Determinantal point process models and statistical inference Fr ed - - PowerPoint PPT Presentation

Introduction Definition Stationary models Approximation Inference Conclusion

Determinantal point process models and statistical inference

Fr´ ed´ eric Lavancier, Laboratoire de Math´ ematiques Jean Leray, Nantes (France) Joint work with Jesper Møller (Aalborg University, Danemark) and Ege Rubak (Aalborg University, Danemark). Some recent results with Christophe Biscio (Nantes).

Introduction Definition Stationary models Approximation Inference Conclusion

1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

Introduction Definition Stationary models Approximation Inference Conclusion

Examples of point pattern datasets

• (a) Spanish towns
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• (b) Kidney cells of

two types (hamster)

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• (c) Japanese pines

Introduction Definition Stationary models Approximation Inference Conclusion

First look at DPPs

Determinantal point processes (DPPs): a class of repulsive (or regular, or inhibitive) point processes. Some examples :

• Poisson

DPP DPP with stronger repulsion

Introduction Definition Stationary models Approximation Inference Conclusion

First look at DPPs

Determinantal point processes (DPPs): a class of repulsive (or regular, or inhibitive) point processes. Some examples :

• Poisson

DPP DPP with stronger repulsion Statistical motivation : Do DPPs constitute a tractable and flexible class of models for repulsive point processes?

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

Gibbs point processes: The usual class when modelling repulsiveness (e.g. Strauss model, Area interaction model).

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

Gibbs point processes: The usual class when modelling repulsiveness (e.g. Strauss model, Area interaction model). In general:

moments are not expressible in closed form; likelihoods involve intractable normalizing constants; elaborate McMC methods are needed for simulations and

approximate likelihood inference;

for infinite Gibbs point processes defined on Rd, ‘things’

become rather complicated (existence and uniqueness)

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

DPPs possess a number of appealing properties:

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

DPPs possess a number of appealing properties: (a) simple conditions for existence;

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known;

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known; (c) the density of the DPP restricted to any compact set S ⊂ Rd is expressible on closed form;

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known; (c) the density of the DPP restricted to any compact set S ⊂ Rd is expressible on closed form; (d) the DPP on any compact set can easily be simulated;

Introduction Definition Stationary models Approximation Inference Conclusion

Gibbs point processes vs DPPs

DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known; (c) the density of the DPP restricted to any compact set S ⊂ Rd is expressible on closed form; (d) the DPP on any compact set can easily be simulated; (e) parametric models are available, and inference can be done by MLEs or using the moments.

Introduction Definition Stationary models Approximation Inference Conclusion

Background

Introduction Definition Stationary models Approximation Inference Conclusion

Background

DPPs were introduced in their general form by O. Macchi

in 1975 to model fermions in quantum mechanics.

Particular cases include the law of the eigenvalues of

certain random matrices (Gaussian Unitary Ensemble, Ginibre Ensemble,...)

Most theoretical studies have been published in the 2000’s. Recent growing interest for statistical applications (in

machine learning, telecommunications, forestry, biology,...)

Introduction Definition Stationary models Approximation Inference Conclusion

1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

Introduction Definition Stationary models Approximation Inference Conclusion

Notation

X : spatial point process on Rd

Introduction Definition Stationary models Approximation Inference Conclusion

Notation

X : spatial point process on Rd For any borel set B ⊆ Rd, XB = X ∩ B.

Introduction Definition Stationary models Approximation Inference Conclusion

Notation

X : spatial point process on Rd For any borel set B ⊆ Rd, XB = X ∩ B. For any integer n > 0, denote ρ(n) the n’th order product density

function of X. Intuitively, ρ(n)(x1, . . . , xn) dx1 · · · dxn is the probability that for each i = 1, . . . , n, X has a point in a region around xi of volume dxi.

Introduction Definition Stationary models Approximation Inference Conclusion

Notation

X : spatial point process on Rd For any borel set B ⊆ Rd, XB = X ∩ B. For any integer n > 0, denote ρ(n) the n’th order product density

function of X. Intuitively, ρ(n)(x1, . . . , xn) dx1 · · · dxn is the probability that for each i = 1, . . . , n, X has a point in a region around xi of volume dxi. In particular ρ = ρ(1) is the intensity function.

