an introduction to spatial point processes · an introduction to spatial point processes...

Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes

An introduction to spatial point processes

Jean-Francois Coeurjolly


1 Examples of spatial data

2 Intensities and Poisson p.p.

3 Summary statistics

4 Models for point processes


A very very brief introduction . . .

The realization x , of a spatial point process defined on a spaceS is a (locally) finite set of objects xi ∈ S .

x = x1, . . . , xn , xi and n are random.

Thank you for your attention !


Spatial data . . .

. . . can be roughly and mainly classified into three categories :

1 Geostatistical data.

2 Lattice data.

3 Spatial point pattern


Geostatistical data

observation of a (e.g.) continuous random variable at fixedlocations

meuse dataset (R package gstat) : topsoil heavy metalconcentrations, at the observation locations, collected in a floodplain of the river Meuse.

100

200

300

400

500

600

100

200

300

400

500

600

Main objective : interpolate the spatial data.


Lattice data (1)

Eire dataset (R packagespdep)

% of people with groupA in eire, observed in 26regions.

The data are aggregatedon the region ⇒random field on anetwork.

Percentage with blood group A in Eire

under 27.9127.91 − 29.2629.26 − 31.02over 31.02


Lattice data (2)

Lennon dataset (Rpackage fields)

Real-valued random field(gray scale image withvalues in [0, 1]).

Defined on the network1, . . . , 2562.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


Lattice data (3)

Over-interpretation : xkcd.


Spatial point pattern (1)

Japanesepines dataset (Rpackage spatstat)

Locations of 65 trees on abounded domain.

S = R2 (equipped with ‖ · ‖).

japanesepines

Questions of interest :

Can we estimate the number of trees per unit volume ?Homogeneous or inhomogeneous ?

Is there any independence, attraction or repulsion betweentrees ?



Longleaf dataset (R packagespatstat)

Locations of 584 treesobserved with theirdiameter at breast height.

S = R2 × R+ (equipped withmax(‖ · ‖, | · |)).

longleaf

Additional scientific questions :

Can the mark explain the intensity of the number of trees ?

Does a large tree tend to have smaller trees close to it ?



Ants dataset (R packagespatstat)

Locations of 97 antscategorised into two species.

S = R2 × 0, 1 (equippedwith the metricmax(‖ · ‖, dM ) for anydistance dM on the markspace).

ants


Competition inside one specie ? between the two species ?



3604 locations of trees observed with spatial covariates(here the elevation field).

S = R2 (equipped with the metric ‖ · ‖), z (·) ∈ R2.


Can the elevation field explain the arrangement of trees ?

Among a large number of spatial covariates, which oneshave the largest influence ?



Spatio-temporal point process on a complex spaceDaily observation of sunspots at the surface of the sun.can be viewed as the realization of a markedspatio-temporal point process on the sphere.S = S2 × R

+ × R+ (state, time, and mark).


Spatial point pattern (6) : eye-movement data

Eye-movement (on an image or video) iscomposed of

sacades : exploratory step, local, veryquick 120ms.

fixations (< 1 of oscillation) ; analysingfixations allows to understand how asubject explores an image ; locations offixations as well as their number arerandom.

Oculo-nimbus project (Univ. Grenoble) : aim to understandmechanisms of newborns vision

Dozens of images

Newborns of 3-, 6-, 9- and12-month + adults controlgroup

' 40 subjects per age group

' 15 − 20 fixations bysubject


Mathematical definition of a spatial point process

Do you really want to look at this ?

S : Polish state space of the point process (equipped with theσ-algebra of Borel sets B).

A configuration of points is denoted x = x1, . . . , xn , . . .. ForB ⊆ S : xB = x ∩ B .

Nlf : space of locally finite configurations, i.e.

x ,n(xB ) < ∞,∀B bounded ⊆ S

equipped with Nlf = σ(x ∈ Nlf ,n(xB ) = m,B ∈ B,B bounded,m ≥ 1

).

Definition

A point process X defined on S is a measurable application defined onsome probability space (Ω,F ,P ) with values on Nlf .

