Spatial point processes: introduction

Aila Sarkka

Mathematical sciencesChalmers University of Technology and University of Gothenburg

Gothenburg, Sweden

Aila Sarkka Spatial point processes: introduction

What is spatial statistics?

Locations of observations (in ≥ 2 dimensions) play an importantrole. Observations are given in the form

[x ,m(x)],

where

x is a location

m(x) is an observation at x , value of some variable

Aila Sarkka Spatial point processes: introduction

Some examples

I x is the location of a cell and m(x) the size or state of health

I x is the location of a picture element and m(x) is the colour

I x is the location of a tree in a forest and m(x) is thehight/diameter/species of the tree

I x is the location of a weather station and m(x) is the meandaily temperature

Aila Sarkka Spatial point processes: introduction

Typical questions

1) Where do we observe something? → x interesting

2) What do we observe somewhere? → m(x) interesting

3) Where and what do we observe? → both x and m(x)interesting

Aila Sarkka Spatial point processes: introduction

Some remarks

I There is no natural ordering in higher (> 1) dimensions.I Spatial dependencies between the observations

I make statistical analysis difficultI are the basis for spatial statistics

Aila Sarkka Spatial point processes: introduction

Some examples of data structures

I tessellation

I fibre data

I lattice data

I point pattern

Aila Sarkka Spatial point processes: introduction

Tessellation: materials science

Aila Sarkka Spatial point processes: introduction

Fibre pattern: epidermal nerve fibers

3151 (normal), skeletonized image

Aila Sarkka Spatial point processes: introduction

Lattice data: poverty level in the different counties inIllinois, Indiana, Michigan, Ohio, and Wisconsin

Aila Sarkka Spatial point processes: introduction

Point pattern: crime incidents in London

Aila Sarkka Spatial point processes: introduction

Spatial point processes

A point process N is a stochastic mechanism or rule to producepoint patterns or realisations according to the distribution of theprocess.

A marked point process is a point process where each point xi ofthe process is assigned a quantity m(xi ), called a mark. Often,marks are integers or real numbers but more general marks canalso be considered.

Diggle (2013)Illian et al. (2008)

R library spatstat (Baddeley and Turner, 2005)

Aila Sarkka Spatial point processes: introduction

Remarks

Remark 1: We assume that all point processes are simple, i.e. thatthere are no multiple points (xi 6= xj if i 6= j).

Remark 2: There is a large literature on point processes in time.There is an overlap of methods for point processes in space and intime but the temporal case is not only a special case of the spatialprocess with d = 1. Time is 1-directional.

Remark 3: To avoid confusion between points of the process andpoints of Rd , the points of the process or point pattern(realization) are called events.

Aila Sarkka Spatial point processes: introduction

Spatial point patterns

clustered completely random regular

Aila Sarkka Spatial point processes: introduction

Examples

I Locations of betacells within a rectangular region in a cat’seye (regular)

I Locations of Finnish pine saplings (clustered)

I Locations of Spanish towns (regular)

I Locations of galaxes (clustered)

Remark: Very different scales, from microscopic to cosmic

Aila Sarkka Spatial point processes: introduction

Marked point patterns

−6 −2 0 2 4 6

−8

−4

02

Finish pines

0 200 400 600 800

02

00

60

01

00

0

0 200 400 600 800

02

00

60

01

00

0

Betacells

Finnish pine saplings: locations and diameters

Beta-type retina cells in the retina of a cat: locations and type(red triangles ”on”, blue circles ”off”)

Aila Sarkka Spatial point processes: introduction

First-order properties (without marks)

(Compare to the mean of a real-valued random varaible.)

The mean number of points of N in B is E(N(B)) (depends on theset B). We use the notation

Λ(B) = E(N(B))

and call Λ the intensity measure.

Under some continuity conditions, a density function λ, called theintensity function, exists, and

Λ(B) =

∫Bλ(x) dx .

Aila Sarkka Spatial point processes: introduction

Some properties of point processes: stationarity andisotropy

I A point process N is stationary (translation invariant) if Nand the translated point process Nx have the samedistribution for all translations x , i.e.

N = {x1, x2, ...} and Nx = {x1 + x , x2 + x , ...}

have the same distribution for all x ∈ Rd .

I A point process is isotropic (rotation invariant) if itscharacteristics are invariant under rotations, i.e.

