Download Random Point Processes

Random Point Processes
Models, Characteristics and Structural Properties
Volker Schmidt
Department of Stochastics
University of Ulm
Söllerhaus–Workshop, March 2006
1
Introduction
Aims
1. Explain some basic ideas for stochastic modelling of
spatial point patterns
stationarity (homogeneity) vs. spatial trends
isotropy (rotational invariance)
complete spatial randomness
interaction between points (clustering, repulsion)
Söllerhaus–Workshop, March 2006
2
Introduction
Aims
1. Explain some basic ideas for stochastic modelling of
spatial point patterns
stationarity (homogeneity) vs. spatial trends
isotropy (rotational invariance)
complete spatial randomness
interaction between points (clustering, repulsion)
2. Describe some basic characteristics of point–process
models
Intensity measure, conditional intensities
Pair correlation function, Ripley’s K-function, etc.
Söllerhaus–Workshop, March 2006
2
Introduction
Aims
1. Explain some basic ideas for stochastic modelling of
spatial point patterns
stationarity (homogeneity) vs. spatial trends
isotropy (rotational invariance)
complete spatial randomness
interaction between points (clustering, repulsion)
2. Describe some basic characteristics of point–process
models
Intensity measure, conditional intensities
Pair correlation function, Ripley’s K-function, etc.
3. Present techniques for the statistical analysis of spatial
point patterns
Söllerhaus–Workshop, March 2006
2
Introduction
Overview
1. Examples of spatial point patterns
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Poisson point processes
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Poisson point processes
Cluster and hard–core processes
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Poisson point processes
Cluster and hard–core processes
Gibbs point processes
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Poisson point processes
Cluster and hard–core processes
Gibbs point processes
3. Characteristics of point processes
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Poisson point processes
Cluster and hard–core processes
Gibbs point processes
3. Characteristics of point processes
4. Some statistical issues
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Poisson point processes
Cluster and hard–core processes
Gibbs point processes
3. Characteristics of point processes
4. Some statistical issues
Nonparametric estimation of model characteristics
Söllerhaus–Workshop, March 2006
3
Introduction
Overview
1. Examples of spatial point patterns
point patterns in networks
other point patterns
2. Models for spatial point patterns
Poisson point processes
Cluster and hard–core processes
Gibbs point processes
3. Characteristics of point processes
4. Some statistical issues
Nonparametric estimation of model characteristics
Maximum pseudolikelihood estimation
Söllerhaus–Workshop, March 2006
3
Examples
Point patterns in networks
Street system of Paris
Söllerhaus–Workshop, March 2006
Cytoskeleton of a leukemia cell
4
Examples
Random tessellations
Poisson line (PLT)
Poisson-Voronoi (PVT)
Poisson-Delaunay (PDT)
Simple and iterated tessellation models
PLT/PLT
Söllerhaus–Workshop, March 2006
PLT/PDT
PVT/PLT
5
Examples
Model fitting for network data
Street system of Paris
Cutout
Tessellation model
(PLT/PLT)
Model fitting for telecommunication networks
Söllerhaus–Workshop, March 2006
6
Examples
Analysis of biological networks
a) Actin network at the cell periphery
b) Lamellipodium
c) Region behind lamellipodia
Actin filament networks
Söllerhaus–Workshop, March 2006
7
Examples
Analysis of biological networks
SEM image (keratin)
Graph structure
Tessellation model
(PVT/PLT)
Model fitting for keratin and actin networks
SEM image (actin)
Söllerhaus–Workshop, March 2006
Graph structure
Tessellation model
(PLT/PDT)
8
Other examples
Modelling of tropical storm tracks
Storm tracks of cyclons over Japan
Estimated intensity field for initial points of
1945-2004
storm tracks over Japan
Söllerhaus–Workshop, March 2006
9
Other examples
Point patterns in biological cell nuclei
Heterochromatin structures in interphase nuclei
3D-reconstruction of NB4 cell nuclei (DNA shown in gray
levels) and centromere distributions (shown in red)
Söllerhaus–Workshop, March 2006
10
Other examples
Point patterns in biological cell nuclei
Projections of the 3D chromocenter distributions of an
undifferentiated (left) NB4 cell and a differentiated (right)
NB4 cell onto the xy-plane
Söllerhaus–Workshop, March 2006
11
Other examples
Point patterns in biological cell nuclei
Capsides of
cytomegalovirus
Söllerhaus–Workshop, March 2006
Extracted point pattern
Model for a point field
(cluster)
12
Models
Basic ideas
1. Mathematical definition of spatial point processes
Let {X1 , X2 , ...} be a sequence of random vectors
with values in 2 and
let X(B) = #{n : Xn ∈ B} denote the number of
„points” Xn located in a set B ⊂ 2 .
Söllerhaus–Workshop, March 2006
13
Models
Basic ideas
1. Mathematical definition of spatial point processes
Let {X1 , X2 , ...} be a sequence of random vectors
with values in 2 and
let X(B) = #{n : Xn ∈ B} denote the number of
„points” Xn located in a set B ⊂ 2 .
If X(B) < ∞ for each bounded set B ⊂ 2 , then
{X1 , X2 , ...} is called a random point process.
Söllerhaus–Workshop, March 2006
13
Models
Basic ideas
1. Mathematical definition of spatial point processes
Let {X1 , X2 , ...} be a sequence of random vectors
with values in 2 and
let X(B) = #{n : Xn ∈ B} denote the number of
„points” Xn located in a set B ⊂ 2 .
If X(B) < ∞ for each bounded set B ⊂ 2 , then
{X1 , X2 , ...} is called a random point process.
2. A point process {X1 , X2 , ...} is called
stationary (homogeneous) if the distribution of
{X1 , X2 , ...} is invariant w.r.t translations of the origin,
d
i.e. {Xn } = {Xn − u} ∀ u ∈
Söllerhaus–Workshop, March 2006
2
13
Models
Basic ideas
1. Mathematical definition of spatial point processes
Let {X1 , X2 , ...} be a sequence of random vectors
with values in 2 and
let X(B) = #{n : Xn ∈ B} denote the number of
„points” Xn located in a set B ⊂ 2 .
If X(B) < ∞ for each bounded set B ⊂ 2 , then
{X1 , X2 , ...} is called a random point process.