Introduction Definition Stationary models Approximation Inference Conclusion

Definition of a determinantal point process

Let C be a function from Rd × Rd → C. Denote [C](x1, . . . , xn) the n × n matrix with entries C(xi, xj). Ex : [C](x1) = C(x1, x1) [C](x1, x2) = C(x1, x1) C(x1, x2) C(x2, x1) C(x2, x2)

• .

Introduction Definition Stationary models Approximation Inference Conclusion

Definition of a determinantal point process

Let C be a function from Rd × Rd → C. Denote [C](x1, . . . , xn) the n × n matrix with entries C(xi, xj). Ex : [C](x1) = C(x1, x1) [C](x1, x2) = C(x1, x1) C(x1, x2) C(x2, x1) C(x2, x2)

• .

Definition X is a determinantal point process with kernel C, denoted X ∼ DPP(C), if its product density functions satisfy ρ(n)(x1, . . . , xn) = det[C](x1, . . . , xn), n = 1, 2, . . .

Introduction Definition Stationary models Approximation Inference Conclusion

Definition of a determinantal point process

Let C be a function from Rd × Rd → C. Denote [C](x1, . . . , xn) the n × n matrix with entries C(xi, xj). Ex : [C](x1) = C(x1, x1) [C](x1, x2) = C(x1, x1) C(x1, x2) C(x2, x1) C(x2, x2)

• .

Definition X is a determinantal point process with kernel C, denoted X ∼ DPP(C), if its product density functions satisfy ρ(n)(x1, . . . , xn) = det[C](x1, . . . , xn), n = 1, 2, . . . The Poisson process with intensity ρ(x) is the special case where C(x, x) = ρ(x) and C(x, y) = 0 if x = y.

Introduction Definition Stationary models Approximation Inference Conclusion

Definition of a determinantal point process

Let C be a function from Rd × Rd → C. Denote [C](x1, . . . , xn) the n × n matrix with entries C(xi, xj). Ex : [C](x1) = C(x1, x1) [C](x1, x2) = C(x1, x1) C(x1, x2) C(x2, x1) C(x2, x2)

• .

Definition X is a determinantal point process with kernel C, denoted X ∼ DPP(C), if its product density functions satisfy ρ(n)(x1, . . . , xn) = det[C](x1, . . . , xn), n = 1, 2, . . . The Poisson process with intensity ρ(x) is the special case where C(x, x) = ρ(x) and C(x, y) = 0 if x = y. Some conditions on C are necessary for existence (see later), at least C must satisfy : for all x1, . . . , xn, det[C](x1, . . . , xn) ≥ 0.

Introduction Definition Stationary models Approximation Inference Conclusion

First properties (if X ∼ DPP(C) exists)

Introduction Definition Stationary models Approximation Inference Conclusion

First properties (if X ∼ DPP(C) exists)

The intensity of X is ρ(x) = C(x, x).

Introduction Definition Stationary models Approximation Inference Conclusion

First properties (if X ∼ DPP(C) exists)

The intensity of X is ρ(x) = C(x, x). The pair correlation function is

g(x, y) := ρ(2)(x, y) ρ(x)ρ(y) = det[C](x, y) C(x, x)C(y, y) = 1 − C(x, y)C(y, x) C(x, x)C(y, y)

Thus g ≤ 1 (i.e. repulsiveness) if C is Hermitian.

Introduction Definition Stationary models Approximation Inference Conclusion

First properties (if X ∼ DPP(C) exists)

The intensity of X is ρ(x) = C(x, x). The pair correlation function is

g(x, y) := ρ(2)(x, y) ρ(x)ρ(y) = det[C](x, y) C(x, x)C(y, y) = 1 − C(x, y)C(y, x) C(x, x)C(y, y)

Thus g ≤ 1 (i.e. repulsiveness) if C is Hermitian. If X ∼ DPP(C), then XB ∼ DPP(CB)

Introduction Definition Stationary models Approximation Inference Conclusion

First properties (if X ∼ DPP(C) exists)

The intensity of X is ρ(x) = C(x, x). The pair correlation function is

g(x, y) := ρ(2)(x, y) ρ(x)ρ(y) = det[C](x, y) C(x, x)C(y, y) = 1 − C(x, y)C(y, x) C(x, x)C(y, y)

Thus g ≤ 1 (i.e. repulsiveness) if C is Hermitian. If X ∼ DPP(C), then XB ∼ DPP(CB) Any smooth transformation or independent thinning of a DPP is

still a DPP with explicit given kernel.