Measurability of X⇔ N (B ) is a r.v. for any bounded B ∈ B.


Theoretical characterization of the distribution of X

Proposition

The distribution of a point process X is determined

1 by the joint distribution of N (B1), . . . ,N (Bm ) for any boundedB1, . . . ,Bm ∈ B and any m ≥ 1.

2 or by its void probabilities, i.e. by

P (N (B ) = 0), for bounded B ∈ B.

For the rest of the talk :

let S = R2 or a bounded domain of R2.

everything can ± be extended to marked spatial point processes,spatio-temporal point processes, manifold-values point processes.


Moment measures

Moments (mean, variance, covariance,. . .) play an important rolein the characterization of a random variable (or a time series,random field) ;

For point processes : moment measures which are related tomoments of counting variables ;

Definition : for n ≥ 1 we define

the n-th order (reduced) moment measure (defined on Sn) by

α(n)(D) = E,∑

u1,...,un∈X

1(u1, . . . , un ∈ D), D ⊆ Sn .

where the , sign means that the n points are pairwise distinct.


Intensity functions

Often (always) assumed that moment measures areabsolutely continuous w.r.t. Lebesgue measures, so insteadof the moment measures, the quantitites of interest (i.e.the ones you should keep in mind !) are :

1 Intensity function : ρ(·) : Rd → R+

ρ(u) = lim|du |→0

P(one event in B (u ,du))|du |

2 Second-order intensity function : ρ(2)(·, ·) : Rd × Rd → R+

ρ(2)(u , v ) = lim|du |,|dv |→0

P(2 distinct events in B (u , du) and B (v ,dv ))|du ||dv |

3 k -th order intensity function . . .


Campbell formula (1)

Valid for any point process (having an intensity function)

Campbell Theorem

For any measurable function h : Rd → R (such that . . .. . .)

E∑u∈X

h(u) =

∫h(u)ρ(u)du .

Examples :

h(u) = 1(u ∈W ), for W ⊂ Rd

EN (W ) =

∫Wρ(u)du

(= ρ|W | if ρ(·) = ρ, homogeneous case

)h(u) = 1(u ∈W )ρ(u)−1, for W ⊂ Rd

E∑

u∈X∩W

ρ(u)−1 = |W |.


Campbell formula (2)

Valid for any point process (having a 2-nd . . .)

Campbell Theorem

For any measurable function h : Rd × Rd → R (such that . . .. . .)

E,∑

u ,v∈X

h(u , v ) =

∫ ∫h(u , v )ρ(2)(u , v )dudv .

Examples :

h(u) = 1(u ∈ A, v ∈ B ) for A,B ⊂ Rd s.t. A ∩ B = ∅

E,∑

u ,v∈X

1(u ∈ A)1(v ∈ B ) = E (N (A)N (B )) =

∫A

∫Bρ(2)(u , v )dudv .

⇒ Modelling/estimating ρ(2) allows to understand/modelcovariances of counting variables.


Covariances of counting variables

For A,B ⊂ Rd s.t. A ∩ B = ∅

Cov (N (A),N (B )) =

∫A

∫B

ρ(2)(u , v ) − ρ(u)ρ(v )

dudv

=

∫A

∫B

ρ(u)ρ(v ) g(u , v ) − 1dudv .

where the function g given by

g(u , v ) =ρ(2)(u , v )ρ(u)ρ(v )

is called the pair correlation function.

The departure of g to 1 will measure some kind of independencefor a point process X (wait for a few slides).

If X is isotropic (i.e the distribution of X in invariant underrotation), then g(u , v ) = g(‖v − u‖).


Poisson point processes

Intuitive definition : X ∼Poisson(S , ρ)

∀m ≥ 1, ∀ bounded and disjoint B1, . . . ,Bm ⊂ S , the r.v.XB1

, . . . ,XBmare independent.

N (B ) ∼ P(∫

Bρ(u)du

)for any bounded A ⊂ S .

Poisson process is the reference model for point processes.

PPP model points without any interaction !