2. A point process {X1 , X2 , ...} is called
stationary (homogeneous) if the distribution of
{X1 , X2 , ...} is invariant w.r.t translations of the origin,
d
i.e. {Xn } = {Xn − u} ∀ u ∈
2
isotropic if the distribution of {Xn } is invariant
w.r.t rotations around the origin
Söllerhaus–Workshop, March 2006
13
Models
Basic ideas
1. Stochastic model vs. single realization
Söllerhaus–Workshop, March 2006
14
Models
Basic ideas
1. Stochastic model vs. single realization
Point processes are mathematical models
Söllerhaus–Workshop, March 2006
14
Models
Basic ideas
1. Stochastic model vs. single realization
Point processes are mathematical models
Observed point patterns are their realizations
Söllerhaus–Workshop, March 2006
14
Models
Basic ideas
1. Stochastic model vs. single realization
Point processes are mathematical models
Observed point patterns are their realizations
2. Some further remarks
Equivalent notions:
point field instead of spatial point process
Söllerhaus–Workshop, March 2006
14
Models
Basic ideas
1. Stochastic model vs. single realization
Point processes are mathematical models
Observed point patterns are their realizations
2. Some further remarks
Equivalent notions:
point field instead of spatial point process
Point processes are not necessarily dynamic
Söllerhaus–Workshop, March 2006
14
Models
Basic ideas
1. Stochastic model vs. single realization
Point processes are mathematical models
Observed point patterns are their realizations
2. Some further remarks
Equivalent notions:
point field instead of spatial point process
Point processes are not necessarily dynamic
Dynamics (w.r.t. time/space) can be added
=> spatial birth-and-death processes
Söllerhaus–Workshop, March 2006
14
Models
Stationary Poisson process
1. Definition of stationary Poisson processes
Poisson distribution of point counts:
X(B) ∼ Poi(λ|B|) for any bounded B ⊂
λ>0
Söllerhaus–Workshop, March 2006
2
and some
15
Models
Stationary Poisson process
1. Definition of stationary Poisson processes
Poisson distribution of point counts:
X(B) ∼ Poi(λ|B|) for any bounded B ⊂
λ>0
2
and some
Independent scattering of points: the point counts
X(B1 ), . . . , X(Bn ) are independent random variables
for any pairwise disjoint sets B1 , . . . , Bn ⊂ 2
Söllerhaus–Workshop, March 2006
15
Models
Stationary Poisson process
1. Definition of stationary Poisson processes
Poisson distribution of point counts:
X(B) ∼ Poi(λ|B|) for any bounded B ⊂
λ>0
2
and some
Independent scattering of points: the point counts
X(B1 ), . . . , X(Bn ) are independent random variables
for any pairwise disjoint sets B1 , . . . , Bn ⊂ 2
2. Basic properties
X(B) = λ|B| =⇒ λ = intensity of points
Void–probabilities: (X(B) = 0) = exp(−λ|B|)
Söllerhaus–Workshop, March 2006
15
Models
Stationary Poisson process
1. Definition of stationary Poisson processes
Poisson distribution of point counts:
X(B) ∼ Poi(λ|B|) for any bounded B ⊂
λ>0
2
and some
Independent scattering of points: the point counts
X(B1 ), . . . , X(Bn ) are independent random variables
for any pairwise disjoint sets B1 , . . . , Bn ⊂ 2
2. Basic properties
X(B) = λ|B| =⇒ λ = intensity of points
Void–probabilities: (X(B) = 0) = exp(−λ|B|)
Conditional uniformity: Given X(B) = n, the locations
of the n points in B are independent and uniformly
distributed random variables
Söllerhaus–Workshop, March 2006
15
Models
Stationary Poisson process
Simulated realization of a stationary Poisson process
Söllerhaus–Workshop, March 2006
16
Models
Stationary Poisson process
Simulated realization of a stationary Poisson process
convincingly shows
Complete spatial randomness
Söllerhaus–Workshop, March 2006
16
Models
Stationary Poisson process
Simulated realization of a stationary Poisson process
convincingly shows
Complete spatial randomness
No spatial trend
Söllerhaus–Workshop, March 2006
16
Models
Stationary Poisson process
Simulated realization of a stationary Poisson process
convincingly shows
Complete spatial randomness
No spatial trend
No interaction between the points
Söllerhaus–Workshop, March 2006
16
Models
General Poisson process
1. Definition of general (non–homogeneous) Poisson
processes
Poisson distribution of point counts:
X(B) ∼ Poi(Λ(B)) for any bounded B ⊂ 2 and
some measure Λ : B( 2 ) → [0, ∞]
Söllerhaus–Workshop, March 2006
17
Models
General Poisson process
1. Definition of general (non–homogeneous) Poisson
processes
Poisson distribution of point counts:
X(B) ∼ Poi(Λ(B)) for any bounded B ⊂ 2 and
some measure Λ : B( 2 ) → [0, ∞]
Independent scattering of points: the point counts
X(B1 ), . . . , X(Bn ) are independent random variables
for any pairwise disjoint sets B1 , . . . , Bn ⊂ 2
Söllerhaus–Workshop, March 2006
17
Models
General Poisson process
1. Definition of general (non–homogeneous) Poisson
processes
Poisson distribution of point counts:
X(B) ∼ Poi(Λ(B)) for any bounded B ⊂ 2 and
some measure Λ : B( 2 ) → [0, ∞]
Independent scattering of points: the point counts
X(B1 ), . . . , X(Bn ) are independent random variables
for any pairwise disjoint sets B1 , . . . , Bn ⊂ 2
2. Basic properties
X(B) = Λ(B) =⇒ Λ = intensity measure
Söllerhaus–Workshop, March 2006
17
Models
General Poisson process
1. Definition of general (non–homogeneous) Poisson
processes
Poisson distribution of point counts:
X(B) ∼ Poi(Λ(B)) for any bounded B ⊂ 2 and
some measure Λ : B( 2 ) → [0, ∞]
Independent scattering of points: the point counts
X(B1 ), . . . , X(Bn ) are independent random variables
for any pairwise disjoint sets B1 , . . . , Bn ⊂ 2
2. Basic properties
X(B) = Λ(B) =⇒ Λ = intensity measure
Void–probabilities: (X(B) = 0) = exp(−Λ(B))
Söllerhaus–Workshop, March 2006
17
Models
General Poisson process
1. Definition of general (non–homogeneous) Poisson
processes
