J-functions for inhomogeneous point processes in space and...

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J-functions for inhomogeneous point

processes in space and time

Marie-Colette van Lieshout

CWI Amsterdam

The Netherlands

J-functions for inhomogeneous point processes in space and time – p. 1/20

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Exploratory analysis for spatial point processes

cells

0.00 0.05 0.10 0.15

0.0

0.2

0.4

0.6

0.8

1.0

envelope(cells, Gest, simulate = expression(runifpoint(42)))

r

G(r)

obsmmeanhilo

plot(envelope(cells, Gest, simulate=expression(runifpoint(42))))

Generating 99 simulations by evaluating expression ...

lty col key label meaning

obs 1 1 obs obs(r) observed value of G(r) for data pattern

mmean 2 2 mmean mean(r) sample mean of G(r) from simulations

hi 1 8 hi hi(r) upper pointwise envelope of G(r) from simulations

lo 1 8 lo lo(r) lower pointwise envelope of G(r) from simulations


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Summary statistics for stationary point processes

Let X be a stationary point process on Rd with intensity ρ > 0.

Popular statistics include

F (t) = P(X ∩B(0, t) 6= ∅)

G(t) = P(X \ {0} ∩B(0, t) 6= ∅ | 0 ∈ X)

K(t) = 1ρE

[

∑

x∈X\{0} 1{x ∈ B(0, t)} | 0 ∈ X]

J(t) = (1−G(t)) / (1− F (t))

where B(0, t) is the closed ball of radius t ≥ 0 centred at the origin.

Write P!0(X ∈ A) = P(X \ {0} ∈ A | 0 ∈ X) for the reduced Palm

distribution of X.


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Summary statistics for inhomogeneous patterns

Define

Kinhom(t) =1

|B|E

[

∑ 6=

x,y∈X

1{x ∈ B} 1 {y ∈ B(x, t)}

ρ(x) ρ(y)

]

regardless of the choice of bounded Borel set B ⊂ Rd with strictly

positive volume |B|, and using the convention a/0 = 0 for a ≥ 0.

Baddeley, Møller and Waagepetersen, SN 2000.

Remark: the definition can be extended to space-time point processes.

Gabriel and Diggle, SN 2009.

Goal: define analogues of F , G, J for inhomogeneous point processes.


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Product densities and correlation functions

The product densities ρ(n) of a simple point process X satisfy

E

[

∑ 6=

x1,...,xn∈Xf(x1, . . . , xn)

]

=

∫

· · ·

∫

f(x1, . . . , xn) ρ(n)(x1, . . . , xn) dx1 · · · dxn

for all measurable f ≥ 0 (6= indicates a sum over n-tuples of distinct

points).

In words: ρ(n)(x1, . . . , xn) dx1 · · · dxn is the infinitesimal probability of

finding points of X at each of dx1, . . ., dxn.

N-point correlation functions are defined recursively by ξ1 ≡ 1 and

ρ(n)(x1, . . . , xn)

ρ(x1) · · · ρ(xn)=

n∑

k=1

∑

D1,...,Dk

ξn(D1)(xD1) · · · ξn(Dk)(xDk )

where {D1, . . . , Dk} is a partition of {1, . . . , n}, xDj = {xi : i ∈ Dj}.


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Summary statistics and product densities

Let X be a stationary point process with intensity ρ > 0. Then

K(t) =∫

B(0,t)

ρ(2)(0,x)

ρ2dx

F (t) = −∑∞

n=1(−1)n

n!

∫

B(0,t)· · ·

∫

B(0,t)ρ(n)(x1, . . . , xn) dx1 · · · dxn

G(t) = −∑∞

n=1(−1)n

n!

∫

B(0,t)· · ·

∫

B(0,t)

ρ(n+1)(0,x1,...,xn)ρ

dx1 · · · dxn

J(t) = 1 +∑∞

n=1(−ρ)n

n!Jn(t)

for

Jn(t) =

∫

B(0,t)

· · ·

∫

B(0,t)

ξn+1(0, x1, . . . , xn) dx1 · · · dxn.

Hence

J(t)− 1 ≈ −ρ (K(t)− |B(0, t)|) .