Introduction Definition Stationary models Approximation Inference Conclusion

First properties (if X ∼ DPP(C) exists)

The intensity of X is ρ(x) = C(x, x). The pair correlation function is

g(x, y) := ρ(2)(x, y) ρ(x)ρ(y) = det[C](x, y) C(x, x)C(y, y) = 1 − C(x, y)C(y, x) C(x, x)C(y, y)

Thus g ≤ 1 (i.e. repulsiveness) if C is Hermitian. If X ∼ DPP(C), then XB ∼ DPP(CB) Any smooth transformation or independent thinning of a DPP is

still a DPP with explicit given kernel.

Given a kernel C, there exists at most one DPP(C).

Introduction Definition Stationary models Approximation Inference Conclusion

Existence

C must be non-negative definite. Henceforth assume (C1) C is a continuous complex covariance function.

Introduction Definition Stationary models Approximation Inference Conclusion

Existence

C must be non-negative definite. Henceforth assume (C1) C is a continuous complex covariance function. By Mercer’s theorem, for any compact set S ⊂ Rd, C restricted to S × S, denoted CS, has a spectral representation, CS(x, y) =

∞

• k=1

λS

k φS k (x)φS k (y),

(x, y) ∈ S × S, where λS

k ≥ 0 and

• S φS

k (x)φS l (x) dx = 1{k=l}.

Introduction Definition Stationary models Approximation Inference Conclusion

Existence

C must be non-negative definite. Henceforth assume (C1) C is a continuous complex covariance function. By Mercer’s theorem, for any compact set S ⊂ Rd, C restricted to S × S, denoted CS, has a spectral representation, CS(x, y) =

∞

• k=1

λS

k φS k (x)φS k (y),

(x, y) ∈ S × S, where λS

k ≥ 0 and

• S φS

k (x)φS l (x) dx = 1{k=l}.

Theorem (Macchi, 1975) Under (C1), existence of DPP(C) is equivalent to : λS

k ≤ 1 for all compact S ⊂ Rd and all k.

Introduction Definition Stationary models Approximation Inference Conclusion

Density on a compact set S

Let X ∼ DPP(C) and S ⊂ Rd be any compact set. Recall that CS(x, y) = ∞

k=1 λS k φS k (x)φS k (y).

Theorem (Macchi (1975)) If λS

k < 1 ∀k, then XS ≪ Poisson(S, 1), with density

f({x1, . . . , xn}) = exp(|S| − D) det[ ˜ C](x1, . . . , xn), where D = − ∞

k=1 log(1 − λS k ) and ˜

C : S × S → C is given by ˜ C(x, y) =

∞

• k=1

λS

k

1 − λS

k

φS

k (x)φS k (y)

Introduction Definition Stationary models Approximation Inference Conclusion

Simulation

Let X ∼ DPP(C). We want to simulate XS for S ⊂ Rd compact. Recall that XS ∼ DPP(CS) with CS(x, y) = ∞

k=1 λS k φS k (x)φS k (y).

Introduction Definition Stationary models Approximation Inference Conclusion

Simulation

Let X ∼ DPP(C). We want to simulate XS for S ⊂ Rd compact. Recall that XS ∼ DPP(CS) with CS(x, y) = ∞

k=1 λS k φS k (x)φS k (y).

Theorem (Hough et al. (2006)) For k ∈ N, let Bk be independent Bernoulli r.v. with mean λS

k .

Define K(x, y) =

∞

• k=1

BkφS

k (x)φS k (y),

(x, y) ∈ S × S. Then DPP(CS) d = DPP(K).

Introduction Definition Stationary models Approximation Inference Conclusion

So simulating XS is equivalent to simulate DPP(K) with K(x, y) =

∞

• k=1

BkφS

k (x)φS k (y),

(x, y) ∈ S × S.

Introduction Definition Stationary models Approximation Inference Conclusion

So simulating XS is equivalent to simulate DPP(K) with K(x, y) =

∞

• k=1

BkφS

k (x)φS k (y),

(x, y) ∈ S × S.

1 Simulate M := sup{k ≥ 0 : Bk = 0} (by the inversion

method).

Introduction Definition Stationary models Approximation Inference Conclusion

So simulating XS is equivalent to simulate DPP(K) with K(x, y) =

∞

• k=1

BkφS

k (x)φS k (y),

(x, y) ∈ S × S.