Poisson distribution of point counts:
X(B) ∼ Poi(Λ(B)) for any bounded B ⊂ 2 and
some measure Λ : B( 2 ) → [0, ∞]
Independent scattering of points: the point counts
X(B1 ), . . . , X(Bn ) are independent random variables
for any pairwise disjoint sets B1 , . . . , Bn ⊂ 2
2. Basic properties
X(B) = Λ(B) =⇒ Λ = intensity measure
Void–probabilities: (X(B) = 0) = exp(−Λ(B))
R
Intensity function λ(x) ≥ 0 if Λ(B) = B λ(x) dx
Söllerhaus–Workshop, March 2006
17
Further basic models
Poisson hardcore process
Realisation of a Poisson hardcore process
Söllerhaus–Workshop, March 2006
18
Further basic models
Poisson hardcore process
Realisation of a Poisson hardcore process
Constructed from stationary Poisson processes (by
random deletion of points)
Söllerhaus–Workshop, March 2006
18
Further basic models
Poisson hardcore process
Realisation of a Poisson hardcore process
Constructed from stationary Poisson processes (by
random deletion of points)
Realizations are relatively regular point patterns (with
smaller spatial variability than in the Poisson case)
Söllerhaus–Workshop, March 2006
18
Further basic models
Poisson hardcore process
Description of the model
Start from a stationary Poisson process (with some
intensity λ > 0)
Söllerhaus–Workshop, March 2006
19
Further basic models
Poisson hardcore process
Description of the model
Start from a stationary Poisson process (with some
intensity λ > 0)
Cancel all those points whose distance to their nearest
neighbor is smaller than some R > 0
Söllerhaus–Workshop, March 2006
19
Further basic models
Poisson hardcore process
Description of the model
Start from a stationary Poisson process (with some
intensity λ > 0)
Cancel all those points whose distance to their nearest
neighbor is smaller than some R > 0
Minimal (hardcore) distance between point–pairs
=> R/2 = hardcore radius
Söllerhaus–Workshop, March 2006
19
Further basic models
Poisson hardcore process
Description of the model
Start from a stationary Poisson process (with some
intensity λ > 0)
Cancel all those points whose distance to their nearest
neighbor is smaller than some R > 0
Minimal (hardcore) distance between point–pairs
=> R/2 = hardcore radius
Spatial interaction between points (mutual repulsion)
Söllerhaus–Workshop, March 2006
19
Further basic models
Poisson hardcore process
Description of the model
Start from a stationary Poisson process (with some
intensity λ > 0)
Cancel all those points whose distance to their nearest
neighbor is smaller than some R > 0
Minimal (hardcore) distance between point–pairs
=> R/2 = hardcore radius
Spatial interaction between points (mutual repulsion)
Two–parametric model (with parameters λ and R)
Söllerhaus–Workshop, March 2006
19
Further basic models
Matern–cluster process
Realization of a Matern-cluster process
Söllerhaus–Workshop, March 2006
20
Further basic models
Matern–cluster process
Realization of a Matern-cluster process
Constructed from stationary Poisson processes (of
so–called cluster centers)
Söllerhaus–Workshop, March 2006
20
Further basic models
Matern–cluster process
Realization of a Matern-cluster process
Constructed from stationary Poisson processes (of
so–called cluster centers)
Realizations are clustered point patterns (with higher
spatial variability than in the Poisson case)
Söllerhaus–Workshop, March 2006
20
Further basic models
Matern–cluster process
Description of the model
Centers of clusters form a stationary Poisson process
(with intensity λ0 > 0)
Söllerhaus–Workshop, March 2006
21
Further basic models
Matern–cluster process
Description of the model
Centers of clusters form a stationary Poisson process
(with intensity λ0 > 0)
Cluster points are within a disc of radius R > 0 (around
the cluster center)
Söllerhaus–Workshop, March 2006
21
Further basic models
Matern–cluster process
Description of the model
Centers of clusters form a stationary Poisson process
(with intensity λ0 > 0)
Cluster points are within a disc of radius R > 0 (around
the cluster center)
Inside these discs stationary Poisson processes are
realized (with intensity λ1 > 0)
Söllerhaus–Workshop, March 2006
21
Further basic models
Matern–cluster process
Description of the model
Centers of clusters form a stationary Poisson process
(with intensity λ0 > 0)
Cluster points are within a disc of radius R > 0 (around
the cluster center)
Inside these discs stationary Poisson processes are
realized (with intensity λ1 > 0)
Spatial interaction between points (mutual attraction,
clustering of points)
Söllerhaus–Workshop, March 2006
21
Further basic models
Matern–cluster process
Description of the model
Centers of clusters form a stationary Poisson process
(with intensity λ0 > 0)
Cluster points are within a disc of radius R > 0 (around
the cluster center)
Inside these discs stationary Poisson processes are
realized (with intensity λ1 > 0)
Spatial interaction between points (mutual attraction,
clustering of points)
Three–parametric model (with parameters λ0 , λ1 and R)
Söllerhaus–Workshop, March 2006
21
More advanced models
Gibbs point processes
1. Define Gibbs processes via conditional intensities:
Söllerhaus–Workshop, March 2006
22
More advanced models
Gibbs point processes
1. Define Gibbs processes via conditional intensities:
For each location u ∈ 2 and for each realization
x = {x1 , x2 , ...} of the point process X = {X1 , X2 , ...},
consider the value λ(u, x) ≥ 0,
Söllerhaus–Workshop, March 2006
22
More advanced models
Gibbs point processes
1. Define Gibbs processes via conditional intensities:
For each location u ∈ 2 and for each realization
x = {x1 , x2 , ...} of the point process X = {X1 , X2 , ...},
consider the value λ(u, x) ≥ 0,
where λ(u, x) du is the conditional probability that
X has a point in du, given the positions of all points
