residuals and goodness-of-fit tests for marked gibbs point...

Residuals and Goodness-of-fit tests for marked Gibbspoint processes

Frederic Lavancier(Laboratoire de Mathematiques Jean Leray, Nantes, France)

Joint work with J.-F. Coeurjolly (Grenoble, France)

27/06/2012

F. Lavancier () Residuals and GoF tests for Gibbs pp 27/06/2012 1 / 19

1 Gibbs point processes

2 Validations through residualsResiduals for spatial point processesAsymptoticsMeasures of departures to the true model

Type of data

Question :−→ Independence or interaction between locations and/or between marks ?

Type of data

Question :−→ Independence or interaction between locations and/or between marks ?

Marked point processes

State space : S := Rd ×M associated to the measure µ = λ⊗ λm.

I λ denotes the Lebesgue measure on Rd .I M is a measurable space, endowed with the probability measure λm.

Rd is the space of points and M the space of marks.

Let denote xm = (x ,m) an element of Rd ×M, i.e. a marked point.

ϕ denotes a locally finite point configuration in S.

ϕΛ is the restriction of ϕ on Λ : ϕΛ = ϕ ∩ Λ.

Ω is the space of locally finite point configurations ϕ.

Definition (marked point process)

A marked point process is a random variable on Ω.

Example :

π : the standard (non-marked) Poisson point process.

Example :

Gibbs measures

Let (VΛ)Λ∈B(Rd ) be a family of energies (or Hamiltonian)

VΛ : Ω −→ R ∪ +∞ϕ 7−→ VΛ(ϕΛ|ϕΛc )

VΛ(ϕ) is the energy of ϕΛ inside Λ given the outside configuration ϕΛc

Definition

A probability measure P on Ω is a Gibbs measure associated to the family(VΛ)Λ if for every bounded set Λ and P-almost every ϕΛc

P(dϕΛ|ϕΛc ) ∝ e−VΛ(ϕΛ|ϕΛc ) πΛ(dϕΛ).

Example : Strauss marked point process, M = r , b

VΛ(ϕ; θ) = θ11|ϕrΛ|+ θ12|ϕb

+ θ21

∑(x r ,y r )

1||y r−x r ||<10 + θ22

∑(xb,yb)

1||yb−xb||<10 + θ23

∑(xb,y r )

1||y r−xb||<10

Gibbs Voronoi tessellation

VΛ(ϕ) =∑

C∈ Vor(ϕ)C∩Λ 6=∅

V1(C ) +∑

C ,C ′∈ Vor(ϕ)C and C ′are neighbors

(C∪C ′)∩Λ 6=∅

V2(C ,C ′)

V1(C ) : deals with the shape of the cell and V2(C ,C ′) = θ |vol(C )− vol(C ′)|.

θ > 0 θ < 0

1 Gibbs point processes

2 Validations through residualsResiduals for spatial point processesAsymptoticsMeasures of departures to the true model

Motivation

In pratice, for a given data set :

1 Propose one reasonable model, i.e. choose (VΛ(.; θ))Λ∈B(R[d]).

2 Estimate θ? (different methods are available)

3 Assess the adequation of the fitted model to the data ⇒ Residuals.

For Gibbs point processes :

1 How to define residuals ?

2 Theoretical properties of residuals ?

3 Construction of Goodness-of-fits tests.

Motivation

In pratice, for a given data set :

1 Propose one reasonable model, i.e. choose (VΛ(.; θ))Λ∈B(R[d]).

2 Estimate θ? (different methods are available)

3 Assess the adequation of the fitted model to the data ⇒ Residuals.

For Gibbs point processes :

1 How to define residuals ?

2 Theoretical properties of residuals ?

3 Construction of Goodness-of-fits tests.

The h-residuals Baddeley, Turner, Møller, Hazelton (05)

h−innovations (in Λ) : for any function h(·, ·; θ) : S× Ω→ R,

IΛ(ϕ; h, θ?) :=∑

xm∈ϕΛ

h(xm, ϕ \ xm; θ?)−∫

Λ×Mh(xm, ϕ; θ?)e−V (xm|ϕ;θ?)µ(dxm)

where V (xm|ϕ; θ) := VΛ(ϕ ∪ xm; θ)− VΛ(ϕ; θ) is the local energy of xm in ϕ.

h−residuals (in Λ) : RΛ(ϕ; h, θ) := IΛ(ϕ; h, θ)

Theoretical Motivation :

Theorem (Georgii-Nguyen-Zessin)

For any h(·, ·; θ) : S× Ω→ R, for any θ ∈ Θ,

(∑xm∈ϕ

h (xm, ϕ \ xm; θ)

)= Eθ?