If ρ(·) = ρ, X is said to be homogeneous which implies

EN (B ) = ρ|B |, VarN (B ) = ρ|B |.


A few realizations on W

u = (u1, u2) ∈Wρ(u) = βe−u1−u

21−.5u

31 , W = [0, 1]2.

ρ = 100, W = [0, 1]2.

ρ(u) = βe2 sin(4πu1u2), W = [−1, 1]2.

(β is adjusted s.t. the mean number of points in W ,∫Wρ(u)du = 200.)

110 points

050

100

150

2040

6080


A few properties of Poisson point processes

Proposition : if X ∼Poisson(W , ρ)

Void probabilities : v (B ) = P (N (B ) = 0) = e−∫B

(ρ(u)du).

For any u , v ∈ Rd ,

ρ(2)(u , v ) = ρ(u)ρ(v )⇒ g(u , v ) =ρ2(u , v )ρ(u)ρ(v )

= 1

(also valid for ρ(k ), k ≥ 1)

Hence, for a general point process X

g(u , v ) < 1 means that two points are less likely to appear at u , vthan for the Poisson model.⇒ characteristic for repulsive patterns

g(u , v ) > 1 means that two points are more likely to appear atu , v than for the Poisson model.⇒ characteristic for clustered patterns


Statistical inference for a Poisson point process

Simulation :

homogeneous case : very simplenon-homogeneous case : a thinning procedure can beefficiently done.

Inference :

consists in estimating ρ, ρ(·; β) or ρ(u) depending on thecontext.All these estimates can be used even if the spatial pointprocess is not Poisson (wait for 2 slides)Asymptotic properties very simple to derive under thePoisson assumption.

Goodness-of-fit tests : tests based on quadrats counting,based on the void probability,. . .


Parametric intensity estimation

Problem : model ρ(u) = ρ(u; β) for β ∈ Rp , p ≥ 1 and estimate β.

Example : forestry dataset

ρ(u; β) = exp (β1 + β2Alt(u) + β3Slope(u))

where Alt(u) and Slope(u) are spatial covariates corresponding tomaps of altitude and slope of elevation.

Assume we have a Poisson point process with intensity ρ(u; β)observed in W .

Poisson likelihood

It can be shown that the log-likelihood for this model writes (up to anormalizing constant)

`W (X; β) =∑

u∈X∩W

log ρ(u; β) −∫W

ρ(u; β)du


Towards estimating equations

Rathburn and Cressie (98) : MLE is consistent and

asymptotically normal when X=Poisson ;

. . .. . .but the procedure is acutally valid for much more generalpoint processes

Evaluate the score function (gradient vector with length p)

`′(X, β) =∑

u∈X∩W

ρ′(u; β)ρ(u; β)

−

∫W

ρ′(u; β)du .

⇒ Campbell form. (valid for any X) : E `′(X, β) = 0

So, `′(X, β) is a nice estimating equation (see RW’s talk).

(Properties for the resulting estimator require more techniques).


Back to Eye-movement data


Example of modelling

log ρ(u ,m; β) = β1Saliency(u) +

4∑m ′=1

(βm

′

0 + βm′

1 Ad(u))1(m′ = m)

where

Saliency(u) : Deterministic model of intensity map.

Ad(u) : binary map built from the top 5% of ρAd(u).

βm′

0 : can be ,, the different number of points per agegroup.

βm′

1 : parameters of interest ; the values are not interesting

but β11 < · · · < β

41 is the cognitive hypothesis to test.


Adjusted contrasts tests

⇒ Significant differences for 5 out of the 6 images.


Objective and classification

Objective :

Define some descriptive statistics for s.p.p. (independently onany model so).

Measure the abundance of points, the clustering or therepulsiveness of a spatial point pattern w.r.t. the Poisson pointprocess.

Classification :

First-order type based on the intensity function.

Second-order type statistics : pair correlation function, Ripley’sK function.

Statistics based on distances : empy space function F ,nearest-neigbour G , J function.