x \ {u} of X outside of the infinitesimal region du
Söllerhaus–Workshop, March 2006
22
More advanced models
Gibbs point processes
1. Define Gibbs processes via conditional intensities:
For each location u ∈ 2 and for each realization
x = {x1 , x2 , ...} of the point process X = {X1 , X2 , ...},
consider the value λ(u, x) ≥ 0,
where λ(u, x) du is the conditional probability that
X has a point in du, given the positions of all points
x \ {u} of X outside of the infinitesimal region du
2. Special cases
stationary Poisson process: λ(u, x) = λ
Söllerhaus–Workshop, March 2006
22
More advanced models
Gibbs point processes
1. Define Gibbs processes via conditional intensities:
For each location u ∈ 2 and for each realization
x = {x1 , x2 , ...} of the point process X = {X1 , X2 , ...},
consider the value λ(u, x) ≥ 0,
where λ(u, x) du is the conditional probability that
X has a point in du, given the positions of all points
x \ {u} of X outside of the infinitesimal region du
2. Special cases
stationary Poisson process: λ(u, x) = λ
general Poisson process: λ(u, x) = λ(u)
Söllerhaus–Workshop, March 2006
22
More advanced models
Gibbs point processes
1. Define Gibbs processes via conditional intensities:
For each location u ∈ 2 and for each realization
x = {x1 , x2 , ...} of the point process X = {X1 , X2 , ...},
consider the value λ(u, x) ≥ 0,
where λ(u, x) du is the conditional probability that
X has a point in du, given the positions of all points
x \ {u} of X outside of the infinitesimal region du
2. Special cases
stationary Poisson process: λ(u, x) = λ
general Poisson process: λ(u, x) = λ(u)
Stationary Strauss process λ(u, x) = λγ t(u,x) , where
t(u, x) = #{n : |u − xn | < r} is the number of points
in x = {xn } that have a distance from u less than r
Söllerhaus–Workshop, March 2006
22
More advanced models
Strauss process
3 Properties of the Strauss process with conditional
intensity λ(u, x) = λγ t(u,x)
Söllerhaus–Workshop, March 2006
23
More advanced models
Strauss process
3 Properties of the Strauss process with conditional
intensity λ(u, x) = λγ t(u,x)
γ ∈ [0, 1] is the interaction parameter, and r > 0 the
interaction radius
Söllerhaus–Workshop, March 2006
23
More advanced models
Strauss process
3 Properties of the Strauss process with conditional
intensity λ(u, x) = λγ t(u,x)
γ ∈ [0, 1] is the interaction parameter, and r > 0 the
interaction radius
Three-parametric model (with parameters λ, γ, r)
Söllerhaus–Workshop, March 2006
23
More advanced models
Strauss process
3 Properties of the Strauss process with conditional
intensity λ(u, x) = λγ t(u,x)
γ ∈ [0, 1] is the interaction parameter, and r > 0 the
interaction radius
Three-parametric model (with parameters λ, γ, r)
4 Special cases:
If γ = 1, then λ(u, x) = λ (Poisson process)
Söllerhaus–Workshop, March 2006
23
More advanced models
Strauss process
3 Properties of the Strauss process with conditional
intensity λ(u, x) = λγ t(u,x)
γ ∈ [0, 1] is the interaction parameter, and r > 0 the
interaction radius
Three-parametric model (with parameters λ, γ, r)
4 Special cases:
If γ = 1, then λ(u, x) = λ (Poisson process)
If γ = 0, then hardcore process
(
0 for t(u, x) > 0
λ(u, x) =
λ for t(u, x) = 0
Söllerhaus–Workshop, March 2006
23
More advanced models
Strauss process
3 Properties of the Strauss process with conditional
intensity λ(u, x) = λγ t(u,x)
γ ∈ [0, 1] is the interaction parameter, and r > 0 the
interaction radius
Three-parametric model (with parameters λ, γ, r)
4 Special cases:
If γ = 1, then λ(u, x) = λ (Poisson process)
If γ = 0, then hardcore process
(
0 for t(u, x) > 0
λ(u, x) =
λ for t(u, x) = 0
If 0 < γ < 1, then softcore process (repulsion)
Söllerhaus–Workshop, March 2006
23
More advanced models
Strauss processes in bounded sets
Probability density w.r.t. stationary Poisson processes
=> simulation algorithm
Söllerhaus–Workshop, March 2006
24
More advanced models
Strauss processes in bounded sets
Probability density w.r.t. stationary Poisson processes
=> simulation algorithm
Consider an open bounded set W ⊂ 2 , a Strauss
process X in W , and a (stationary) Poisson process
XPoi in W with XPoi (W ) = 1
Söllerhaus–Workshop, March 2006
24
More advanced models
Strauss processes in bounded sets
Probability density w.r.t. stationary Poisson processes
=> simulation algorithm
Söllerhaus–Workshop, March 2006
Consider an open bounded set W ⊂ 2 , a Strauss
process X in W , and a (stationary) Poisson process
XPoi in W with XPoi (W ) = 1
R
Then, (X ∈ A) = A f (x) (XPoi ∈ dx) for some (local)
probability density f (x) with
24
More advanced models
Strauss processes in bounded sets
Probability density w.r.t. stationary Poisson processes
=> simulation algorithm
Consider an open bounded set W ⊂ 2 , a Strauss
process X in W , and a (stationary) Poisson process
XPoi in W with XPoi (W ) = 1
R
Then, (X ∈ A) = A f (x) (XPoi ∈ dx) for some (local)
probability density f (x) with
f (x) ∼ λn(x) γ s(x)
Söllerhaus–Workshop, March 2006
24
More advanced models
Strauss processes in bounded sets
Probability density w.r.t. stationary Poisson processes
=> simulation algorithm
Consider an open bounded set W ⊂ 2 , a Strauss
process X in W , and a (stationary) Poisson process
XPoi in W with XPoi (W ) = 1
R
Then, (X ∈ A) = A f (x) (XPoi ∈ dx) for some (local)
probability density f (x) with
f (x) ∼ λn(x) γ s(x)
where n(x) is the number of points of x ⊂ W , and
Söllerhaus–Workshop, March 2006
24
More advanced models
Strauss processes in bounded sets
Probability density w.r.t. stationary Poisson processes
=> simulation algorithm
Consider an open bounded set W ⊂ 2 , a Strauss
process X in W , and a (stationary) Poisson process
XPoi in W with XPoi (W ) = 1
R
Then, (X ∈ A) = A f (x) (XPoi ∈ dx) for some (local)
probability density f (x) with