1 Simulate M := sup{k ≥ 0 : Bk = 0} (by the inversion

method).

2 Given M = m, generate B1, . . . , Bm−1

Introduction Definition Stationary models Approximation Inference Conclusion

So simulating XS is equivalent to simulate DPP(K) with K(x, y) =

∞

• k=1

BkφS

k (x)φS k (y),

(x, y) ∈ S × S.

1 Simulate M := sup{k ≥ 0 : Bk = 0} (by the inversion

method).

2 Given M = m, generate B1, . . . , Bm−1 3 simulate DPP(K) given B1, . . . , BM and M = m.

The kernel K becomes w.l.o.g. K(x, y) =

n

• k=1

φS

k (x)φS k (y)

and DPP(K) is a determinantal projection process.

Introduction Definition Stationary models Approximation Inference Conclusion

Simulation of a determinantal projection process: K(x, y) =

n

• k=1

φS

k (x)φS k (y) = v(y)∗v(x),

v(x) = (φS

1 (x), . . . , φS n(x))T .

DPP(K) has a.s. n points (X1, . . . , Xn) that can be generated by the following Gram-Schmidt procedure sample Xn from the density pn(x) = v(x)2/n; set e1 = v(Xn)/v(Xn); for i = (n − 1) to 1 do sample Xi from the density (given Xi+1, . . . , Xn) : pi(x) = 1 i  v(x)2 −

n−i

• j=1

|e∗

jv(x)|2

  , x ∈ S set wi = v(Xi) − n−i

j=1

• e∗

jv(Xi)

• ej, en−i+1 = wi/wi

Introduction Definition Stationary models Approximation Inference Conclusion

Summary

Therefore, given a kernel C :

condition for existence of DPP(C) are known* all moments of DPP(C) are explicitly known the density of DPP(C) on any compact set is known* DPP(C) can be easily and quickly simulated on any

compact set* * if the spectral representation of CS is known on any S (see later for an approximation).

Introduction Definition Stationary models Approximation Inference Conclusion

1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

Introduction Definition Stationary models Approximation Inference Conclusion

Stationary models

Consider a stationary kernel : C(x, y) = C0(x − y), x, y ∈ Rd.

Introduction Definition Stationary models Approximation Inference Conclusion

Stationary models

Consider a stationary kernel : C(x, y) = C0(x − y), x, y ∈ Rd. Recall (C1): C0 is a continuous covariance function. Moreover, if C0 ∈ L2(Rd) we can define its Fourier transform ϕ(x) =

• C0(t)e−2πix·t dt,

x ∈ Rd.

Introduction Definition Stationary models Approximation Inference Conclusion

Stationary models

Consider a stationary kernel : C(x, y) = C0(x − y), x, y ∈ Rd. Recall (C1): C0 is a continuous covariance function. Moreover, if C0 ∈ L2(Rd) we can define its Fourier transform ϕ(x) =

• C0(t)e−2πix·t dt,

x ∈ Rd. Theorem Under (C1), if C0 ∈ L2(Rd), then existence of DPP(C0) is equivalent to ϕ ≤ 1.

Introduction Definition Stationary models Approximation Inference Conclusion

Stationary models

Consider a stationary kernel : C(x, y) = C0(x − y), x, y ∈ Rd. Recall (C1): C0 is a continuous covariance function. Moreover, if C0 ∈ L2(Rd) we can define its Fourier transform ϕ(x) =

• C0(t)e−2πix·t dt,

x ∈ Rd. Theorem Under (C1), if C0 ∈ L2(Rd), then existence of DPP(C0) is equivalent to ϕ ≤ 1. To construct parametric families of DPP : Consider parametric families of C0 and rescale so that ϕ ≤ 1. → This will induce restriction on the parameter space.

Introduction Definition Stationary models Approximation Inference Conclusion

Several parametric families of covariance functions are available, with closed form expressions for their Fourier transform.

Introduction Definition Stationary models Approximation Inference Conclusion

Several parametric families of covariance functions are available, with closed form expressions for their Fourier transform.

For d = 2, the circular covariance function with range α is given by

C0(x) = ρ 2 π

• arccos(x/α) − x/α
• 1 − (x/α)2
• 1x<α.

