , . . . , Yn−1 , U1 , . . . , Un−1
qP
n
Construct Xn = (Rn cos Un , Rn sin Un ) with Rn =
k =1 Yk /(πλ)
Page 45
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
I
Algorithm:
I
I
Simulate Yn ∼ Exp(1), Un ∼ U[0, 2π) independent of
Y1 , . . . , Yn−1 , U1 , . . . , Un−1
qP
n
Construct Xn = (Rn cos Un , Rn sin Un ) with Rn =
k =1 Yk /(πλ)
Page 46
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Radial simulation of Poisson processes
I
Algorithm:
I
I
I
Simulate Yn ∼ Exp(1), Un ∼ U[0, 2π) independent of
Y1 , . . . , Yn−1 , U1 , . . . , Un−1
qP
n
Construct Xn = (Rn cos Un , Rn sin Un ) with Rn =
k =1 Yk /(πλ)
√
Stop if Rn > a/ 2, where a is the side length of the sampling window
Page 47
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Slivnyak’s theorem
Theorem
Let X be a stationary Poisson process with some intensity λ > 0. Then
P(X 0 ∈ A) = P(X ∪ {o} ∈ A) ,
where X 0 is the Palm version of X , i.e., X 0 is distributed according to the Palm
distribution PX0 of X .
Page 47
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Slivnyak’s theorem
Theorem
Let X be a stationary Poisson process with some intensity λ > 0. Then
P(X 0 ∈ A) = P(X ∪ {o} ∈ A) ,
where X 0 is the Palm version of X , i.e., X 0 is distributed according to the Palm
distribution PX0 of X .
Proof Consider void probabilities P(X 0 (C) = 0), C ⊂ R2 compact. Then
P(X 0 ({o}) = 1) = P(X ({o}) = 0) = 1 by definition. Furthermore, if o 6∈ C,
then
P(X 0 C) = 0)
=
lim P(X (C) = 0 | X (B(o, ε)) = 1)
ε&0
Page 47
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Slivnyak’s theorem
Theorem
Let X be a stationary Poisson process with some intensity λ > 0. Then
P(X 0 ∈ A) = P(X ∪ {o} ∈ A) ,
where X 0 is the Palm version of X , i.e., X 0 is distributed according to the Palm
distribution PX0 of X .
Proof Consider void probabilities P(X 0 (C) = 0), C ⊂ R2 compact. Then
P(X 0 ({o}) = 1) = P(X ({o}) = 0) = 1 by definition. Furthermore, if o 6∈ C,
then
P(X 0 C) = 0)
=
=
=
lim P(X (C) = 0 | X (B(o, ε)) = 1)
ε&0
P(X (C) = 0)P(X (B(o, ε)\C) = 1)
ε&0
P(X (B(o, ε)) = 1
P(X (C) = 0) .
lim
Page 48
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
The typical Voronoi cell
I
Let
I
{Xn } be a stationary Poisson process and T = {Ξn } the induced
Poisson-Voronoi tessellation (PVT), i.e.,
Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n}
\
=
H(Xn , Xk )
k ∈N:k 6=n
with half planes H(Xn , Xk ) = {x ∈ R2 : |x − Xn | ≤ |x − Xk |}
Page 48
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
The typical Voronoi cell
I
Let
I
{Xn } be a stationary Poisson process and T = {Ξn } the induced
Poisson-Voronoi tessellation (PVT), i.e.,
Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n}
\
=
H(Xn , Xk )
k ∈N:k 6=n
I
with half planes H(Xn , Xk ) = {x ∈ R2 : |x − Xn | ≤ |x − Xk |}
Ξ∗ be the typical cell of T , i.e., Ξ∗ is the typical mark of {(Xn , Ξn − Xn )}
Page 48
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
The typical Voronoi cell
I
Let
I
{Xn } be a stationary Poisson process and T = {Ξn } the induced
Poisson-Voronoi tessellation (PVT), i.e.,
Ξn = {x ∈ R2 : |x − Xn | ≤ |x − Xk | ∀k 6= n}
\
=
H(Xn , Xk )
k ∈N:k 6=n
I
I
with half planes H(Xn , Xk ) = {x ∈ R2 : |x − Xn | ≤ |x − Xk |}
Ξ∗ be the typical cell of T , i.e., Ξ∗ is the typical mark of {(Xn , Ξn − Xn )}
Slivnyak’s theorem yields
Ξ∗ =
∞
\
n=1
H(o, Xn )
Page 49
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical cell of PVT
I
Algorithm:
I
Place point at o Simulate points X1 , X2 , X3 of Poisson process X radially
Page 50
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical cell of PVT