f (x) ∼ λn(x) γ s(x)
where n(x) is the number of points of x ⊂ W , and
s(x) the number of pairs x, x0 ∈ x with |x − x0 | < r
Söllerhaus–Workshop, March 2006
24
More advanced models
Strauss processes in bounded sets
MCMC simulation by spatial birth-and-death processes
Söllerhaus–Workshop, March 2006
25
More advanced models
Strauss processes in bounded sets
MCMC simulation by spatial birth-and-death processes
b)
a)
Initial configuration
Söllerhaus–Workshop, March 2006
Birth of a point
25
More advanced models
Strauss processes in bounded sets
MCMC simulation by spatial birth-and-death processes
b)
a)
Initial configuration
Birth of a point
d)
c)
Birth of another point
Söllerhaus–Workshop, March 2006
Death of a point
25
More advanced models
Strauss hardcore process
1. Definition of Strauss hardcore processes:
Söllerhaus–Workshop, March 2006
26
More advanced models
Strauss hardcore process
1. Definition of Strauss hardcore processes:
For some λ, γ, r, R > 0 with r < R, the conditional
intensity λ(u, x) is given by
(
0
if tr (u, x) > 0
λ(u, x) =
λγ tR (u,x) if tr (u, x) = 0
where ts (u, x) = #{n : |u − xn | < s}
Söllerhaus–Workshop, March 2006
26
More advanced models
Strauss hardcore process
1. Definition of Strauss hardcore processes:
For some λ, γ, r, R > 0 with r < R, the conditional
intensity λ(u, x) is given by
(
0
if tr (u, x) > 0
λ(u, x) =
λγ tR (u,x) if tr (u, x) = 0
where ts (u, x) = #{n : |u − xn | < s}
2. Properties:
Söllerhaus–Workshop, March 2006
26
More advanced models
Strauss hardcore process
1. Definition of Strauss hardcore processes:
For some λ, γ, r, R > 0 with r < R, the conditional
intensity λ(u, x) is given by
(
0
if tr (u, x) > 0
λ(u, x) =
λγ tR (u,x) if tr (u, x) = 0
where ts (u, x) = #{n : |u − xn | < s}
2. Properties:
Minimal interpoint distance r (hardcore radius r/2)
Söllerhaus–Workshop, March 2006
26
More advanced models
Strauss hardcore process
1. Definition of Strauss hardcore processes:
For some λ, γ, r, R > 0 with r < R, the conditional
intensity λ(u, x) is given by
(
0
if tr (u, x) > 0
λ(u, x) =
λγ tR (u,x) if tr (u, x) = 0
where ts (u, x) = #{n : |u − xn | < s}
2. Properties:
Minimal interpoint distance r (hardcore radius r/2)
(r, R) = interval of interaction distances
(attraction/clustering if γ > 1, repulsion if γ < 1)
Söllerhaus–Workshop, March 2006
26
More advanced models
Strauss hardcore process
1. Definition of Strauss hardcore processes:
For some λ, γ, r, R > 0 with r < R, the conditional
intensity λ(u, x) is given by
(
0
if tr (u, x) > 0
λ(u, x) =
λγ tR (u,x) if tr (u, x) = 0
where ts (u, x) = #{n : |u − xn | < s}
2. Properties:
Minimal interpoint distance r (hardcore radius r/2)
(r, R) = interval of interaction distances
(attraction/clustering if γ > 1, repulsion if γ < 1)
Four-parametric model (with parameters λ, γ, r, R)
Söllerhaus–Workshop, March 2006
26
More advanced models
Strauss hardcore processes in bounded sets
Probability density f (x) w.r.t. stationary Poisson
process in an open bounded set W ⊂ 2 :
Z
(X ∈ A) =
f (x) (XPoi ∈ dx)
A
Söllerhaus–Workshop, March 2006
27
More advanced models
Strauss hardcore processes in bounded sets
Probability density f (x) w.r.t. stationary Poisson
process in an open bounded set W ⊂ 2 :
Z
(X ∈ A) =
f (x) (XPoi ∈ dx)
A
MCMC simulation by spatial birth-and-death processes
Söllerhaus–Workshop, March 2006
27
More advanced models
Strauss hardcore processes in bounded sets
Probability density f (x) w.r.t. stationary Poisson
process in an open bounded set W ⊂ 2 :
Z
(X ∈ A) =
f (x) (XPoi ∈ dx)
A
MCMC simulation by spatial birth-and-death processes
Initial configuration
Söllerhaus–Workshop, March 2006
Death of a point
Death of another point
27
Characteristics of point processes
Intensity measure
Consider an arbitrary point process {X1 , X2 , . . .} ⊂
Söllerhaus–Workshop, March 2006
2
28
Characteristics of point processes
Intensity measure
Consider an arbitrary point process {X1 , X2 , . . .} ⊂
2
Intensity measure Λ(B) = X(B)
Söllerhaus–Workshop, March 2006
28
Characteristics of point processes
Intensity measure
Consider an arbitrary point process {X1 , X2 , . . .} ⊂
2
Intensity measure Λ(B) = X(B)
Stationary case Λ(B) = λ|B|, where λ = intensity
Söllerhaus–Workshop, March 2006
28
Characteristics of point processes
Intensity measure
Consider an arbitrary point process {X1 , X2 , . . .} ⊂
2
Intensity measure Λ(B) = X(B)
Stationary case Λ(B) = λ|B|, where λ = intensity
λ = 0.01
Söllerhaus–Workshop, March 2006
λ = 0.1
28
Characteristics of point processes
Factorial moment measure; product density
Consider the second-order factorial moment measure
α(2) : B( 2 ) ⊗ B( 2 ) → [0, ∞] with
Söllerhaus–Workshop, March 2006
29
Characteristics of point processes
Factorial moment measure; product density
Consider the second-order factorial moment measure
α(2) : B( 2 ) ⊗ B( 2 ) → [0, ∞] with
α(2) (B1 × B2 ) = #{(i, j) : Xi ∈ B1 , Xj ∈ B2 ∀ i 6= j}
Söllerhaus–Workshop, March 2006
29
Characteristics of point processes
Factorial moment measure; product density
Consider the second-order factorial moment measure
α(2) : B( 2 ) ⊗ B( 2 ) → [0, ∞] with
α(2) (B1 × B2 ) = #{(i, j) : Xi ∈ B1 , Xj ∈ B2 ∀ i 6= j}
X(B1 )X(B2 ) = α(2) (B1 × B2 ) + Λ(B1 ∩ B2 )
Then
Söllerhaus–Workshop, March 2006
29
Characteristics of point processes
Factorial moment measure; product density
Consider the second-order factorial moment measure
α(2) : B( 2 ) ⊗ B( 2 ) → [0, ∞] with
α(2) (B1 × B2 ) = #{(i, j) : Xi ∈ B1 , Xj ∈ B2 ∀ i 6= j}
X(B1 )X(B2 ) = α(2) (B1 × B2 ) + Λ(B1 ∩ B2 )
Then
and
Söllerhaus–Workshop, March 2006
VarX(B) =
α(2) (B
× B) + Λ(B) − Λ(B)
2
29
Characteristics of point processes
Factorial moment measure; product density
Consider the second-order factorial moment measure
α(2) : B( 2 ) ⊗ B( 2 ) → [0, ∞] with