DPP(C0) exists iff ϕ ≤ 1 ⇔ ρα2 ≤ 4/π. ⇒ Tradeoff between the intensity ρ and the range of repulsion α.

Introduction Definition Stationary models Approximation Inference Conclusion

Several parametric families of covariance functions are available, with closed form expressions for their Fourier transform.

For d = 2, the circular covariance function with range α is given by

C0(x) = ρ 2 π

• arccos(x/α) − x/α
• 1 − (x/α)2
• 1x<α.

DPP(C0) exists iff ϕ ≤ 1 ⇔ ρα2 ≤ 4/π. ⇒ Tradeoff between the intensity ρ and the range of repulsion α.

Whittle-Mat´

ern (includes Exponential and Gaussian) : C0(x) = ρ 21−ν Γ(ν) x/ανKν(x/α), x ∈ Rd, DPP(C0) exists iff ρ ≤

Γ(ν) Γ(ν+d/2)(2√πα)d

Introduction Definition Stationary models Approximation Inference Conclusion

Several parametric families of covariance functions are available, with closed form expressions for their Fourier transform.

For d = 2, the circular covariance function with range α is given by

C0(x) = ρ 2 π

• arccos(x/α) − x/α
• 1 − (x/α)2
• 1x<α.

DPP(C0) exists iff ϕ ≤ 1 ⇔ ρα2 ≤ 4/π. ⇒ Tradeoff between the intensity ρ and the range of repulsion α.

Whittle-Mat´

ern (includes Exponential and Gaussian) : C0(x) = ρ 21−ν Γ(ν) x/ανKν(x/α), x ∈ Rd, DPP(C0) exists iff ρ ≤

Γ(ν) Γ(ν+d/2)(2√πα)d Generalized Cauchy

C0(x) = ρ (1 + x/α2)ν+d/2 , x ∈ Rd. DPP(C0) exists iff ρ ≤

Γ(ν+d/2) Γ(ν)(√πα)d

Introduction Definition Stationary models Approximation Inference Conclusion

Pair correlation functions of DPP(C0) for previous models when ρ = 1 and α = αmax(ν): In blue : C0 is the circular covariance function. In red : C0 is Whittle-Mat´ ern, for different values of ν In green : C0 is generalized Cauchy, for different values of ν

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 r = |x − y| g(r)

Introduction Definition Stationary models Approximation Inference Conclusion

Are the previous parametric families ”rich enough”? How repulsive a stationary DPP can be?

• Less repulsive DPP = Poisson Point Process.
• What is the most repulsive DPP?

We introduce criteria of repulsiveness based on the pair

correlation function g.

Introduction Definition Stationary models Approximation Inference Conclusion

How repulsive a stationary DPP can be?

Introduction Definition Stationary models Approximation Inference Conclusion

How repulsive a stationary DPP can be?

Criteria of repulsiveness based on the pair correlation function.

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 Attractive zone Repulsive zone

Introduction Definition Stationary models Approximation Inference Conclusion

How repulsive a stationary DPP can be?

Criteria of repulsiveness based on the pair correlation function.

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 Attractive zone Repulsive zone

Let X and Y be two DPPs with the same intensity ρ and pcf gX, gY . Definition

• X is globally more repulsive than Y if ”gX has a larger blue zone”,

i.e.

• (1 − gX) ≥
• (1 − gY ).

Introduction Definition Stationary models Approximation Inference Conclusion

How repulsive a stationary DPP can be?

Criteria of repulsiveness based on the pair correlation function.

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 Attractive zone Repulsive zone

Let X and Y be two DPPs with the same intensity ρ and pcf gX, gY . Definition

• X is globally more repulsive than Y if ”gX has a larger blue zone”,

i.e.

• (1 − gX) ≥
• (1 − gY ).
• X is locally more repulsive than Y if ”gX is more flat near 0”,

i.e. gX(0) = gY (0) = 0, ∇gX(0) = ∇gY (0) = 0 and ∆gX(0) ≤ ∆gY (0). Note that for a hardcore point process: g(0) = ∇g(0) = ∆g(0) = 0.