I
Algorithm:
I
Intersect halfplanes H(o, X1 ), H(o, X2 ) and H(o, X3 )
Page 51
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical cell of PVT
I
Algorithm:
I
Simulate further points of X and intersect halfplanes ⇒ inital cell
Page 52
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical cell of PVT
I
Algorithm:
I
Simulate further points of X and intersect initial cell with halfplanes
Page 53
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical cell of PVT
I
Algorithm:
I
Simulate further points of X and intersect initial cell with halfplanes
Page 54
Mathematical tools for spatial networks on various length scales |
Local simulation of typical Voronoi cells
Local simulation of the typical cell of PVT
I
Algorithm:
I
Stop if |Xn | > 2 maxi=1,...,m |Vm |, where V1 , . . . , Vm are the vertices of the
current modification of the initial cell
Page 55
Mathematical tools for spatial networks on various length scales |
Contents
Introduction
Point processes and Palm calculus
Random tessellations
Local simulation of typical Voronoi cells
Cox processes on random tessellations
Multiscale network modeling (Outlook to part II)
Cox processes on random tessellations
Page 56
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
I
Let
I
I
λ` > 0 any fixed number,
T a random tessellation with edge set T (1) .
Page 56
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
I
Let
I
I
I
λ` > 0 any fixed number,
T a random tessellation with edge set T (1) .
Λ a random measure with Λ(B) = λ` ν1 (B ∩ T (1) ) for B ∈ B(R2 ).
Page 56
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
I
Let
I
I
I
I
λ` > 0 any fixed number,
T a random tessellation with edge set T (1) .
Λ a random measure with Λ(B) = λ` ν1 (B ∩ T (1) ) for B ∈ B(R2 ).
X the Cox process with random intensity measure Λ
Page 56
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
I
Let
I
I
I
I
λ` > 0 any fixed number,
T a random tessellation with edge set T (1) .
Λ a random measure with Λ(B) = λ` ν1 (B ∩ T (1) ) for B ∈ B(R2 ).
X the Cox process with random intensity measure Λ
Then, X is called a Cox process on T (1) with linear intensity λ` .
Page 56
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
I
Let
I
I
I
I
λ` > 0 any fixed number,
T a random tessellation with edge set T (1) .
Λ a random measure with Λ(B) = λ` ν1 (B ∩ T (1) ) for B ∈ B(R2 ).
X the Cox process with random intensity measure Λ
Then, X is called a Cox process on T (1) with linear intensity λ` .
I
If T is stationary with γ = ν1 (T (1) ∩ [0, 1)2 ), then X is stationary with
intensity λ = λ` γ.
Page 56
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
I
Let
I
I
I
I
λ` > 0 any fixed number,
T a random tessellation with edge set T (1) .
Λ a random measure with Λ(B) = λ` ν1 (B ∩ T (1) ) for B ∈ B(R2 ).
X the Cox process with random intensity measure Λ
Then, X is called a Cox process on T (1) with linear intensity λ` .
I
If T is stationary with γ = ν1 (T (1) ∩ [0, 1)2 ), then X is stationary with
intensity λ = λ` γ.
I
Let X be a Cox process on T (1)
I
Then, X is a (conditional) Poisson process with intensity measure
µ( · ) = λ` ν1 ( · ∩ T (1) ) given T (1)
Page 56
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
I
Let
I
I
I
I
λ` > 0 any fixed number,
T a random tessellation with edge set T (1) .
Λ a random measure with Λ(B) = λ` ν1 (B ∩ T (1) ) for B ∈ B(R2 ).
X the Cox process with random intensity measure Λ
Then, X is called a Cox process on T (1) with linear intensity λ` .
I
If T is stationary with γ = ν1 (T (1) ∩ [0, 1)2 ), then X is stationary with
intensity λ = λ` γ.