(We assume that ρ and ρ(2) exist in the rest of the talk)


Ripley’s K function (isotropic and planar case)

Definition : let r ≥ 0

K (r ) =ρ−1 E(number of extra events within distance r of a randomly chosen event

)=ρ−1E

(N (B (0, r ) \ 0)|0 ∈ X )

L(r ) =√K (r )/π

Properties :

Under the Poisson assumption, K (r ) = πr2 ; L(r ) = r .

If K (r ) > πr2 or L(r ) > r (resp. K (r ) < πr2 or L(r ) < r) wesuspect clustering (regularity) at distances lower than r .

Application in practice :

define a grid r values : r1, . . . , rI ;

find an estimator of K (ri ) or L(ri ), say K (ri ) and L(ri ) ;

Plot e.g. (ri , L(ri )) and compare with the Poisson case.


Edge corrected estimation of the K function

Definition

We define

the border-corrected estimate as

KBC (r ) =1

ρ

1

N (Wr )

∑u∈Wr

N (B (u , r )) − 1

where Wr = u ∈W : B (u , r ) ⊆W is the erosion of W by r .

the translation-corrected estimate as

KTC (r ) =1

ρ2

,∑u ,v∈XW

1(v − u ∈ B )|W ∩Wv−u |

where Wu = W + u = u + v : v ∈W .

Remark : everything extends to 2nd-order reweighted stationary point processes ;

asymptotic properties depend on mixing conditions,. . .


Example of L function for a Poisson point pattern

106

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

r

Linh

om(r)

Linhomobs(r)Linhom(r)Linhomhi(r)Linhomlo(r)

The enveloppes are constructed using a Monte-Carloapproach under the Poisson assumption.

⇒ we don’t reject the Poisson assumption.


Example of L function for a repulsive point pattern

113

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

r

Linh

om(r)


⇒ the point pattern does not come from the realization ofa homogeneous Poisson point process.

exhibits repulsion at short distances (r ≤ .05)


Example of L function for a clustered point pattern

Xth

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

r

Linh

om(r)


⇒ the point pattern does not come from the realization ofa homogeneous Poisson point process.

exhibits attraction at short distances (r ≤ .08).


Statistics based on distances : F , G and J functions

Assume X is stationary (definitions can be extended in the general case)

Definition

The empty space function is defined by

F (r ) = P (d (0,X ) ≤ r ) = P (N (B (0, r )) > 0), r > 0.

The nearest-neighbour distribution function is

G(r ) = P (d (0,X \ 0) ≤ r |0 ∈ X )

J -function : J (r ) = (1 −G(r ))/(1 − F (r )), r > 0.

Poisson case : ∀r > 0, F (r ) = G(r ) = 1 − e−πr2, J (r ) = 1.

F (r ) < Fpois (r ), G(r ) > Gpois (r ), J (r ) < 1 : attraction at dist. < r .

F (r ) > Fpois (r ), G(r ) < Gpois (r ), J (r ) > 1 : repulsion at dist. < r .


Non-parametric estimation of F , G and J

As for the K and L functions, several edge corrections exist. We focus here only on

the border correction. We assume that X is observed on a bounded window W

with positive volume.

Definition

Let I ⊆W be a finite regular grid of points and n(I ) itscardinality. Then, the (border corrected) estimator of F is

F (r ) =1

n(Ir )

∑u∈Ir

1(d (u ,X ) ≤ r )

where Ir = I ∩Wr .

The (border corrected) estimator of G is

G(r ) =1

N (Wr )

∑u∈X∩Wr

1(d (u ,X \ u) ≤ r )


Application to a clustered point pattern dataXth

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

r

F(r)

Fobs(r)F(r)Fhi(r)Flo(r)

0.000 0.005 0.010 0.015 0.020

0.0

0.2

0.4

0.6

0.8

r

G(r)

Gobs(r)G(r)Ghi(r)Glo(r)

0.00 0.02 0.04 0.06 0.08 0.10

05

1015

2025

r

J(r)

Jobs(r)J(r)Jhi(r)Jlo(r)


More realistic models than the Poisson point process

We can distinguish several classes of models for spatial pointprocesses. Among them :

1 Cox point processes (which include Poisson Cluster pointprocesses,. . . ).

2 Gibbs point processes.Strong links with statistical physics

3 Determinantal point processes.Links with random matrices


An attempt to classify these models . . .