Introduction Definition Stationary models Approximation Inference Conclusion

Let Jν be the Bessel function of the first kind with order ν. Proposition There exists a unique DPP that is both the most globally and the most locally repulsive DPP with intensity ρ. Its kernel in dimension d = 2 is given by: C0(x) = √ρ J1(2√πρ||x||)

√π||x||

and ϕ(x) = 1||x||2≤ρπ−1

Introduction Definition Stationary models Approximation Inference Conclusion

Let Jν be the Bessel function of the first kind with order ν. Proposition There exists a unique DPP that is both the most globally and the most locally repulsive DPP with intensity ρ. Its kernel in dimension d = 2 is given by: C0(x) = √ρ J1(2√πρ||x||)

√π||x||

and ϕ(x) = 1||x||2≤ρπ−1

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 r = |x − y| g(r)

Introduction Definition Stationary models Approximation Inference Conclusion

New parametric families of kernels

Introduction Definition Stationary models Approximation Inference Conclusion

New parametric families of kernels

• Laguerre-Gauss family, for n ∈ N and 0 < α ≤ αmax(d, ρ, n)

C0(x) ∝ L

d 2

n−1

1 n

• x

α

• 2

e− 1

n x α 2

ϕ(x) ∝ e−n(παx)2 n−1

• k=0

(π√nαx)2k k! − → covers all possible degrees of repulsiveness.

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 r = |x − y| g(r)

Introduction Definition Stationary models Approximation Inference Conclusion

• Generalized-Sinc family, for σ ≥ 0, 0 < α ≤ αmax(d, ρ, σ)

C0(x) ∝ J σ+d

2

• 2 x

α

• σ+d

2

• 2 x

α

• σ+d

2

σ+d

2

ϕ(x) ∝

• 1 − 2(παx)2

σ + d σ

2

+

− → covers all possible degrees of repulsiveness. − → ϕ is compactly supported.

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 r = |x − y| g(r)

Introduction Definition Stationary models Approximation Inference Conclusion

A mixing property for stationary DPPs

A point process is said to be Brillinger mixing if for any k ≥ 2, its k-th reduced factorial cumulant moment measure has a finite total variation. Theorem Let C0 ∈ L2(Rd) be a continuous covariance function such that ϕ ≤ 1, then DPP(C0) is Brillinger mixing. This mixing property is a first step towards asymptotic statistics for DPPs.

Introduction Definition Stationary models Approximation Inference Conclusion

1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

Introduction Definition Stationary models Approximation Inference Conclusion

Approximation of stationary DPP’s models

Consider a stationary kernel C0 and X ∼ DPP(C0).

• The simulation and the density of XS requires the expansion

CS(x, y) = C0(y − x) =

∞

• k=1

λS

k φS k (x)φS k (y),

(x, y) ∈ S × S, but in general λS

k and φS k are not expressible on closed form.

Introduction Definition Stationary models Approximation Inference Conclusion

Approximation of stationary DPP’s models

Consider a stationary kernel C0 and X ∼ DPP(C0).

• The simulation and the density of XS requires the expansion

CS(x, y) = C0(y − x) =

∞

• k=1

λS

k φS k (x)φS k (y),

(x, y) ∈ S × S, but in general λS

k and φS k are not expressible on closed form.

• Consider w.l.g. the unit box S = [− 1

2, 1 2]d and the Fourier expansion

C0(y − x) =

• k∈Zd

cke2πik·(y−x), y − x ∈ S.

Introduction Definition Stationary models Approximation Inference Conclusion

Approximation of stationary DPP’s models

Consider a stationary kernel C0 and X ∼ DPP(C0).

• The simulation and the density of XS requires the expansion

CS(x, y) = C0(y − x) =

∞

• k=1

λS

k φS k (x)φS k (y),

(x, y) ∈ S × S, but in general λS

k and φS k are not expressible on closed form.

• Consider w.l.g. the unit box S = [− 1

2, 1 2]d and the Fourier expansion

C0(y − x) =

• k∈Zd

cke2πik·(y−x), y − x ∈ S. The Fourier coefficients are ck =

• S

C0(u)e−2πik·u du

Introduction Definition Stationary models Approximation Inference Conclusion

Approximation of stationary DPP’s models

Consider a stationary kernel C0 and X ∼ DPP(C0).

• The simulation and the density of XS requires the expansion

CS(x, y) = C0(y − x) =

∞

• k=1

λS

k φS k (x)φS k (y),

(x, y) ∈ S × S, but in general λS

k and φS k are not expressible on closed form.