I
Let X be a Cox process on T (1)
I
I
Then, X is a (conditional) Poisson process with intensity measure
µ( · ) = λ` ν1 ( · ∩ T (1) ) given T (1)
and the oints of X are placed as linear Poisson processes of intensity λ` on
the edges of T (1)
Page 57
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on random tessellations
Examples
Realizations of Cox processes on the edge sets
of various random tessellations
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
I
Consider a stationary Cox process X whose random intensity measure Λ
is concentrated on the edge set T (1) of a stationary tessellation T .
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
I
I
Consider a stationary Cox process X whose random intensity measure Λ
is concentrated on the edge set T (1) of a stationary tessellation T .
Use Slivnyak’s theorem, which stays that
I
the Palm version X 0 of X is a Cox process
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
I
I
Consider a stationary Cox process X whose random intensity measure Λ
is concentrated on the edge set T (1) of a stationary tessellation T .
Use Slivnyak’s theorem, which stays that
I
I
the Palm version X 0 of X is a Cox process
e (1) of
whose random intensity measure Λ0 is concentrated on the edge set T
(1)
e
a conditional version T of T , given that o ∈ T .
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
I
I
Consider a stationary Cox process X whose random intensity measure Λ
is concentrated on the edge set T (1) of a stationary tessellation T .
Use Slivnyak’s theorem, which stays that
I
I
I
the Palm version X 0 of X is a Cox process
e (1) of
whose random intensity measure Λ0 is concentrated on the edge set T
(1)
e
a conditional version T of T , given that o ∈ T .
e.
Use a suitable representation of T
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
I
I
Consider a stationary Cox process X whose random intensity measure Λ
is concentrated on the edge set T (1) of a stationary tessellation T .
Use Slivnyak’s theorem, which stays that
I
I
I
the Palm version X 0 of X is a Cox process
e (1) of
whose random intensity measure Λ0 is concentrated on the edge set T
(1)
e
a conditional version T of T , given that o ∈ T .
e.
Use a suitable representation of T
Then,
I
e (under the condition that o ∈ T (1) )
simulate the underlying tessellation T
(1)
e
and points of the Cox process on T (approximatively) radially,
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
I
I
Consider a stationary Cox process X whose random intensity measure Λ
is concentrated on the edge set T (1) of a stationary tessellation T .
Use Slivnyak’s theorem, which stays that
I
I
I
the Palm version X 0 of X is a Cox process
e (1) of
whose random intensity measure Λ0 is concentrated on the edge set T
(1)
e
a conditional version T of T , given that o ∈ T .
e.
Use a suitable representation of T
Then,
I
I
e (under the condition that o ∈ T (1) )
simulate the underlying tessellation T
(1)
e
and points of the Cox process on T (approximatively) radially,
e and new points of the Cox process on T
e (1) in an
add new edges of T
alternating fashion
Page 58
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Local simulation of typical Cox-Voronoi cells
General idea
I
I
Consider a stationary Cox process X whose random intensity measure Λ
is concentrated on the edge set T (1) of a stationary tessellation T .
Use Slivnyak’s theorem, which stays that
I
I
I
the Palm version X 0 of X is a Cox process
e (1) of
whose random intensity measure Λ0 is concentrated on the edge set T
(1)
e
a conditional version T of T , given that o ∈ T .
e.
Use a suitable representation of T
Then,
I
I
I
e (under the condition that o ∈ T (1) )
simulate the underlying tessellation T
(1)
e
and points of the Cox process on T (approximatively) radially,
e and new points of the Cox process on T
e (1) in an
add new edges of T
alternating fashion
Find a good stopping rule to reduce runtime.
Page 59
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Slivnyak’s theorem for Cox processes
Theorem
Let X be a Cox process with stationary random intensity measure Λ.
