point processes - purdue university

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

Point Processes

Tonglin Zhang

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

Outline

Outline

I Simulated Examples

I Interesting Problems

I Analysis under Stationarity

I Analysis under Nonstationarity

I Likelihood Analysis

I Stochastic Integral

I Asymtotic Frameworks


Simulated Examples

Poisson Processes

Let S be the study area. A poisson process is derived ifN(A1), · · · ,N(Ak) are iid Poisson random variable with meanµ(A1), · · · , µ(Ak) if A1, · · · ,Ak are disjoint subsets of S.Extensions:

I Cox process: µ is a random measure (e.g. log µ is given by aGaussian random field).

I Mixed poisson process (mean measure is given by Yµ, whereY is a nongenative random variable).


Simulated Examples

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Homogeneous Poisson Process

x

y

Figure : A Homogeneous Poisson Point Process on [0, 1]2.


Simulated Examples

Neyman Scott Cluster Process

A Neyman Scott cluster process has:I A parent process and an off-spring process.

I The parent process is Poisson.

I The off-spring process is derived from the parent process:each parent point generate a Poisson number of off-springpoints around it.

I Example: Let s1, · · · , sm be parent point. Then,I There are m clusters.I Cluster 1 is around s1; cluster 2 is around s2; and so on.


Simulated Examples

0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

Homogeneous Poisson Cluster Process

x

y

Figure : A Homogeneous Neyman Scott Cluster Process on [0, 1]2.


Simulated Examples

How to generate an inhomogeneous Poisson point process

Let f be a PDF on s. Then, we canI Generate n ∼ Poisson(b).

I Generate n observations independently from a PDF f . Letthem be s1, · · · , sn.

I Then, {s1, · · · , sm} is a sample from Poisson point processwith intensity

λ(s) = bf (s).


Simulated Examples

Other Processes

Compound Point ProcessI Generate n from a nonnegative discreate random varible.

I Generate n observations independent from f .


Interesting Research Problems

I Stationarity: λ(s) is a constant; λ2(s1, s2) = λ2(s1 − s2);g(s1, s2) = g(s1 − s2); and etc.

I Isotropic: Stationarity, and λ2(s1, s2) = λ2(∥s1 − s2∥).I Nonstationary process: estimation of intensity functions.

I Stationary temporal point process: return (or recurrence)intervals.

I Second-order analysis: K -function, L-function, Pair correlationfunction.

I Return Intervals.

I Marked point processes: relationship between points andmarks.


Analysis under Stationarity

The K -Function (Ripley 1976)

Let {s1, · · · , sn} be observed points from a stationary pointprocess with intensity λ. K -function describes the second-orderproperties, which is

K (t) = 2π

∫ t

0ug(u)du,

where u is the pair correlation function.



We can estimate

K̂ (t) =1

|A|λ2

n∑i=1

∑j ̸=i

wij I (dij ≤ t),

where A is a disc, wij is the weight for edge correction, and

dij = ∥si − sj∥ (λ may be replaced as λ̂ if λ is unknown).

There is a local version, which is called the local K -function, whichis

K̂local(t; si ) =1

|A|λ∑j ̸=i

wij I (dij ≤ t).



If N is a homogeneous Poisson point process, then

K (t) = 2π

∫ t

0udu = πt2.

For any s in S, let

Us,t = {s′ : 0 < ∥s− s′} ≤ t}.

If there is a point at s, then

E [N(Us,t)|s] = πt2.

Therefore, the L-function

L(t) =

√K (t)

π− t

is useful.Tonglin Zhang, Department of Statistics, Purdue University Spatial Statistics for Point and Lattice Data (Part II)


If N is a homogeneous cluster point process with cluster size k,then

g(u) ≈ k.

Therefore,K (t) ≈ kπt2.

Then,E [N(Us,t)|s] ≈ kπt2.

There isL(t) ≈ (k − 1)t > 0.

For any A ⊂ S , there is

E [N(A)] = λ|A|

andV [N(A)] ≈ kE [N(A)] = kλ|A|



Pair Correlation Function

If N is a Neyman cluster process, then we roughly have

g(u) ≈ k < 1

for some k . Then,L(t) < 0.



Pair Correlation Function

If N is strong stationary, then (under for conditions for weakdependence) for any A, we roughtly have

V [N(A)] = ϕE [N(A)]

where

ϕ = λ

∫Rd

[g(u)− 1]du+ 1.



A Simulated Example

We evaluate the value ϕ = V [N(A)]/E [N(A)] based on thefollowing example.

I N is a cluster process on [0, 1]2 with cluster size k .

I The cluster is determined by a bivariate normal with center atits parent point and standard deviance σ.

I The parameters are (λ, σ, k).

I A is a rectangle at (0.2, 0.2), (0.2, 0.8), (0.8, 0.2), and(0.8, 0.8).

I I simulated 1000 times.



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Example for Dispersion Effect

x

y

Figure : An Example for Dispersion Effect under Stationarity



Table : Averages under Stationarity in 10, 000-run simulations

λ σ k E [N(A)] V [N(A)] ϕ

1000 0.005 4 359.872 1413.705 3.9281000 0.005 2 360.600 695.630 1.9291000 0.005 1 360.188 355.049 0.9861000 0.001 4 359.594 1426.759 3.9681000 0.001 2 360.050 710.812 1.9741000 0.001 1 360.346 368.514 1.0235000 0.005 4 1800.936 7131.298 3.9605000 0.005 2 1799.848 3582.422 1.9905000 0.005 1 1800.289 1769.029 0.9835000 0.001 4 1800.547 7301.358 4.0555000 0.001 2 1800.527 3561.236 1.9785000 0.001 1 1800.356 1823.486 1.013


Annalysis under Stationarity

Return Intervals.