α(2) (B1 × B2 ) = #{(i, j) : Xi ∈ B1 , Xj ∈ B2 ∀ i 6= j}
X(B1 )X(B2 ) = α(2) (B1 × B2 ) + Λ(B1 ∩ B2 )
Then
and
VarX(B) =
α(2) (B
R
× B) + Λ(B) − Λ(B)
R
2
(2)
Often
1 × B2 ) = B1 B2 ρ (u1 , u2 ) du1 du2
ρ(2) (u1 , u2 ) = product density
α(2) (B
Söllerhaus–Workshop, March 2006
29
Characteristics of point processes
Factorial moment measure; product density
Consider the second-order factorial moment measure
α(2) : B( 2 ) ⊗ B( 2 ) → [0, ∞] with
α(2) (B1 × B2 ) = #{(i, j) : Xi ∈ B1 , Xj ∈ B2 ∀ i 6= j}
X(B1 )X(B2 ) = α(2) (B1 × B2 ) + Λ(B1 ∩ B2 )
Then
and
VarX(B) =
α(2) (B
R
× B) + Λ(B) − Λ(B)
R
2
(2)
Often
1 × B2 ) = B1 B2 ρ (u1 , u2 ) du1 du2
ρ(2) (u1 , u2 ) = product density
α(2) (B
ρ(2) (u1 , u2 ) du1 du2 ≈ probability that in each of the sets
du1 , du2 there is at least one point of {Xn }
Söllerhaus–Workshop, March 2006
29
Characteristics of point processes
Factorial moment measure; product density
Consider the second-order factorial moment measure
α(2) : B( 2 ) ⊗ B( 2 ) → [0, ∞] with
α(2) (B1 × B2 ) = #{(i, j) : Xi ∈ B1 , Xj ∈ B2 ∀ i 6= j}
X(B1 )X(B2 ) = α(2) (B1 × B2 ) + Λ(B1 ∩ B2 )
Then
and
VarX(B) =
α(2) (B
R
× B) + Λ(B) − Λ(B)
R
2
(2)
Often
1 × B2 ) = B1 B2 ρ (u1 , u2 ) du1 du2
ρ(2) (u1 , u2 ) = product density
α(2) (B
ρ(2) (u1 , u2 ) du1 du2 ≈ probability that in each of the sets
du1 , du2 there is at least one point of {Xn }
In the stationary case: ρ(2) (u1 , u2 ) = ρ(2) (u1 − u2 )
Söllerhaus–Workshop, March 2006
29
Characteristics of motion-invariant point processes
Pair correlation function
For stationary and isotropic point processes:
ρ(2) (u1 , u2 ) = ρ(2) (s) where s = |u1 − u2 |
Söllerhaus–Workshop, March 2006
30
Characteristics of motion-invariant point processes
Pair correlation function
For stationary and isotropic point processes:
ρ(2) (u1 , u2 ) = ρ(2) (s) where s = |u1 − u2 |
In the Poisson case: ρ(2) (s) = λ2
Söllerhaus–Workshop, March 2006
30
Characteristics of motion-invariant point processes
Pair correlation function
For stationary and isotropic point processes:
ρ(2) (u1 , u2 ) = ρ(2) (s) where s = |u1 − u2 |
In the Poisson case: ρ(2) (s) = λ2
Pair correlation function:
Söllerhaus–Workshop, March 2006
g(s) = ρ(2) (s) / λ2
30
Characteristics of motion-invariant point processes
Pair correlation function
For stationary and isotropic point processes:
ρ(2) (u1 , u2 ) = ρ(2) (s) where s = |u1 − u2 |
In the Poisson case: ρ(2) (s) = λ2
Pair correlation function:
g(s) = ρ(2) (s) / λ2
Examples: Poisson case: g(s) ≡ 1
Söllerhaus–Workshop, March 2006
30
Characteristics of motion-invariant point processes
Pair correlation function
For stationary and isotropic point processes:
ρ(2) (u1 , u2 ) = ρ(2) (s) where s = |u1 − u2 |
In the Poisson case: ρ(2) (s) = λ2
Pair correlation function:
g(s) = ρ(2) (s) / λ2
Examples: Poisson case: g(s) ≡ 1
whereas g(s) > (<)1 indicates clustering (repulsion)
Söllerhaus–Workshop, March 2006
30
Characteristics of motion-invariant point processes
Pair correlation function
For stationary and isotropic point processes:
ρ(2) (u1 , u2 ) = ρ(2) (s) where s = |u1 − u2 |
In the Poisson case: ρ(2) (s) = λ2
Pair correlation function:
g(s) = ρ(2) (s) / λ2
Examples: Poisson case: g(s) ≡ 1
whereas g(s) > (<)1 indicates clustering (repulsion)
Matern-cluster process g(s) > 1 for s ≤ 2R
Söllerhaus–Workshop, March 2006
30
Characteristics of motion-invariant point processes
Pair correlation function
For stationary and isotropic point processes:
ρ(2) (u1 , u2 ) = ρ(2) (s) where s = |u1 − u2 |
In the Poisson case: ρ(2) (s) = λ2
Pair correlation function:
g(s) = ρ(2) (s) / λ2
Examples: Poisson case: g(s) ≡ 1
whereas g(s) > (<)1 indicates clustering (repulsion)
Matern-cluster process g(s) > 1 for s ≤ 2R
Hardcore processes g(s) = 0 for s < dmin
where dmin = minimal interpoint distance
Söllerhaus–Workshop, March 2006
30
Characteristics of motion-invariant point processes
Reduced moment measure; Ripley’s K-function
Instead of using ρ(2) (s) or g(s)Z, we can write
α(2) (B1 × B2 ) = λ2
where
Söllerhaus–Workshop, March 2006
λ2 K(B)
=
R
B1
K(B2 − u) du
(2) (x) dx
ρ
B
31
Characteristics of motion-invariant point processes
Reduced moment measure; Ripley’s K-function
Instead of using ρ(2) (s) or g(s)Z, we can write
α(2) (B1 × B2 ) = λ2
where
λ2 K(B)
=
R
B1
K(B2 − u) du
(2) (x) dx
ρ
B
Furthermore, K(s) = K(Bs (o)) is Ripley’s K-function,
where Bs (o) = {u ∈ R2 : |u| ≤ s} for s > 0
Söllerhaus–Workshop, March 2006
31
Characteristics of motion-invariant point processes
Reduced moment measure; Ripley’s K-function
Instead of using ρ(2) (s) or g(s)Z, we can write
α(2) (B1 × B2 ) = λ2
where
λ2 K(B)
=
R
B1
K(B2 − u) du
(2) (x) dx
ρ
B
Furthermore, K(s) = K(Bs (o)) is Ripley’s K-function,
where Bs (o) = {u ∈ R2 : |u| ≤ s} for s > 0
λK(s) = mean number of points within a ball of radius s
centred at the „typical” point of {Xn }, which itself is not
counted =⇒ K = reduced moment measure
Söllerhaus–Workshop, March 2006
31
Characteristics of motion-invariant point processes
Reduced moment measure; Ripley’s K-function
Instead of using ρ(2) (s) or g(s)Z, we can write
α(2) (B1 × B2 ) = λ2
where
λ2 K(B)
=
R
B1
K(B2 − u) du
(2) (x) dx
ρ
B
Furthermore, K(s) = K(Bs (o)) is Ripley’s K-function,
where Bs (o) = {u ∈ R2 : |u| ≤ s} for s > 0
λK(s) = mean number of points within a ball of radius s
centred at the „typical” point of {Xn }, which itself is not
counted =⇒ K = reduced moment measure
In the Poisson case: K(s) = πs2
Söllerhaus–Workshop, March 2006
(= |Bs (o)|)
31
Characteristics of motion-invariant point processes
Reduced moment measure; Ripley’s K-function
Instead of using ρ(2) (s) or g(s)Z, we can write
α(2) (B1 × B2 ) = λ2
where
λ2 K(B)
=
R
B1
K(B2 − u) du
(2) (x) dx
ρ
B
Furthermore, K(s) = K(Bs (o)) is Ripley’s K-function,
where Bs (o) = {u ∈ R2 : |u| ≤ s} for s > 0