N = {x1, x2, ...} and rNx = {rx1, rx2, ...}

have the same distribution for any rotation r around theorigin.

I If a point process is both stationary and isotropic, it is calledmotion-invariant.

Aila Sarkka Spatial point processes: introduction

First-order properties

If N is stationary, then

Λ(B) = λ|B|,

where 0 < λ <∞ is called the intensity of N and |B| is thearea/volume of B.

λ is the mean number of points of N per unit area/volume, i.e.

λ =Λ(B)

|B|=

E(N(B))

|B|.

Aila Sarkka Spatial point processes: introduction

Summary functions: nearest neighbour distance function

I Let D1 be the distance from an arbitrary event to the nearestother event

I The nearest neighbour distance function is defined as

G (r) = P(D1 ≤ r)

I If the pattern is completely spatially random (CSR),

G (r) = 1− exp(−λπr2)

I For regular patterns G (r) tends to lie below and for clusteredpatterns above the CSR curve.

Aila Sarkka Spatial point processes: introduction

Summary functions: empty space statistics

I Let D2 denote the distance from an arbitrary point to thenearest event

I The empty space function can be defined as

F (r) = P(D2 ≤ r)

I If the pattern is completely spatially random,

F (r) = 1− exp(−λπr2)

I For regular patterns F (r) tends to lie above and for clusteredpatterns below the CSR curve.

Aila Sarkka Spatial point processes: introduction

Summary: combination of G and F

I Using G and F we can define the so-called J function as

J(r) =1− G (r)

1− F (r)

(whenever 1− F (r) > 0)

I If the pattern is completely spatially random,

J(r) ≡ 1

I For regular patterns J(r) > 1 and for clustered patternsJ(r) < 1.

Aila Sarkka Spatial point processes: introduction

Ripley’s K function

I The 2nd order properties (compare to variance) of astationary and isotropic point process can be characterized byRipley’s K function (Ripley, 1977)

λK (r) = E[# further events within distance r of an arbitrary event].

I If the pattern is completely spatially random,

K (r) = πr2.

I For regular patterns K (r) < πr2 and for clustered patternsK (r) > πr2.

Aila Sarkka Spatial point processes: introduction

Ripley’s K function (continues)

Often, a variance stabilizing and centered version of the K function(Besag, 1977) is used, namely (in 2D)

L(r)− r =√K (r)/π − r ,

which equals 0 under CSR. Values less than zero indicate regularityand values larger than zero clustering.

Aila Sarkka Spatial point processes: introduction

Note on estimation of G , F , J and K

I Typically, a point pattern is observed in a (bounded)observation window and points outside the window are notobserved.

I Estimators (except for J(r)) need to be edge-corrected

I Edge correction methods include plus sampling, minussampling, Ripley’s isotropic correction and translation(stationary) correction

Aila Sarkka Spatial point processes: introduction

Poisson process

A point process is a homogeneous Poisson process (CSR) if

(P1) for some λ > 0 and any finite region B, N(B) has a Poissondistribution with mean λ|B|

(P2) given N(B) = n, the events in B form an independent randomsample from the uniform distribution on B

Inhomogeneous Poisson process: intensity λ (in homogeneousPoisson process) replaced by an intensity function λ(x)

Aila Sarkka Spatial point processes: introduction

Realization of a homogeneous Poisson process

simPoisson

Poisson process with intensity 100

Aila Sarkka Spatial point processes: introduction

Summary statistics

allstats(simPoisson)

0.00 0.02 0.04 0.06 0.08

0.0

0.2

0.4

0.6

0.8

F function

F^

km(r)

F^

bord(r)

F^

cs(r)

Fpois(r)

0.00 0.02 0.04 0.06 0.08

0.0

0.2

0.4

0.6

0.8

G function

G^

km(r)

G^

bord(r)

G^

han(r)

Gpois(r)

0.00 0.02 0.04 0.06 0.08

0.7

0.8

0.9

1.0

1.1

1.2

J function

J^

km(r)

J^

han(r)

J^

rs(r)

Jpois(r)

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00.0

50.1

00.1

50.2

0

K function

K^

iso(r)

K^

trans(r)

K^

bord(r)

Kpois(r)

Aila Sarkka Spatial point processes: introduction

Neyman-Scott cluster process (Poisson cluster process)

Cluster processes are models for aggregated spatial point patterns

For Neyman-Scott cluster process

(NS1) parent events form a Poisson process with intensity λ

(NS2) each parent produces a random number S of daughters(offsprings), realized independently and identically for eachparent according to some probability distribution

(NS3) the locations of the daughters relative to their parent areindependently and identically distributed according to somebivariate distribution

The cluster process consists only of the daughter points.