Page 59
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Slivnyak’s theorem for Cox processes
Theorem
Let X be a Cox process with stationary random intensity measure Λ. Then,
the distribution of the Palm version X 0 of X is given by
e ∪ {o} ∈ A) ,
P(X 0 ∈ A) = P(X
Page 59
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Slivnyak’s theorem for Cox processes
Theorem
Let X be a Cox process with stationary random intensity measure Λ. Then,
the distribution of the Palm version X 0 of X is given by
e ∪ {o} ∈ A) ,
P(X 0 ∈ A) = P(X
e is a Cox process whose driving measure is the Palm version Λ0 of Λ.
where X
Page 59
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Slivnyak’s theorem for Cox processes
Theorem
Let X be a Cox process with stationary random intensity measure Λ. Then,
the distribution of the Palm version X 0 of X is given by
e ∪ {o} ∈ A) ,
P(X 0 ∈ A) = P(X
e is a Cox process whose driving measure is the Palm version Λ0 of Λ.
where X
Example: Let Λ( · ) = λ` ν1 ( · ∩ T (1) ) be concentrated on the edge set T (1) of
some stationary tessellation T with (length) intensity γ > 0.
Page 59
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Slivnyak’s theorem for Cox processes
Theorem
Let X be a Cox process with stationary random intensity measure Λ. Then,
the distribution of the Palm version X 0 of X is given by
e ∪ {o} ∈ A) ,
P(X 0 ∈ A) = P(X
e is a Cox process whose driving measure is the Palm version Λ0 of Λ.
where X
Example: Let Λ( · ) = λ` ν1 ( · ∩ T (1) ) be concentrated on the edge set T (1) of
some stationary tessellation T with (length) intensity γ > 0. Then,
I
the distribution PΛ0 of Λ0 is given by
Z
1
PΛ0 (A) = E
1IA (Λ( · + x)) ν1 (dx) ,
γ
T (1) ∩[0,1)2
A∈N.
Page 59
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Slivnyak’s theorem for Cox processes
Theorem
Let X be a Cox process with stationary random intensity measure Λ. Then,
the distribution of the Palm version X 0 of X is given by
e ∪ {o} ∈ A) ,
P(X 0 ∈ A) = P(X
e is a Cox process whose driving measure is the Palm version Λ0 of Λ.
where X
Example: Let Λ( · ) = λ` ν1 ( · ∩ T (1) ) be concentrated on the edge set T (1) of
some stationary tessellation T with (length) intensity γ > 0. Then,
I
I
the distribution PΛ0 of Λ0 is given by
Z
1
PΛ0 (A) = E
1IA (Λ( · + x)) ν1 (dx) ,
γ
T (1) ∩[0,1)2
A∈N.
e (1) ), where T
e can be regarded as
Thus, Λ0 is given by Λ0 ( · ) = λ` ν1 ( · ∩ T
conditional version of T under the condition that o ∈ T (1) .
Page 60
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on Poisson-Voronoi tessellations
Cox process on PVT and its Voronoi tessellation
Page 61
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for Poisson-Voronoi tessellations
Representation of T
Theorem
√
Let T be a PVT with intensity γ = 2 λ induced by a stationary Poisson
process with intensity λ.
Page 61
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for Poisson-Voronoi tessellations
Representation of T
Theorem
√
Let T be a PVT with intensity γ = 2 λ induced by a stationary Poisson
e 2 and Φ be independent random variables,
process with intensity λ. Let R 2 , R
where
I
R 2 gamma distributed with parameters 1.5 (shape) and 1/(λπ) (scale),
e 2 beta distributed with parameters 1 and 1/2,
R
I
Φ uniformly distributed on [0, 2π).
I
Page 61
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for Poisson-Voronoi tessellations
Representation of T
Theorem
√
Let T be a PVT with intensity γ = 2 λ induced by a stationary Poisson
e 2 and Φ be independent random variables,
process with intensity λ. Let R 2 , R
where
I
I
R 2 gamma distributed with parameters 1.5 (shape) and 1/(λπ) (scale),
e 2 beta distributed with parameters 1 and 1/2,
R
Φ uniformly distributed on [0, 2π).
e is the Voronoi tessellation induced by {Xn }∞ , where
Then T
n=1
I
Page 61
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for Poisson-Voronoi tessellations
Representation of T
Theorem
√
Let T be a PVT with intensity γ = 2 λ induced by a stationary Poisson
e 2 and Φ be independent random variables,
process with intensity λ. Let R 2 , R
where
I
I
R 2 gamma distributed with parameters 1.5 (shape) and 1/(λπ) (scale),
e 2 beta distributed with parameters 1 and 1/2,
R
Φ uniformly distributed on [0, 2π).