λK(s) = mean number of points within a ball of radius s
centred at the „typical” point of {Xn }, which itself is not
counted =⇒ K = reduced moment measure
In the Poisson case: K(s) = πs2
(= |Bs (o)|)
Thus, K(s) > (<)πs2 indicates clustering (repulsion)
Söllerhaus–Workshop, March 2006
31
Characteristics of motion-invariant point processes
Other functions
Spherical contact distribution function
H(s) = 1 − X(Bs (o)) = 0 ,
Söllerhaus–Workshop, March 2006
s>0
32
Characteristics of motion-invariant point processes
Other functions
Spherical contact distribution function
H(s) = 1 − X(Bs (o)) = 0 ,
s>0
D(s) = 1 − lim
ε↓0
Söllerhaus–Workshop, March 2006
Nearest-neighbor distance distribution function
X(Bs (o) \ Bε (o)) = 0 | X(Bε (o)) > 0
32
Characteristics of motion-invariant point processes
Other functions
Spherical contact distribution function
H(s) = 1 − X(Bs (o)) = 0 ,
s>0
D(s) = 1 − lim
ε↓0
Nearest-neighbor distance distribution function
X(Bs (o) \ Bε (o)) = 0 | X(Bε (o)) > 0
The J-function is then defined by
1 − D(s)
J(s) =
1 − H(s)
Söllerhaus–Workshop, March 2006
for any s > 0 with H(s) < 1
32
Characteristics of motion-invariant point processes
Other functions
Spherical contact distribution function
H(s) = 1 − X(Bs (o)) = 0 ,
s>0
D(s) = 1 − lim
ε↓0
Nearest-neighbor distance distribution function
X(Bs (o) \ Bε (o)) = 0 | X(Bε (o)) > 0
The J-function is then defined by
1 − D(s)
J(s) =
1 − H(s)
for any s > 0 with H(s) < 1
In the Poisson case: J(s) ≡ 1
Söllerhaus–Workshop, March 2006
32
Characteristics of motion-invariant point processes
Other functions
Spherical contact distribution function
H(s) = 1 − X(Bs (o)) = 0 ,
s>0
D(s) = 1 − lim
ε↓0
Nearest-neighbor distance distribution function
X(Bs (o) \ Bε (o)) = 0 | X(Bε (o)) > 0
The J-function is then defined by
1 − D(s)
J(s) =
1 − H(s)
for any s > 0 with H(s) < 1
In the Poisson case: J(s) ≡ 1
whereas J(s) > (<)1 indicates clustering (repulsion)
Söllerhaus–Workshop, March 2006
32
Estimation of model characteristics
Intensity
Suppose that
the point process {Xn } is stationary with intensity λ
Söllerhaus–Workshop, March 2006
33
Estimation of model characteristics
Intensity
Suppose that
the point process {Xn } is stationary with intensity λ
and can be observed in the bounded sampling
window W ⊂ 2
Söllerhaus–Workshop, March 2006
33
Estimation of model characteristics
Intensity
Suppose that
the point process {Xn } is stationary with intensity λ
and can be observed in the bounded sampling
window W ⊂ 2
b = X(W )/|W | is a natural estimator for λ
Then, λ
Söllerhaus–Workshop, March 2006
33
Estimation of model characteristics
Intensity
Suppose that
the point process {Xn } is stationary with intensity λ
and can be observed in the bounded sampling
window W ⊂ 2
b = X(W )/|W | is a natural estimator for λ
Then, λ
Properties:
b is unbiased, i.e., λ
b=λ
λ
Söllerhaus–Workshop, March 2006
33
Estimation of model characteristics
Intensity
Suppose that
the point process {Xn } is stationary with intensity λ
and can be observed in the bounded sampling
window W ⊂ 2
b = X(W )/|W | is a natural estimator for λ
Then, λ
Properties:
b is unbiased, i.e., λ
b=λ
λ
b → λ with probability 1
If {Xn } is ergodic, then λ
b is strongly consistent
(as |W | → ∞), i.e. λ
Söllerhaus–Workshop, March 2006
33
Estimation of model characteristics
K-function
Recall that λK(s) = mean number of points within a ball of
radius s centered at the „typical” point of {Xn }, which itself
is not counted
Söllerhaus–Workshop, March 2006
34
Estimation of model characteristics
K-function
Recall that λK(s) = mean number of points within a ball of
radius s centered at the „typical” point of {Xn }, which itself
is not counted
Edge effects occurring in the estimation of λK(r)
Söllerhaus–Workshop, March 2006
34
Estimation of model characteristics
K-function
Therefore, estimation of
by „weighted” average
Söllerhaus–Workshop, March 2006
λ2 K(s)
=
R
(2) (x) dx
ρ
Bs (o)
35
Estimation of model characteristics
K-function
Therefore, estimation of
by „weighted” average
λ2 K(s)
=
R
(2) (x) dx
ρ
Bs (o)
Edge-corrected estimator for λ2 K(s):
X
1(|Xi − Xj | < s)
\
2
λ K(s) =
|(W + Xi ) ∩ (W + Xj )|
Xi ,Xj ∈W,i6=j
Söllerhaus–Workshop, March 2006
35
Estimation of model characteristics
K-function
Therefore, estimation of
by „weighted” average
λ2 K(s)
=
R
(2) (x) dx
ρ
Bs (o)
Edge-corrected estimator for λ2 K(s):
X
1(|Xi − Xj | < s)
\
2
λ K(s) =
|(W + Xi ) ∩ (W + Xj )|
Xi ,Xj ∈W,i6=j
2 K(s) is unbiased, i.e., λ\
2 K(s) = λ2 K(s)
Notice that λ\
Söllerhaus–Workshop, March 2006
35
Estimation of model characteristics
K-function
Therefore, estimation of
by „weighted” average
λ2 K(s)
=
R
(2) (x) dx
ρ
Bs (o)
Edge-corrected estimator for λ2 K(s):
X
1(|Xi − Xj | < s)
\
2
λ K(s) =
|(W + Xi ) ∩ (W + Xj )|
Xi ,Xj ∈W,i6=j
2 K(s) is unbiased, i.e., λ\
2 K(s) = λ2 K(s)
Notice that λ\
=> Edge-corrected estimator for K(s):
X
1(|Xi − Xj | < s)
1
[
K(s) =
λb2 Xi ,Xj ∈W, i6=j |(W + Xi ) ∩ (W + Xj )|
where λb2 = X(W )(X(W ) − 1)/|W |
Söllerhaus–Workshop, March 2006
35
Estimation of model characteristics
Further edge-corrected estimators
Product density ρ(2) (s):
X
1
ρb(2) (s) =
2πr
Xi ,Xj ∈W, i6=j
where k :
Söllerhaus–Workshop, March 2006
→
k(|Xi − Xj | − s)
|(W + Xi ) ∩ (W + Xj )|
is some kernel function.
36
Estimation of model characteristics
Further edge-corrected estimators
Product density ρ(2) (s):
X
1
ρb(2) (s) =
2πr
Xi ,Xj ∈W, i6=j
where k :
→
k(|Xi − Xj | − s)
|(W + Xi ) ∩ (W + Xj )|
is some kernel function.
Pair correlation function g(s):
gb(s) = ρb(2) (s)/λb2
Söllerhaus–Workshop, March 2006
36
Estimation of model characteristics
Further edge-corrected estimators
Product density ρ(2) (s):
X
1
ρb(2) (s) =
2πr
Xi ,Xj ∈W, i6=j
where k :
→
k(|Xi − Xj | − s)
|(W + Xi ) ∩ (W + Xj )|
is some kernel function.