(∫Rd×M

h (xm, ϕ; θ) e−V (xm|ϕ;θ?)µ(dxm)

Conjecture from BTMH : RΛ(ϕ; h, θ) ≈ 0 and |Λ|1/2RΛ(ϕ; h, θ) ≈ Gaussian

IΛ(ϕ; h, θ?) :=∑

xm∈ϕΛ

h(xm, ϕ \ xm; θ?)−∫

(∑xm∈ϕ

h (xm, ϕ \ xm; θ)

)= Eθ?

(∫Rd×M

IΛ(ϕ; h, θ?) :=∑

xm∈ϕΛ

h(xm, ϕ \ xm; θ?)−∫

(∑xm∈ϕ

h (xm, ϕ \ xm; θ)

)= Eθ?

(∫Rd×M

Examples of residuals

RΛ(ϕ; h, θ) :=∑

xm∈ϕΛ

h(xm, ϕ \ xm; θ)−∫

Λ×Mh(xm, ϕ; θ)e−V(xm|ϕ;bθ)µ(dxm)

1 Raw residuals : h(xm, ϕ; θ) := 1

2 Inverse residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ)

3 Pearson residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ)/2

4 Residuals based on the empty space function :

For r > 0, the choice hr (xm, ϕ, θ) := eV (xm|ϕ;θ)1||xm−ϕ||<r leads to

RΛ(ϕ; hr , θ) = FParam(r)− FNonParam(r)

where F (r) := P (ϕ ∩ B(0, r) 6= ∅) is the empty space function.

Examples of residuals

RΛ(ϕ; h, θ) :=∑

xm∈ϕΛ

h(xm, ϕ \ xm; θ)−∫

Λ×Mh(xm, ϕ; θ)e−V(xm|ϕ;bθ)µ(dxm)

1 Raw residuals : h(xm, ϕ; θ) := 1

2 Inverse residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ)

3 Pearson residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ)/2

4 Residuals based on the empty space function :

For r > 0, the choice hr (xm, ϕ, θ) := eV (xm|ϕ;θ)1||xm−ϕ||<r leads to

RΛ(ϕ; hr , θ) = FParam(r)− FNonParam(r)

where F (r) := P (ϕ ∩ B(0, r) 6= ∅) is the empty space function.

Asymptotic frameworksAim : asymptotic behavior of the residuals, i.e. as Λ→ Rd

RΛ(ϕ; h, θ) ≈ 0 and |Λ|1/2RΛ(ϕ; h, θ) ≈ Gaussian

In 2 frameworks :

1. computed for different test functions ⇒ joint behavior in law

2. computed on several disjoint sub-domains ⇒ joint behavior in law

=⇒ Construction of Goodness of Fit Tests.

Main assumptions :

Existence and stationarity of Pθ, for all θ ∈ Θ

Locality assumption : ∃D > 0, V (xm|ϕ) = V (xm|ϕB(x,D))

Some regularity assumptions on V and h

For θ an estimate of θ? computed on Λ :

as |Λ| → Rd ,

a.s.−→ θ?

|Λ|1/2(θ − θ?)d−→ T for some r.v. T

In 2 frameworks :

Main assumptions :

as |Λ| → Rd ,

a.s.−→ θ?

In 2 frameworks :

Main assumptions :

as |Λ| → Rd ,

a.s.−→ θ?

In 2 frameworks :

Main assumptions :

as |Λ| → Rd ,

a.s.−→ θ?

In 2 frameworks :

Main assumptions :

as |Λ| → Rd ,

a.s.−→ θ?

Framework 1 ((h1, . . . , hs)−residuals on Λ))8>>>>>>>>>><>>>>>>>>>>:

RΛ(ϕ, h1, bθ)

RΛ(ϕ, hj , bθ)

RΛ(ϕ, hs , bθ)

Take s different test-functions hj

Compute each hj -residuals from Λ

Consider the vector R1 :=(

RΛ(ϕ, hj , θ))

j=1,...,s.