• Consider w.l.g. the unit box S = [− 1

2, 1 2]d and the Fourier expansion

C0(y − x) =

• k∈Zd

cke2πik·(y−x), y − x ∈ S. The Fourier coefficients are ck =

• S

C0(u)e−2πik·u du ≈

• Rd C0(u)e−2πik·u du = ϕ(k)

which is a good approximation if C0(u) ≈ 0 for |u| > 1

2.

Introduction Definition Stationary models Approximation Inference Conclusion

Approximation of stationary DPP’s models

Consider a stationary kernel C0 and X ∼ DPP(C0).

• The simulation and the density of XS requires the expansion

CS(x, y) = C0(y − x) =

∞

• k=1

λS

k φS k (x)φS k (y),

(x, y) ∈ S × S, but in general λS

k and φS k are not expressible on closed form.

• Consider w.l.g. the unit box S = [− 1

2, 1 2]d and the Fourier expansion

C0(y − x) =

• k∈Zd

cke2πik·(y−x), y − x ∈ S. The Fourier coefficients are ck =

• S

C0(u)e−2πik·u du ≈

• Rd C0(u)e−2πik·u du = ϕ(k)

which is a good approximation if C0(u) ≈ 0 for |u| > 1

2.

• Example: For the circular covariance, this is true whenever ρ|S| > 5.

Introduction Definition Stationary models Approximation Inference Conclusion

Approximation of stationary models

So, DPP(C0) on S can be approximated by DPP(Capp,0) with Capp,0(y − x) =

• k∈Zd

ϕ(k)e2πi(y−x)·k, x, y ∈ S, where ϕ is the Fourier transform of C0. This kernel approximation allows us

to simulate DPP(C0) on S, by simulating DPP(Capp,0) to compute the (approximated) density of DPP(C0) on S.

Turns out to be a very good approximation in most cases.

Introduction Definition Stationary models Approximation Inference Conclusion

Exemples of approximations

− Solid lines : theoretical pair correlation function

• Circles : pair correlation from the approximated kernel

0.00 0.01 0.02 0.03 0.04 0.05 0.0 0.2 0.4 0.6 0.8 1.0

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ν = 0.25, ρmax = 324 ν = 0.50, ρmax = 337 ν = 1.00, ρmax = 329 ν = 2.00, ρmax = 315 ν = ∞, ρmax = 293

0.00 0.01 0.02 0.03 0.04 0.05 0.0 0.2 0.4 0.6 0.8 1.0

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ν = 0.50, ρmax = 232 ν = 1.00, ρmax = 275 ν = 2.00, ρmax = 294 ν = 4.00, ρmax = 298 ν = ∞, ρmax = 293

Whittle-Mat´ ern Generalized Cauchy

Introduction Definition Stationary models Approximation Inference Conclusion

1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

Introduction Definition Stationary models Approximation Inference Conclusion

Consider a stationary and isotropic parametric DPP(C0), with C0(x − y) = ρRψ(x − y), where Rψ(0) = 1 and ψ is some parameter

Introduction Definition Stationary models Approximation Inference Conclusion

Consider a stationary and isotropic parametric DPP(C0), with C0(x − y) = ρRψ(x − y), where Rψ(0) = 1 and ψ is some parameter First and second moments are easily deduced:

ρ = intensity. Pair correlation function:

g(x, y) = 1 − Rψ(x − y)2.

Introduction Definition Stationary models Approximation Inference Conclusion

Consider a stationary and isotropic parametric DPP(C0), with C0(x − y) = ρRψ(x − y), where Rψ(0) = 1 and ψ is some parameter First and second moments are easily deduced:

ρ = intensity. Pair correlation function:

g(x, y) = 1 − Rψ(x − y)2.

Ripley’s K-function is given in terms of Rψ:

If e.g. d = 2, Kψ(r) :=

• x≤r
• 1 − Rψ(x)2

dx = πr2−2π r tRψ(t)2 dt.

Introduction Definition Stationary models Approximation Inference Conclusion

Parameter estimation

When C0(x − y) = ρRψ(x − y) estimate

1 using the kernel approximation for the likelihood

(ρ, ψ) by MLE

• r ρ by #{obs. points}/[area of obs. window]

and ψ by MLE;

2 or using the moments

ρ by #{obs. points}/[area of obs. window] and ψ by minimum contrast estimation e.g. ˆ ψ = argminψ rmax

• K(r) −
• Kψ(r)
• 2

dr → Consistent and asymptotically normal

Introduction Definition Stationary models Approximation Inference Conclusion

Non homogeneous case

A point process is second-order intensity-reweighted stationary if

its intensity ρ(x) is not constant its pcf is invariant by translation : g(x, y) = g0(x − y)

For a DPP, this means that the kernel can be written: C(x, y) =

• ρ(x)Rθ(x − y)
• ρ(y)

where ρ(x) = C(x, x) and Rθ(0) = 1.