Model Allows to model Are moments Density w.r.t.explicit ? Poisson ?

Cox attraction yes no

Gibbs repulsion no yesbut also attraction

Determinantal repulsion yes yes

This classification is really important since the

methodologies to infer these models will be based either onmoment methods or on conditional densities w.r.t. Poisson pointprocess.

asymptotic results require different tools : e.g. CLT based onmixing conditions (for Cox, determinantal point process) or on a“martingale-type” condition for Gibbs point process.


Cox point processes (1)

Definition

Suppose that Z = Z (u) : u ∈ S is a nonnegative random field so thatwith probability one, u → Z (u) is a locally integrable function. If theconditional distribution of X given Z is a Poisson process on S withintensity function Z , then X is said to be a Cox process driven by Z .

It is straightforwardly seen that

1 Provided Z (u) has finite expectation and variance for any u ∈ S

ρ(u) = EZ (u), ρ(2)(u , η) = E[Z (u)Z (η)], g(u , η) =E[Z (u)Z (η)]ρ(u)ρ(η)

.

2 The void probabilities are given by

v (B ) = E exp

(−

∫B

Z (u)du)

for bounded B ⊆ S .


Cox point processes (2) : Neymann-Scott process

Definition

Let C ∼Poisson(Rd , κ). Conditional on C , let Xc ∼Poisson(Rd , ρc) beindependent Poisson processes for any c ∈ C where

ρc(u) = αk (u − c)

where α > 0 is a parameter and k is a kernel (i.e. for all c ∈ Rd ,u → k (u − c) is a density function). Then X = ∪c∈CXc is aNeymann-Scott process with cluster centres C and clustersXc , c ∈ C .

X is a Cox process on Rd driven by Z (u) =∑

c∈C αk (u − c).

When k is the Gaussian kernel, X is called the Thomas process.


Four realizations of Thomas point processes

κ = 50, σ = 0.03, α = 5

κ = 100, σ = 0.03, α = 5

κ = 50, σ = 0.01, α = 5

κ = 100, σ = 0.01, α = 5


Cox point processes (4) : Log-Gaussian Cox processes

Definition

Let X be a Cox process on Rd driven by Z = expY where Y is aGaussian random field. Then, X is said to be a log Gaussian Coxprocess (LGCP).

Basic properties : let m and c denote the mean function and thecovariance function of Y

1 the intensition function of X is

ρ(u) = exp (m(u) + c(u , u)/2) .

2 The pair correlation function g of X is

g(u , η) = exp(c(u , u)).


Four realizations of (stationary) LGCP point processes

with exponentialcorrelation function (δ = 1).

The mean m of theGaussian process is suchthat ρ = exp(m + σ2/2).

σ = 2.5, α = 0.01, ρ = 100

σ = 2.5, α = 0.005, ρ = 100

σ = 2.5, α = 0.01, ρ = 200

σ = 2.5, α = 0.005, ρ = 200


Cox point processes (5) : parametric estimation method

For most of the models, the likelihood is not available butmoments are accessible.

Then the idea is then to estimate θ using a minimum contrastapproach : i.e. define θ as the minimizer of∫ r2

r1

∣∣∣∣K (r )q −Kθ(r )q∣∣∣∣2 dr or

∫ r2

r1

∣∣∣g(r )q − gθ(r )q∣∣∣2 dr

where

K (r ) and g(r ) are the nonparametric estimates of K (r ) andg(r ).where [r1, r2] is a set of r fixed values.q is a power parameter (adviced in the literature to be setto q = 1/4 or 1/2).


Gibbs point process (1)

We focus on the case S bounded.