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Intensity-reweighted moment stationarity

Let X be a simple point process on Rd for which product densities of all

orders exist and infx ρ(x) = ρ̄ > 0. If the ξn are translation invariant, that

is,

ξn(x1 + a, . . . , xn + a) = ξn(x1, . . . , xn)

for almost all x1, . . . , xn ∈ Rd and all a ∈ Rd, X is intensity-reweighted

moment stationary.

Examples

• Poisson point processes;

• location dependent thinning of a stationary point process;

• log Gaussian Cox processes driven by a Gaussian random field with

translation invariant covariance function.


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Inhomogeneous J-function

Let X be an intensity-reweighted moment stationary point process. Set

Jn(t) =

∫

B(0,t)

· · ·

∫

B(0,t)

ξn+1(0, x1, . . . , xn) dx1 · · · dxn

and define

Jinhom(t) = 1 +

∞∑

n=1

(−ρ̄)n

n!Jn(t).

Remarks

• Jinhom(t) > 1 indicates inhibition at range t;

• Jinhom(t) < 1 suggests clustering;

• Jinhom(t)− 1 ≈ −ρ̄ (Kinhom(t)− |B(0, t)|);

• the definition does not depend on the choice of origin.


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The generating functional

For measurable v : Rd → [0, 1] that are identically 1 except on some

bounded subset of Rd,

G(v) = E

[

∏

x∈X

v(x)

]

where by convention an empty product is taken to be 1.

Properties

• the distribution of X is determined uniquely by G;

• suppose product densities of all orders exist and let u : Rd → [0, 1] be

measurable with bounded support. Then

G(1−u) = 1+∞∑

n=1

(−1)n

n!

∫

· · ·

∫

u(x1) · · ·u(xn) ρ(n)(x1, . . . , xn) dx1 · · · dxn

(provided the series converges).


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Jinhom in terms of generating functionals

Write, for t ≥ 0 and a ∈ Rd,

uat (x) =ρ̄ 1{x ∈ B(a, t)}

ρ(x), x ∈ Rd.

Then

Jinhom(t) =G!a (1− uat )

G (1− uat )

where G!a is the generating functional of the reduced Palm distribution

P!a at a, G that of P itself.

Note: G!a (1− uat ) and G (1− uat ) do not depend on the choice of a and

lend themselves to estimation.


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Remarks

For stationary X, uat (x) = 1{x ∈ B(a, t)}, hence

G (1− uat ) = P(X ∩B(a, t) = ∅) = 1− F (t)

G!a (1− uat ) = P!a(X ∩B(a, t) = ∅) = 1−G(t)

Consequently, one retrieves the classic definition of the J-function.

For intensity-reweighted moment stationary X, we get counterparts of

the F - and G-functions:

Finhom(t) = 1−G (1− uat ) ;

Ginhom(t) = 1−G!a (1− uat )

which do not depend on the choice of origin a and are easy to use in

practice.


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Minus sampling estimators

In practice, X is observed in some bounded window W .

Let L ⊆ W be a finite point grid. Set

̂1− Finhom(t) =

∑

lk∈L∩W⊖t

∏

x∈X∩B(lk,t)

[

1− ρ̄ρ(x)

]

#L ∩W⊖t,

where W⊖t is the eroded set. Similarly, set

̂1−Ginhom(t) =

∑

xk∈X∩W⊖t

∏

x∈X\{xk}∩B(xk,t)

[

1− ρ̄ρ(x)

]

#X ∩W⊖t.

Proposition: ̂Finhom(t) is unbiased, ̂Ginhom(t) is ratio-unbiased.


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Pakistani earthquakes

Pakistan is regularly affected by earthquakes due to the subduction of

the Indo-Australian continental plate under the Eurasian plate.

There are two convergence zones: One crosses the country from its

South-West border with the Arabic Sea to Kashmir in the North-East, the

other crosses the northern part of the country in the East-West direction.

Two major earthquakes were recorded during 1973–2008

• 1997: magnitude 7.3 along the SW to NE zone; about seventy

casualties;

• 2005: magnitude 7.6 in Kashmir; devastating with at least 86, 000

casualties.

Restrict to shallow quakes (depth less than 70 km) of magnitude at least

4.5 as deeper or weaker ones may not be felt.