Pair correlation function g(s):
gb(s) = ρb(2) (s)/λb2
where λb2 = X(W )(X(W ) − 1)/|W |
Söllerhaus–Workshop, March 2006
36
Estimation of model characteristics
Further edge-corrected estimators
b :
Spherical contact distribution function H(s)
S
|W Bs (o) ∪ Xn ∈W Bs (Xn )|
b
H(s)
=
|W Bs (o)|
Söllerhaus–Workshop, March 2006
37
Estimation of model characteristics
Further edge-corrected estimators
b :
Spherical contact distribution function H(s)
S
|W Bs (o) ∪ Xn ∈W Bs (Xn )|
b
H(s)
=
|W Bs (o)|
b :
Nearest-neighbor distance distribution function D(s)
b
D(s)
=
Söllerhaus–Workshop, March 2006
X
Xn ∈W
1(Xn ∈ W Bd(Xn) (o)) 1(d(Xn ) < s)
|W Bd(Xn ) (o)|
37
Estimation of model characteristics
Further edge-corrected estimators
b :
Spherical contact distribution function H(s)
S
|W Bs (o) ∪ Xn ∈W Bs (Xn )|
b
H(s)
=
|W Bs (o)|
b :
Nearest-neighbor distance distribution function D(s)
b
D(s)
=
X
Xn ∈W
1(Xn ∈ W Bd(Xn) (o)) 1(d(Xn ) < s)
|W Bd(Xn ) (o)|
where d(Xn ) is the distance from Xn to its nearest
neighbor
Söllerhaus–Workshop, March 2006
37
Maximum pseudolikelihood estimation
Berman-Turner device
Suppose that
{Xn } is a Gibbs process with conditional intensity
λθ (u, x) which depends on some parameter θ
Söllerhaus–Workshop, March 2006
38
Maximum pseudolikelihood estimation
Berman-Turner device
Suppose that
{Xn } is a Gibbs process with conditional intensity
λθ (u, x) which depends on some parameter θ
and that the realization x = {x1 , . . . , xn } ⊂ W of {Xn }
is observed
Söllerhaus–Workshop, March 2006
38
Maximum pseudolikelihood estimation
Berman-Turner device
Suppose that
{Xn } is a Gibbs process with conditional intensity
λθ (u, x) which depends on some parameter θ
and that the realization x = {x1 , . . . , xn } ⊂ W of {Xn }
is observed
Consider the pseudolikelihood
PL(θ; x) =
n
Y
i=1
Söllerhaus–Workshop, March 2006
Z
λθ (xi , x) exp −
W
λθ (u, x) du
38
Maximum pseudolikelihood estimation
Berman-Turner device
Suppose that
{Xn } is a Gibbs process with conditional intensity
λθ (u, x) which depends on some parameter θ
and that the realization x = {x1 , . . . , xn } ⊂ W of {Xn }
is observed
Consider the pseudolikelihood
PL(θ; x) =
n
Y
i=1
Z
λθ (xi , x) exp −
W
λθ (u, x) du
The maximum pseudolikelihood estimate θb of θ is the
value which maximizes PL(θ; x)
Söllerhaus–Workshop, March 2006
38
Maximum pseudolikelihood estimation
Berman-Turner device
To determine the maximum θb, discretise the integral
Z
m
X
λθ (u, x) du ≈
wj λθ (uj , x)
W
Söllerhaus–Workshop, March 2006
j=1
39
Maximum pseudolikelihood estimation
Berman-Turner device
To determine the maximum θb, discretise the integral
Z
m
X
λθ (u, x) du ≈
wj λθ (uj , x)
W
j=1
where uj ∈ W are „quadrature points” and wj ≥ 0 the
associated „quadrature weights”
Söllerhaus–Workshop, March 2006
39
Maximum pseudolikelihood estimation
Berman-Turner device
To determine the maximum θb, discretise the integral
Z
m
X
λθ (u, x) du ≈
wj λθ (uj , x)
W
j=1
where uj ∈ W are „quadrature points” and wj ≥ 0 the
associated „quadrature weights”
The Berman-Turner device involves choosing a set of
quadrature points {uj }
Söllerhaus–Workshop, March 2006
39
Maximum pseudolikelihood estimation
Berman-Turner device
To determine the maximum θb, discretise the integral
Z
m
X
λθ (u, x) du ≈
wj λθ (uj , x)
W
j=1
where uj ∈ W are „quadrature points” and wj ≥ 0 the
associated „quadrature weights”
The Berman-Turner device involves choosing a set of
quadrature points {uj }
which includes all the data points xj as well as some
„dummy” points
Söllerhaus–Workshop, March 2006
39
Maximum pseudolikelihood estimation
Berman-Turner device
To determine the maximum θb, discretise the integral
Z
m
X
λθ (u, x) du ≈
wj λθ (uj , x)
W
j=1
where uj ∈ W are „quadrature points” and wj ≥ 0 the
associated „quadrature weights”
The Berman-Turner device involves choosing a set of
quadrature points {uj }
which includes all the data points xj as well as some
„dummy” points
Let zj = 1 if uj is a data point, and zj = 0 if uj is a
dummy point
Söllerhaus–Workshop, March 2006
39
Maximum pseudolikelihood estimation
Berman-Turner device
Then
log PL(θ; x) =
=
m X
j=1
m
X
j=1
zj log λθ (uj , x) − wj λθ (uj , x)
wj (yj log λj − λj )
where yj = zj /wj and λj = λθ (uj , x)
Söllerhaus–Workshop, March 2006
40
Maximum pseudolikelihood estimation
Berman-Turner device
Then
log PL(θ; x) =
=
m X
j=1
m
X
j=1
zj log λθ (uj , x) − wj λθ (uj , x)
wj (yj log λj − λj )
where yj = zj /wj and λj = λθ (uj , x)
This is the log likelihood of m independent Poisson
random variables Yj with means λj and responses yj .
Söllerhaus–Workshop, March 2006
40
Maximum pseudolikelihood estimation
Berman-Turner device
Then
log PL(θ; x) =
=
m X
j=1
m
X
j=1
zj log λθ (uj , x) − wj λθ (uj , x)
wj (yj log λj − λj )
where yj = zj /wj and λj = λθ (uj , x)
This is the log likelihood of m independent Poisson
random variables Yj with means λj and responses yj .
Thus, standard statistical software for fitting
generalized linear models can be used to compute θb
Söllerhaus–Workshop, March 2006
40
References
A. Baddeley, P. Gregori, J. Mateu, R. Stoica, D. Stoyan (eds.): Case Studies in
Spatial Point Process Modeling. Lecture Notes in Statistics 185, Springer, New York
2006.
J. Moeller, R.P. Waagepetersen: Statistical Inference and Simulation for Spatial Point
Processes. Monographs on Statistics and Applied Probability 100, Chapman and
Hall, Boca Raton 2004.
D. Stoyan, W.S. Kendall, J. Mecke: Stochastic Geometry and Its Applications
(2nd ed.). J. Wiley & Sons, Chichester 1995.
D. Stoyan, H. Stoyan: Fractals, Random Shapes and Point Fields. J. Wiley & Sons,
Chichester 1994.
Söllerhaus–Workshop, March 2006
41