Definition

A finite point process X on a bounded domain S is said to be a Gibbspoint process if it admits a density f w.r.t. a Poisson point processwith unit rate, i.e. for any F ⊆ Nf

P (X ∈ F ) =∑n≥0

e−|S |

n!

∫S

. . .

∫S

1(x1, . . . , xn ∈ F )f (x1, . . . , xn )dx1 . . . dxn

where the term n = 0 is read as exp(−|S |)1(∅ ∈ F )f (∅).

Gpp can be viewed as a perturbation of a point process.

f is easily interpretable ' weight w.r.t. a Poisson process.

f specified up to an unknown constant f = c−1h with

c =∑n≥0

exp(−|S |)n!

∫S

. . .

∫S

h(x1, . . . , xn )dx1 . . . dxn = E[h(Y )]


Gibbs point process (2) : the most well-known classDefinition

An istotropic and homogeneous parwise interaction point process hasa density of the form (for any x ∈ Nf )

f (x ) ∝ βn(x )∏u ,v ⊆x

φ2(‖v − u‖)

where φ2 : R+∗ → R

+ is called the interaction function.

The main example is the Strauss point process defined by

f (x ) ∝ βn(x )γsR(x ) where sR(x ) =∑u ,v ∈x

1(‖v − u‖ ≤ R)

where β > 0,R < ∞, γ is called the interaction parameter :

γ = 1 : homogeneous Poisson point process with intensity β.

0 < γ < 1 : repulsive point process.

γ = 0 : hard-core process with hard-core R.

γ > 1 : the model is not well-defined.


Realizations of a Strauss point process

(simulation of spatial Gibbspoint processes can be doneusing spatial birth-and-deathprocess or using MCMC withreversible jumps, see Møllerand Waagepetersen fordetails)

β = 100, γ = 0, R = 0.075

β = 100, γ = 0.3, R = 0.075

β = 100, γ = 0.6, R = 0.075

β = 100, γ = 1, R = 0.075


Gibbs point processes (3) : inference

Likelihood unavailable : normalizing constant unknown, momentsnot expressible (e.g. in the stationary case ρ = Eλ(0,X )).

Models (even when S = Rd) can be defined through thePapangelou conditional intensity

λ(u , x ) =f (x ∪ u)f (x )

, x ∈ Nlf , u ∈ S .

Key-concept since several alternatives methods exist based onλ (and not on f ) including the pseudo-likelihood

LPLW (x ; θ) =∑u∈xW

λ(x , x \ u; θ) −∫W

λ(u , x ; θ)du .

Approaches and diagnostic tools use Georgii-Nguyen-Zessinformula

E∑u∈X

h(u ,X \ u) =

∫E(h(u ,X )λ(u ,X ))du .


Conclusion

The anaysis of spatial point pattern

very large domain of research including probability,mathematical statistics, applied statistics

own specific models, methodologies and software(s) to deal with.

is involved in more and more applied fields : economy, biology,physics, hydrology, environmentrics,. . .

Still a lot of challenges

Modelling : the “true model”, problems of existence, phasetransition.

Many classical statistical methodologies need to be adapted (andproved) to s.p.p. : robust methods, resampling techniques,multiple hypothesis testing.

High-dimensional problems : S = Rd with d large, selection ofvariables, regularization methods,. . .

Space-time point processes.


References

A. Baddeley and R. Turner.Spatstat : an R package for analyzing spatial point patterns.Journal of Statistical Software, 12 :1–42, 2005.

N. Cressie.Statistics for spatial data.John Wiley and Sons, Inc, 1993.

P. J. Diggle.Statistical Analysis of Spatial Point Patterns.Arnold, London, second edition, 2003.

X. Guyon.Random Fields on a Network.Springer-Verlag, New York, 1991.

J. Illian, A. Penttinen, H. Stoyan, and D. Stoyan.Statistical Analysis and Modelling of Spatial Point Patterns.Statistics in Practice. Wiley, Chichester, 2008.

J. Møller and R. P. Waagepetersen.Statistical Inference and Simulation for Spatial Point Processes.Chapman and Hall/CRC, Boca Raton, 2004.

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