e is the Voronoi tessellation induced by {Xn }∞ , where
Then T
n=1
p
2
e 2 R 2 , RR)
e
I X1 and X2 are given by the points X1 = ( R − R
and
p
2
2
2
e
e
X2 = ( R − R R , −RR), respectively, rotated around o with angle Φ,
I
Page 61
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for Poisson-Voronoi tessellations
Representation of T
Theorem
√
Let T be a PVT with intensity γ = 2 λ induced by a stationary Poisson
e 2 and Φ be independent random variables,
process with intensity λ. Let R 2 , R
where
I
I
R 2 gamma distributed with parameters 1.5 (shape) and 1/(λπ) (scale),
e 2 beta distributed with parameters 1 and 1/2,
R
Φ uniformly distributed on [0, 2π).
e is the Voronoi tessellation induced by {Xn }∞ , where
Then T
n=1
p
2
e 2 R 2 , RR)
e
I X1 and X2 are given by the points X1 = ( R − R
and
p
2
2
2
e
e
X2 = ( R − R R , −RR), respectively, rotated around o with angle Φ,
I
I
{Xn }∞
n=3 is distributed according to a stationary Poisson process in
R2 \B(o, r ) with intensity λ given R = r .
Page 61
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for Poisson-Voronoi tessellations
Representation of T
Theorem
√
Let T be a PVT with intensity γ = 2 λ induced by a stationary Poisson
e 2 and Φ be independent random variables,
process with intensity λ. Let R 2 , R
where
I
I
R 2 gamma distributed with parameters 1.5 (shape) and 1/(λπ) (scale),
e 2 beta distributed with parameters 1 and 1/2,
R
Φ uniformly distributed on [0, 2π).
e is the Voronoi tessellation induced by {Xn }∞ , where
Then T
n=1
p
2
e 2 R 2 , RR)
e
I X1 and X2 are given by the points X1 = ( R − R
and
p
2
2
2
e
e
X2 = ( R − R R , −RR), respectively, rotated around o with angle Φ,
I
I
{Xn }∞
n=3 is distributed according to a stationary Poisson process in
R2 \B(o, r ) with intensity λ given R = r .
Proof See Baumstark & Last (2007)
Page 62
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PVT
Line segment through the origin with the generating points X1 and X2 ,
e
where R1 = R R
Page 63
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PVT
I
Simulate two points X1 and X2 (grey) generating the segment through o,
Page 63
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PVT
I
I
Simulate two points X1 and X2 (grey) generating the segment through o,
Simulate points X3 , X4 , . . . of a stationary Poisson process in R2 \B(o, r )
with intensity λ given R = r .
Page 64
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PVT
Place points on the edges of underlying Voronoi cells and construct Initial cell
Page 65
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PVT
Intersect initial cell by bisectors
Page 66
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PVT
Stop
Page 67
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PVT
Stopping criterion
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
I
T ∗ the conditional version of T under the Palm distribution with respect to
the vertices of T ,
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
I
T ∗ the conditional version of T under the Palm distribution with respect to
the vertices of T ,
I
E ∗ the edge star of T ∗ at o, and
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
I
T ∗ the conditional version of T under the Palm distribution with respect to
the vertices of T ,
I
E ∗ the edge star of T ∗ at o, and
e the conditonal version of T given that o ∈ T (1) .
T
I
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
I
T ∗ the conditional version of T under the Palm distribution with respect to
the vertices of T ,
I
E ∗ the edge star of T ∗ at o, and
e the conditonal version of T given that o ∈ T (1) .
T
I
Then, for any measurable function h : NF → [0, ∞),
e) =
Eh(T
1
∗
∗
E
ν
(E
)
h(T
−
Z
)
,
1
E ν1 (E ∗ )
where the random variable Z is uniformly distributed on E ∗ given T ∗ .
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
I
T ∗ the conditional version of T under the Palm distribution with respect to
the vertices of T ,
I
E ∗ the edge star of T ∗ at o, and
e the conditonal version of T given that o ∈ T (1) .
T
I
Then, for any measurable function h : NF → [0, ∞),
e) =
Eh(T
1
∗
∗
E
ν
(E
)
h(T
−
Z
)
,
1
E ν1 (E ∗ )
where the random variable Z is uniformly distributed on E ∗ given T ∗ .