Aila Sarkka Spatial point processes: introduction

Examples of Neyman-Scott processes

Matern cluster process: The points in a cluster are independentlyand uniformly scattered in a disc (ball) of radius R centered at theparent point.

Thomas process: The distribution of the daughter points aroundthe parent point is a radially symmetric normal distribution withvariance σ2.

Aila Sarkka Spatial point processes: introduction

Realization of a Matern cluster process

simClust simPoisson

Left: Parent point intensity 20, cluster radius 0.05, averagenumber of daughter points per cluster 5Right: Poisson process with intensity 100

Aila Sarkka Spatial point processes: introduction

Summary statistics

allstats(simClust)

0.00 0.05 0.10 0.15

0.0

0.2

0.4

0.6

0.8

1.0

F function

F^

km(r)

F^

bord(r)

F^

cs(r)

Fpois(r)

0.00 0.01 0.02 0.03 0.04

0.0

0.2

0.4

0.6

0.8

G function

G^

km(r)

G^

bord(r)

G^

han(r)

Gpois(r)

0.00 0.05 0.10 0.15

0.0

0.2

0.4

0.6

0.8

1.0

J function

J^

km(r)

J^

han(r)

J^

rs(r)

Jpois(r)

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00.0

50.1

00.1

50.2

0

K function

K^

iso(r)

K^

trans(r)

K^

bord(r)

Kpois(r)

Aila Sarkka Spatial point processes: introduction

Matern hard-core processes

I Hard-core processes are models for regular spatial pointpatterns

I There is a minimum allowed distance, called hard-coredistance δ, between any two points

I Matern I hard-core process: A Poisson process with intensityλ is thinned by delating all pairs of points that are at distanceless than the hard-core radius apart.

I Matern II hard-core process: Events of a Poisson process aregiven ”marks” from (e.g.) the uniform U(0, 1) distribution.An event x with mark m is retained if b(x , δ) contains nopoints of the process that has a mark smaller than m.

Aila Sarkka Spatial point processes: introduction

Realization of a Matern I hard-core process

simHC simPoisson

Left: Hard-core process with the initial Poisson intensity 300,hard-core radius 0.04Right: Poisson process with intensity 100

Aila Sarkka Spatial point processes: introduction

Summary statistics

allstats(simHC)

0.00 0.02 0.04 0.06 0.08

0.0

0.2

0.4

0.6

0.8

F function

F^

km(r)

F^

bord(r)

F^

cs(r)

Fpois(r)

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

G function

G^

km(r)

G^

bord(r)

G^

han(r)

Gpois(r)

0.00 0.02 0.04 0.06 0.08

1.0

1.2

1.4

1.6

1.8

J function

J^

km(r)

J^

han(r)

J^

rs(r)

Jpois(r)

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00.0

50.1

00.1

50.2

0

K function

K^

iso(r)

K^

trans(r)

K^

bord(r)

Kpois(r)

Aila Sarkka Spatial point processes: introduction

Summary

I You have gotten a short introduction to the analysis ofhomogeneous (stationary) and isotropic spatial point patterns

I Hard to check stationarity/isotropy

I If stationarity/isotropy cannot be assumed, the summarystatistics and models have to be modified to the newconditions

Aila Sarkka Spatial point processes: introduction

References

I Baddeley, A.,Turner, R. Spatstat: an R package for analyzingspatial point patterns. J. Stat. Softw. 12 (2005) 1-42.

I Besag, J.E. Comment on “Modelling spatial patterns” by B.D. Ripley. Journal of the Royal Statistical Society B(Methodological) 39 (1977) 193-195.

I Diggle, P. J. Statistical analysis of spatial and spatio-temporalpoint patterns. Chapman and Hall/CRC Press, Boca Raton(2013).

I Illian, J., Penttinen, A., Stoyan, H., Stoyan, D. StatisticalAnalysis and Modelling of Spatial Point Patterns. Chichester:Wiley (2008).

I Ripley, B.D. Modelling spatial patterns. Journal of the RoyalStatistical Society B 39 (1977) 172-212.

Aila Sarkka Spatial point processes: introduction