Theorem (Coeurjolly, L.)

|Λ|−1R1a.s.→ 0 and |Λ|−1/2 R1

d→ N (0,Σ1), as Λ→ Rd ,

where Σ1 is a matrix depending on h, V and the asymptotic behavior of θ

Corollary (Empty space function type test)

Test 1 : Under some conditions ensuring Σ1 positive-definite

|Λ|−1 ‖Σ−1/21 R1‖2 d→ χ2

where Σ1 is a ”natural” estimate of Σ1

RΛ(ϕ, h1, bθ)

RΛ(ϕ, hj , bθ)

RΛ(ϕ, hs , bθ)

RΛ(ϕ, hj , θ))

j=1,...,s.

|Λ|−1R1a.s.→ 0 and |Λ|−1/2 R1

d→ N (0,Σ1), as Λ→ Rd ,

|Λ|−1 ‖Σ−1/21 R1‖2 d→ χ2

RΛ(ϕ, h1, bθ)

RΛ(ϕ, hj , bθ)

RΛ(ϕ, hs , bθ)

RΛ(ϕ, hj , θ))

j=1,...,s.

|Λ|−1R1a.s.→ 0 and |Λ|−1/2 R1

d→ N (0,Σ1), as Λ→ Rd ,

|Λ|−1 ‖Σ−1/21 R1‖2 d→ χ2

Framework 2 (h−residuals on subdomains of Λ)

Λ is splitted in a finite number of sub-domains :

Λ = ∪j∈JΛj , where |Λi | = |Λj |, ∀i , j .

Consider the vector of residuals

R2 :=(

RΛj (ϕ, h, θ))

j∈J.

I the same h in each sub-domainI θ is computed from the whole domain Λ

|Λj |−1R2a.s.→ 0 and |Λj |−1/2 R2

d→ N (0,Σ2), as Λ→ Rd ,

Σ2 = λInn I|J | + |J |−1(λRes − λInn) J (J = eeT , e = (1, . . . , 1)T ),

λInn ≥ 0 is a covariance depending on h and V ,

λRes ≥ 0 is a covariance depending on h, V and the asympt. behavior of θ.

Framework 2 (h−residuals on subdomains of Λ)

Λ is splitted in a finite number of sub-domains :

Λ = ∪j∈JΛj , where |Λi | = |Λj |, ∀i , j .

Consider the vector of residuals

R2 :=(

RΛj (ϕ, h, θ))

j∈J.

I the same h in each sub-domainI θ is computed from the whole domain Λ

|Λj |−1R2a.s.→ 0 and |Λj |−1/2 R2

d→ N (0,Σ2), as Λ→ Rd ,

Σ2 = λInn I|J | + |J |−1(λRes − λInn) J (J = eeT , e = (1, . . . , 1)T ),

λInn ≥ 0 is a covariance depending on h and V ,

λRes ≥ 0 is a covariance depending on h, V and the asympt. behavior of θ.

Quadrat-counting based tests

Corollary (Coeurjolly, L.)

Test 2 : If λInn > 0 and λRes > 0 then

|Λj |−1 ‖Σ−1/22 R2‖2 d→ χ2

with Σ−1/22 = 1√bλInn

I|J | + ( 1√bλRes

− 1√bλInn

where λInn and λRes are “natural” estimates of λInn and λRes .

Test 3 : If λInn > 0 then

|Λj |−1 λ−1Inn × ‖R2 − R2‖2 d→ χ2

|J |−1,

where λInn is a “natural” estimates of λInn.

Quadrat-counting based tests

Corollary (Coeurjolly, L.)

Test 2 : If λInn > 0 and λRes > 0 then

|Λj |−1 ‖Σ−1/22 R2‖2 d→ χ2

with Σ−1/22 = 1√bλInn

− 1√bλInn

where λInn and λRes are “natural” estimates of λInn and λRes .

Test 3 : If λInn > 0 then

|Λj |−1 λ−1Inn × ‖R2 − R2‖2 d→ χ2

|J |−1,

where λInn is a “natural” estimates of λInn.

