marked point processes - purdue universityhuang251/point2.pdf · marked poisson point process 0.2...

Title: Spatial Statistics for Point Processes and Lattice Data (Part III)

Marked Point Processes

Tonglin Zhang

Tonglin Zhang, Department of Statistics, Purdue University Spatial Statistics for Point and Lattice Data (Part III)

Outline

Outline

I Description and Definition

I Research Problems

I Testing Independence

I A Few Famous Models

I First-Order and Second-Order Analyses

I Asymptotic Frameworks


Description and Definition

Description

A marked point process is composed of a point process andassociate marks, which can be expressed as

{(si ,mi ) : i = 1, · · · , n},

where s1, · · · , sn are locations and m1, · · · ,mn are associatedmarks.



A Simulated Example

I did simulation:I Generate Homogeneous Poisson point process with

λ(s) = 1000 on [0, 1]2.

I Generate mi ∼iid Exp(1) at each si .

I Then, the marked point process {(si ,mi )} is derived.



1

2

3

4

5

6

Figure : A Simulated Example of Marked Point Processes on [0, 1]2.



Definition

A marked point process N can be understood as a pure pointprocess on S ×M, where S is the domain of points and M is thedomain of marks, such that

N(A×M) < ∞

if A is bounded.



Therefore, one can define the first-order intensity function as

λ(s,m) = lim|ds×dm|→0

E [N(ds× dm)]

|ds1 × dm1|.

The second-order intensity function is

λ2((s1,m1), (s2,m2))

= lim|ds1×dm1|,|ds2×dm2|→0

E [N(ds1 × dm1)N(ds2 × dm2)]

|ds1 × dm1||ds2 × dm2|.

Similarly, we can define the kth-order intensity function λk .



Based on joint intensity functions, we can define marginal intensityfunctions. For example, we can define

λs(s) =

∫M

λ(s,m)dm

as the marginal (first-order) intensity for points

λm(m) =

∫Sλ(s,m)ds

as the marginal (first-order) intensity for marks. Then,

f (m|s) = λ(s,m)

λs(s)

is the conditional density of marks.



In addition, we can also define

λ2,s(s1, s2) =

∫M

∫M

λ2((s1,m1), (s2,m2))dm2dm1

and

λ2,m(m1,m2) =

∫S

∫Sλ2((s1,m1), (s2,m2))ds2ds1.

Then,

f2(m1,m2|s1, s2) =λ2((s1,m1), (s2,m2))

λ2,s(s1, s2)

is the bivariate conditional density of marks.



A Simulated Example

Let S = [0, 1]2 and M = R+. The intensity function of points isλs(s) = κβ(2, 2). The density of marks is f (m|s) = exp(ω), whereω = 10(1 + ∥s− s0∥) with s0 = (0.5, 0.5). We consider the markedPoisson and cluster point process, respectively. We chooseκ = 1000 and the size of cluster is Poisson with mean 5.



Marked Poisson Point Process

0.2

0.4

0.6

0.8

1

1.2

Figure : A Simulated Example of Poisson Marked Poisson Processes on[0, 1]2.



Marked Cluster Point Process

0.2

0.4

0.6

0.8

1

1.2

Figure : A Simulated Example of Poisson Cluster Marked PoissonProcesses on [0, 1]2.


Research Problems

Research Problems

Althouth many people have done research on pure point processes,only a few people have worked on marked Point processes. Themost interesting problem is to investigate the relationship betweenpoints and marks, which may include

I testing independence;

I modeling relationship between points and marks;

I first-order and second-order analyses;

I return intervals; and

I parametric and nonparametric analysis.


Testing Independence


There are two ways to describe independence between marks andpoints. The first one uses the Janossy measure and the second oneuses Intensity functions.