Let N be a stationary point process on R. Let τ be the time forthe next occurrence. If N is Poisson, then the average return timeof a point is

E (τ) =1

λ.

If N is a cluster process, then we should consider the return ofparent points. It is about

E (τ) =1

kλ.

If there is a return, then it is a return of cluster.


Annalysis under Nonstationarity

Assume N is not stationary with first-order intensity λ(s) andsecond-order intensity λ2(s1, s2). Recently, most research focuseson the second-order intensity-reweighted stationary or isotropicprocoess, where the first is defined by

g(s1, s2) =λ2(s1, s2)

λ(s1)λ(s2)= g(s1 − s2)

and the second is defined by

g(s2, s2) = g(∥s1 − s2∥).


Analysis under Nonstationarity

For any A ⊆ S, there is

V [N(A)] ≈ (ϕ− 1)E 2[N(A)] + E [N(A)]

for a centain ϕ.If N is an inhomogeneous cluster process, then we roughly have

V [N(A)] ≈ kE [N(A)],

where k is the cluster size. Therefore, an inhomogeneous clusterprocess is not a second-order intensity-reweighted stationaryprocess.



If λ(s) is given, then the K -function is estimated by

K̂ (t, λ) =1

|A|

n∑i=1

∑j ̸=i

wij I (dij ≤ t)

λ(si )λ(sj),

where A is a disc and wij is edge-correction.



One should

I Estimate λ(s);

I Estimate λ2(s1, s2) by estimating g(u) or K (u);I It is hard to provide nonparametric estimation for both the

first-order and the second-order intensity functions. Therefore,one often (e.g. Guan (2009) JASA, 1482-1491) considers:

I A parametric analysis for λ(s); andI a nonparametric analysis for λ2(s1, s2).



A Simulated Example

We generate data from cluster process with λ(s) = bβ(2, 2).

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Inhomogeneous Poisson Cluster Process

x

y

Figure : An Example for Dispersion Effect Under Nonstationarity



Table : Averages under Nonstationarity in 10, 000 Simulations

b σ k E [N(A)] V [N(A)] ϕ

1000 0.005 4 472.797 1857.836 3.9291000 0.005 2 472.561 935.265 1.9791000 0.005 1 472.442 467.561 0.9901000 0.001 4 471.662 1876.383 3.9781000 0.001 2 472.554 957.339 2.0261000 0.001 1 472.861 482.426 1.0205000 0.005 4 2362.713 9401.313 3.9795000 0.005 2 2362.378 4717.934 1.9975000 0.005 1 2363.058 2343.867 0.9925000 0.001 4 2362.672 9274.963 3.9265000 0.001 2 2361.637 4607.397 1.9515000 0.001 1 2363.759 2387.218 1.010


Likelihood Analysis

If N is an inhomogeneous Poisson process with intensity functionλθ(s), then the loglikelihood function is

ℓ(θ) =n∑

i=1

log λθ(si )−∫Sλθ(u)du.

If N is not Poisson, the above is the composite loglikelihoodfunction, which can also provide a consistent estimator.


Stochastic Integral

Let f (s) be a function. We can use Stochastic Integral method as

U =n∑

i=1

f (si ) =

∫Sf (s)N(ds).

Then,

E (U) =

∫Sf (s)λ(s)ds

and

E (U2) =

∫S

∫Sf (s)f (s′)λ2(s, s

′)ds′ds+

∫Sf 2(s)λ(s)ds.

Therefore,

V (U) =

∫S

∫Sf (s)f (s′)[g(s, s′)−1]λ(s)λ(s′)ds′ds+

∫Sf 2(s)λ(s)ds.


Stochastic Integral

For a bivariate function f (s, s′), there is

U =

∫S

∫Sf (s, s′)N(ds)N(s′).

Then,

E (U) =

∫S

∫Sf (s, s′)λ2(s, s

′)dsds′ +

∫Sf (s, s)λ(s)ds.


Stochastic Integral

In addition, there is

E (U2) =

∫S

∫S

∫S

∫Sf (s, s′)f (s′′, s′′′)λ4(s, s

′, s′′, s′′′)dsds′ds′′ds′′′

+

∫S

∫S

∫S[2f (s, s′)f (s′′, s′′) + f (s, s′)f (s, s′′)

+ f (s, s′)f (s′′, s′) + 2f (s, s′)f (s′′, s)]λ3(s, s′, s′′)dsds′ds′′

+

∫S

∫S[f (s, s)f (s′, s′) + f (s, s′)f (s, s′)

+ f (s, s′)f (s′, s)]λ2(s, s′)dsds′

+

∫Sf 2(s, s)λ(s)ds.


Stochastic Integral

For ℓ(θ), there is

ℓ(θ) =

∫Slog λθ(s)N(ds)−

∫Sλθ(u)du.

Then, we have

∂ℓ(θ)

∂θj=

∫S

1

λθ(s)

∂λθ(s)

∂θjN(ds)−

∫S

∂λθ(u)

∂θjdu.

Thus,

E [∂ℓ(θ)

∂θj] = 0.


Asymptotic Frameworks

There are two different types of asymtotic frameworks:

I Increasing domain: S increases to Rd but intensity functionsdo not vary (focused by most research).

I Fixed domain: S does not vary but intensity functions goes toinfnity (e.g. λ(s) = bλ0(s) as b → ∞, rarely discussed).


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