Application to Poisson-Delaunay tessellations
e can be expressed by the distribution of (T ∗ , E ∗ ).
I The distribution of T
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
I
T ∗ the conditional version of T under the Palm distribution with respect to
the vertices of T ,
I
E ∗ the edge star of T ∗ at o, and
e the conditonal version of T given that o ∈ T (1) .
T
I
Then, for any measurable function h : NF → [0, ∞),
e) =
Eh(T
1
∗
∗
E
ν
(E
)
h(T
−
Z
)
,
1
E ν1 (E ∗ )
where the random variable Z is uniformly distributed on E ∗ given T ∗ .
Application to Poisson-Delaunay tessellations
e can be expressed by the distribution of (T ∗ , E ∗ ).
I The distribution of T
I If T is a PDT, then the vertices of T form a stationary Poisson process
Page 68
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
e for stationary tessellations
General representation of T
Theorem
Let T be an arbitrary stationary tessellation,
I
T ∗ the conditional version of T under the Palm distribution with respect to
the vertices of T ,
I
E ∗ the edge star of T ∗ at o, and
e the conditonal version of T given that o ∈ T (1) .
T
I
Then, for any measurable function h : NF → [0, ∞),
e) =
Eh(T
1
∗
∗
E
ν
(E
)
h(T
−
Z
)
,
1
E ν1 (E ∗ )
where the random variable Z is uniformly distributed on E ∗ given T ∗ .
Application to Poisson-Delaunay tessellations
e can be expressed by the distribution of (T ∗ , E ∗ ).
I The distribution of T
I If T is a PDT, then the vertices of T form a stationary Poisson process
I and (T ∗ , E ∗ ) can be easily simulated using Slivnyak’s theorem.
Page 69
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on Poisson-Delaunay tessellations
Cox process on PDT and its Voronoi tessellation
Page 70
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PDT
Start: Simulate typical edge star E ∗ using Slivnyak’s theorem
Page 71
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PDT
Initial cell
Page 72
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PDT
Cell cut by bisectors
Page 73
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PDT
√
Stop: Weight cell characteristic by ν1 (E ∗ )/Eν1 (E ∗ ) = ν1 (E ∗ )/(64/(3π λ))
Page 74
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Cox processes on Poisson line tessellations
Cox process on PLT and its Voronoi tessellation
Page 75
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PLT
Theorem
Let
I
T (1) the edge set of a stationary PLT of intensity γ,
I
`(Φ) the line with o ∈ `(Φ) and direction Φ ∼ U[0, π) independent of T (1) ,
e the conditional version of T given that o ∈ T (1) ,
T
I
Page 75
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PLT
Theorem
Let
I
T (1) the edge set of a stationary PLT of intensity γ,
I
`(Φ) the line with o ∈ `(Φ) and direction Φ ∼ U[0, π) independent of T (1) ,
e the conditional version of T given that o ∈ T (1) ,
T
I
D
e (1) =
then T
T (1) ∪ `(Φ).
Page 75
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PLT
Theorem
Let
I
T (1) the edge set of a stationary PLT of intensity γ,
I
`(Φ) the line with o ∈ `(Φ) and direction Φ ∼ U[0, π) independent of T (1) ,
e the conditional version of T given that o ∈ T (1) ,
T
I
D
e (1) =
then T
T (1) ∪ `(Φ).
Proof Slivnyak’s theorem
Remark : Note that T (1) =
S
n∈Z
`(Φn , Rn ), where
Page 75
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PLT
Theorem
Let
I
T (1) the edge set of a stationary PLT of intensity γ,
I
`(Φ) the line with o ∈ `(Φ) and direction Φ ∼ U[0, π) independent of T (1) ,
e the conditional version of T given that o ∈ T (1) ,
T
I
D
e (1) =
then T
T (1) ∪ `(Φ).
Proof Slivnyak’s theorem
Remark : Note that T (1) =
I
S
n∈Z
`(Φn , Rn ), where
{Rn } a stationary Poisson process in R,
Page 75
Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PLT
Theorem
Let
I
T (1) the edge set of a stationary PLT of intensity γ,
I
`(Φ) the line with o ∈ `(Φ) and direction Φ ∼ U[0, π) independent of T (1) ,
e the conditional version of T given that o ∈ T (1) ,
T
I
D
e (1) =
then T
T (1) ∪ `(Φ).