Definition of Independence

If the Janossy measure is used, then one uses that

P((s1,m1) ∈ A1 × B1, · · · , (sn,mn) ∈ An × Bn)

=P(s1 ∈ A1, · · · , sn ∈ An)P(m1 ∈ B1, · · · ,Bn ∈ Bn),

for any n ≥ 1, A1, · · · ,An ⊆ S, and B1, · · · ,Bn ⊆ M. This isoften used as the definition of independence.



Definition of Separability

If the intensity functions are used, then one uses that

λk((s1,m1), · · · , (sk ,mk))

λk,s(s1, · · · , sk)λk,m(m1, · · · ,mk)

is constant. This is often called separability, where λk,s and λk,m

are the marginal intensity for points and marks, respectively.

Question: What is the relationship between these two definitions?




I have proposed a Kolmogorov-Smirnov test to assessindependence.

I The test considers the null hypothesis

H0 : P(s×m ∈ A× B) = P(s ∈ A)P(m ∈ B)

for any A ∈ B(S) and B ∈ B(M).

I It is enough to consider a collection of subsets in A ∈ B(S)and B ∈ B(M) such that the test statistic is

Tn =√n supA∈A,B∈B

|N(A× B)

n− N(A×M)

n

N(S × B)

n|.

The null hypothesis is rejected if Tn is large.



I Let κ = E (n) = E [N(S ×M)]. If κ is large, then Tn weaklyconverges to a Brownian pillow on S ×M whose distributionis unknown.

I However, if we define A by a transformation F from S to Ras A ∈ A if and only if

A = F−1((−∞, t] : t ∈ R),

thenTn

D→ W2,

where W2 is the standard two-dimensional Brownian pillow on[0, 1]2.

I W2 is an extension from the standard Brownian bridgeW = W1 on R2. However, the quantile function of W2 isunknown. I derive it using a simulation method.



There are some other methods.

I Schoenberg (2004) considers a test based on marked Poissonpoint processes.

I Guan (2007) proposes a stationary tests.

I Schlather, Ribeiro, and Diggle (2004) consider a random fieldmethods.

I All of these methods contain unknown parameters orfunctions to be estimated.



If independence is accepted, then we can

I model points and marks separately;

I ignore the relationship between points and marks;

I predict points and marks independently.

If points and marks are dependent, then we should consider theirrelationship. We can use

I Intensity dependent models; and

I Location dependent models.


A Few Famous Models

Intensity-Dependent Models

I Ho and Stoyan (2008, Statistics and Probability Letters,1194-1199) propose (for normal marks)

m(s) = a+ bλs(s) + ϵ(s),

where ϵ(s) is a white noise; and

λs(s) = exp(α+ βS(s));m(s) = S(s) + ϵ(s),

where S(s) is a Gaussian random field.

I Myllymaki and Penttinen (2009, Statistica Neerlandica,450-473) consider an intensity-dependent marking model as

m(s)|λs(s) ∼ Fm(·|λ(s)).


A Few Famous Models

Location-Dependent Models

It is possible to consider a model with

E [m(s)] = fθ(s)(m)

This is also called the location dependent model. It has beenconsidered. The idea can be thought as motivated from the GWR(geographical weighted regression).


First-Order and Second-Order Analyses

First-Order Analysis

We can still use the method of Stochastic Integral for thefirst-order analysis, which is based on an expression of∫

S

∫M

f (s,m)N(ds× dm).

For example, if λ(s,m) = λθ(s,m), then the composite likelihoodfunction

ℓ(θ) =

∫S

∫M

log λθ(s,m)N(ds× dm)−∫S

∫M

λθ(s,m)dmds

can be used.


First-Order and Second-Order Analyses

Second-Order Analysis

One can also consider a method to estimate λ2((s1,m1), (s2,m2)).However, there is no other measurement for the second-orderproperties.


Asymptotic Frameworks

There are still two types of asymptotic frameworks:

I Increasing domain.

I Fixed domain.


marked point processes - purdue universityhuang251/point2.pdf · marked poisson point process 0.2...

Documents