ConclusionTest 1 : different test functions (h1, . . . , hs) in the same domain

|Λ|−1 ‖Σ−1/21 R1‖2 d→ χ2

−→ deep goodness of fit assessment through (h1, . . . , hs)

−→ Σ1 is a full matrix : not easy to estimate, Σ−1/21 could be very instable.

−→ Σ1 depends on the asymptotic behavior of θn

Test 2 : h-residuals computed on |J | sub-domains

|Λj |−1 ‖Σ−1/22 R2‖2 d→ χ2

−→ Σ−1/22 = 1√bλInn

− 1√bλInn

)J is explicit.

−→ λRes depends on the asymptotic behavior of θn.−→ condition λRes > 0 could be restrictive and not easy to check.

Test 3 : h-residuals computed on |J | sub-domains

|Λj |−1 λ−1Inn × ‖R2 − R2‖2 d→ χ2

|J |−1.

−→ λInn > 0 easy to check and not restrictive.

−→ Generalization of the Poisson dispersion test.F. Lavancier () Residuals and GoF tests for Gibbs pp 27/06/2012 16 / 19

Short simulation in the simplest case : quadrat test (Test 3) with h = 1

True model : Strauss model on Λ = [0, L]2

VΛ(ϕ; θ) = θ1|ϕΛ|+ θ2

∑(x ,y)∈ϕΛ

1||y−x ||<R

In blue : Estimated level of the test when the I-type error is 5%

→ Based on 500 realisations and |J | = 4 or |J | = 9 sub-domains

|J | = 4 |J | = 9L = 1 L = 2 L = 3 L = 1 L = 2 L = 3

Strauss model, R = .05e−θ1 = 200, e−θ2 = 1 6.3% 6% 5.5% 5.8% 5.8% 5.6%

e−θ1 = 300, e−θ2 = 0.3 4.8% 5.4% 5.0% 4.0% 4.8% 5.4%e−θ1 = 300, e−θ2 = 0.7 4.8% 5.0% 5.4% 4.4% 4.5% 5.5%

Inhomogeneous Strauss model, R = .05e−θ1 = 300β(·), e−θ2 = 0.3 12.0% 68.4% 100.0% 9.0% 68.2% 100.0%

Neymann-Scott processρ = 30, r = .6 20.2% 36.2% 38.2% 19.6% 45.0% 51.8%ρ = 30, r = .2 53.8% 56.2% 57.8% 65.4% 83.8% 86.2%

Pairwise step function, (R1,R2) = (.06, .1)e−θ1 = 500, e−θ2 = .3, e−θ3 = .7 1.6% 2.4% 3.0% 1.0% 2.8% 2.8%e−θ1 = 500, e−θ2 = .7, e−θ3 = .3 3.2% 4.2% 5.2% 2.8% 5% 4.4%

A. Baddeley, R. Turner, J. Møller and M. Hazelton, (2005) Residualanalysis for spatial point processes, J.R.S.S. B, 65, 617-666.

J.-F. Coeurjolly, F. Lavancier, (2012) Residuals and Goodness-of-fittests for stationary marked Gibbs point processes, to appear inJ.R.S.S. B (arXiv :1002.0857).

Theoretical Ingredient

Let Xn,i , n ∈ N, i ∈ Zd , be a triangular array field in a measurable space S. Forn ∈ N, let Kn ⊂ Zd and for k ∈ Kn, assume

Zn,k = fn,k (Xn,k+i , i ∈ I0) ,

where I0 = i ∈ Zd , |i | ≤ 1 and fn,k : SI0 → Rp. Let Sn =∑

k∈KnZn,k . If

(i) c3 := supn∈N supk∈KnE|Zn,k |3 <∞,

(ii) ∀n ∈ N, ∀k ∈ Kn, E(Zn,k |Xn,j , j 6= k) = 0,

(iii) |Kn| → +∞ as n→∞,

(iv) There exists a symmetric matrix Σ ≥ 0 such that

∥∥∥∥∥∥|Kn|−1∑k∈Kn

∑j∈Bk (1)∩Kn

Zn,kZn,jT − Σ

∥∥∥∥∥∥→ 0,

then |Kn|−1/2Snd−→ N (0,Σ) as n→∞.

residuals and goodness-of-fit tests for marked gibbs point...

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