Proof Slivnyak’s theorem
Remark : Note that T (1) =
S
n∈Z
`(Φn , Rn ), where
I
{Rn } a stationary Poisson process in R,
I
{Φn } an i.i.d. sequence independent of {Rn } with Φn ∼ U[0, π), and
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Mathematical tools for spatial networks on various length scales |
Cox processes on random tessellations
Typical Voronoi cell of Cox processes on PLT
Theorem
Let
I
T (1) the edge set of a stationary PLT of intensity γ,
I
`(Φ) the line with o ∈ `(Φ) and direction Φ ∼ U[0, π) independent of T (1) ,
e the conditional version of T given that o ∈ T (1) ,
T
I
D
e (1) =
then T
T (1) ∪ `(Φ).
Proof Slivnyak’s theorem
Remark : Note that T (1) =
I
S
n∈Z
`(Φn , Rn ), where
{Rn } a stationary Poisson process in R,
I
{Φn } an i.i.d. sequence independent of {Rn } with Φn ∼ U[0, π), and
I
`(Φn , Rn ) = {(x, y ) ∈ R2 : x sin Φn − y cos Φn = Rn }.
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Mathematical tools for spatial networks on various length scales |
Contents
Introduction
Point processes and Palm calculus
Random tessellations
Local simulation of typical Voronoi cells
Cox processes on random tessellations
Multiscale network modeling (Outlook to part II)
Multiscale network modeling (Outlook to part II)
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
shortest-path lengths along the edge system
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
I
uncovered cell area (e.g., the area where the signal-to-interference ratio is
below a given threshold)
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
I
I
uncovered cell area (e.g., the area where the signal-to-interference ratio is
below a given threshold)
uncovered boundary length of cells (e.g., regions where handover of mobile
users might be problematic), etc.
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
uncovered cell area (e.g., the area where the signal-to-interference ratio is
below a given threshold)
uncovered boundary length of cells (e.g., regions where handover of mobile
users might be problematic), etc.
Develop a virtual network testing tool by
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
uncovered cell area (e.g., the area where the signal-to-interference ratio is
below a given threshold)
uncovered boundary length of cells (e.g., regions where handover of mobile
users might be problematic), etc.
Develop a virtual network testing tool by
I
providing a formula library of analytical (simulation-based, parametric)
approximation formulas
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
uncovered cell area (e.g., the area where the signal-to-interference ratio is
below a given threshold)
uncovered boundary length of cells (e.g., regions where handover of mobile
users might be problematic), etc.
Develop a virtual network testing tool by
I
I
providing a formula library of analytical (simulation-based, parametric)
approximation formulas
which express the distributions of network performance chararacteristics in
terms of model parameters for
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
uncovered cell area (e.g., the area where the signal-to-interference ratio is
below a given threshold)
uncovered boundary length of cells (e.g., regions where handover of mobile
users might be problematic), etc.
Develop a virtual network testing tool by
I
I
I
providing a formula library of analytical (simulation-based, parametric)
approximation formulas
which express the distributions of network performance chararacteristics in
terms of model parameters for
a wide spectrum of multiscale tessellation models, and
Page 77
Mathematical tools for spatial networks on various length scales |
Multiscale network modeling (Outlook to part II)
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
I Insert random graphs into cells (wired networks) and compute the
distribution of
I
I
I
Insert full-dimensional random sets into cells (wireless networks) and
compute the distribution of
I
I
I
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
uncovered cell area (e.g., the area where the signal-to-interference ratio is
below a given threshold)
uncovered boundary length of cells (e.g., regions where handover of mobile
users might be problematic), etc.
Develop a virtual network testing tool by
I
I
I
I
providing a formula library of analytical (simulation-based, parametric)
approximation formulas
which express the distributions of network performance chararacteristics in
terms of model parameters for
a wide spectrum of multiscale tessellation models, and
a wide spectrum of model parameters
Page 78
Mathematical tools for spatial networks on various length scales |
Appendix
Thank you for your attention!