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Pakistani earthquakes: intensity function

• To avoid edge effects, include earthquakes within a distance of

about one degree from the Pakistan border;

• aggregate into a single pattern, but exclude the major earthquake

years;

• calculate the kernel estimator of intensity (isotropic Gaussian kernel

with standard deviation 0.5, approximately 50 km).

Pooled earthquake intensity

010

2030

4050

60


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Earthquake data 2005–2007

Note: a KPSS test indicates the intensity pattern persists over the years.

2005 2006 2007

Question: can the data be explained by a series of inhomogeneous

Poisson processes?


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Inhomogeneous J-functions 2005–2007

0.0 0.5 1.0 1.5

0.85

0.90

0.95

1.00

2005

0.0 0.5 1.0 1.50.

980.

991.

001.

011.

02

2006

0.0 0.5 1.0 1.5

0.98

0.99

1.00

1.01

1.02

2007

Ĵinhom for the locations of shallow earthquakes of magnitude at least 4.5

in 2005–2007 with upper and lower envelopes based on 19 independent

realisations of an inhomogeneous Poisson process.

Conclusion: evidence of clustering over and beyond that explained by

the spatial variation.


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Space-time point processes

Let Y be a simple point process on Rd × R equipped with the supremum

distance

d((x, s), (y, r)) = max{||x− y||, |s− r|}.

One may define Jn and JSTinhom as before. However, it is more natural to

scale space and time differently. Then

Jn(t1, t2) =

∫

B(0,t1)

∫ t2

−t2

. . .

∫

B(0,t1)

∫ t2

−t2

ξn+1(0, y1, . . . , yn)n∏

i=1

dyi

and

JSTinhom(t1, t2)− 1 ≈ −ρ̄

∫

B(0,t1)

∫ t2

−t2

ξ2((0, 0), (x, s)) dx ds,

which corresponds to the K∗ST -approach of Gabriel and Diggle (2009).


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Marked point processes

Let X be a point process with continuous marks in a bounded set

M ⊂ R. Assume that the ξn are translation invariant in the spatial

component and set, for Borel sets A,B ⊂ M ,

JABn (t) =1

|A|

∫

A

∫

(B(0,t)×B)⊗nξn+1((0, b), y1, . . . , yn) db

n∏

i=1

dxi dmi

where yi = (xi,mi) are marked points.

Write XA for the (unmarked) point process of locations in X which are

marked by a point in A. Then

• XA and XB independent ⇒ JABinhom ≡ 1;

• XA and XAc independent ⇒ JAMinhom(t) = J

XAinhom(t);

• If the points in X are randomly labelled with probability density ν(m),

m ∈ M , then JABinhom(t) = Jthinhom(t), the inhomogeneous J-function of

the thinning of XM with retention probability p =∫

Bν(m) dm.


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References

1. M.N.M. van Lieshout. A J-function for inhomogeneous point

processes. Statistica Neerlandica, 65:183–201, 2011.

2. M.N.M. van Lieshout and A. Stein. Earthquake modelling at the

country level using aggregated spatio-temporal point processes.

Mathematical Geosciences, 44:309–326, 2012.

3. K. Türkyilmaz, M.N.M. van Lieshout and A. Stein. Comparing the

Hawkes and trigger process models for aftershock sequences

following the 2005 Kashmir earthquake. Mathematical Geosciences,

45:149–164, 2013.

4. O. Cronie and M.N.M. van Lieshout. A J-function for inhomogeneous

spatio-temporal point processes. Submitted.

5. O. Cronie and M.N.M. van Lieshout. Summary statistics for

inhomogeneous marked point processes. In preparation.


Exploratory analysis for spatial point processesSummary statistics for stationary point processesSummary statistics for inhomogeneous patternsProduct densities and correlation functionsSummary statistics and product densitiesIntensity-reweighted moment stationarityInhomogeneous $J$-functionThe generating functional$J_{m {inhom}}$ in terms of generating functionalsRemarksMinus sampling estimatorsPakistani earthquakesPakistani earthquakes: intensity functionEarthquake data 2005--2007Inhomogeneous $J$-functions 2005--2007Space-time point processesMarked point processesReferences

J-functions for inhomogeneous point processes in space and...

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