Introduction Definition Stationary models Approximation Inference Conclusion

Non homogeneous case

A point process is second-order intensity-reweighted stationary if

its intensity ρ(x) is not constant its pcf is invariant by translation : g(x, y) = g0(x − y)

For a DPP, this means that the kernel can be written: C(x, y) =

• ρ(x)Rθ(x − y)
• ρ(y)

where ρ(x) = C(x, x) and Rθ(0) = 1. Existence : if ρ(x) ≤ ρmax and if DPP(ρmaxRθ(x − y)) is well-defined, then DPP(C) is an independent thinning with retention probability ρ(x)/ρmax, so it is well defined.

Introduction Definition Stationary models Approximation Inference Conclusion

Non homogeneous case

A point process is second-order intensity-reweighted stationary if

its intensity ρ(x) is not constant its pcf is invariant by translation : g(x, y) = g0(x − y)

For a DPP, this means that the kernel can be written: C(x, y) =

• ρ(x)Rθ(x − y)
• ρ(y)

where ρ(x) = C(x, x) and Rθ(0) = 1. Existence : if ρ(x) ≤ ρmax and if DPP(ρmaxRθ(x − y)) is well-defined, then DPP(C) is an independent thinning with retention probability ρ(x)/ρmax, so it is well defined. Inference : 1- Fit a parametric model to ρ(x) (e.g. depending on covariables), by Poisson likelihood. 2- Estimate θ by minimum contrast method based e.g. on gθ(x, y) = 1 − Rθ(x − y)2 or on Kθ(r).

Introduction Definition Stationary models Approximation Inference Conclusion

1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

Introduction Definition Stationary models Approximation Inference Conclusion

Spanish Towns dataset

Ripley (1988) fitted a 4 parameters Strauss hardcore model. We fit a DPP(ρRα,ν) model with 3 parameters:

• ρ : intensity
• Rα,ν : Whittle-Mat´

ern correlation function (2 parameters)

• 2

4 6 8 10 −1.5 −1.0 −0.5 0.0

Strauss Matern Data

1 2 3 0.0 0.2 0.4 0.6 0.8

Strauss Matern Data

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8

Strauss Matern Data

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 2 3 4 5 6

Strauss Matern Data

Introduction Definition Stationary models Approximation Inference Conclusion

Hamster cells dataset

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• X1 (o) : 266 dividing (living) cells

X2 (+) : 77 pyknotic (dying) cells. Are pyknotic cells a random labelling? If so and if X1 ∪ X2 is a DPP, then X1 and X2 are DPPs with the same kernel, up to the intensities. We fit DPP(ρRα), where Rα : Gaussian kernel, to

• X1 ∪ X2 : ˆ

α = 0.0181

• X1 : ˆ

α1 = 0.0188

• X2 : ˆ

α2 = 0.0082 We test H0 : α = α1 = α2 by simulations based on the statistic T = |ˆ α − ˆ α1||ˆ α − ˆ α2|. The p-value is 0.55 and the assumption of random labelling is not rejected.

Introduction Definition Stationary models Approximation Inference Conclusion

Japanese pines dataset

A non homogeneous DPP(

• ρ(x)R(x − y)
• ρ(y)) is fitted.
• log ρ(x) is a cubic polynomial in x, estimated by Poisson likelihood.
• R is a Gaussian kernel, estimated by contrast method based on g.

1 2 3 4

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• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 r ginhom(r) Empirical Gaussian

0.0 0.5 1.0 1.5 2.0 2.5 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

r Linhom(r) − r Pointwise Rank Data

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0

r Ginhom(r) Pointwise Rank Data

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0

r Finhom(r) Pointwise Rank Data

Introduction Definition Stationary models Approximation Inference Conclusion

1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

Introduction Definition Stationary models Approximation Inference Conclusion

Conclusion

• DPPs provide flexible parametric models of repulsive point

processes.

Introduction Definition Stationary models Approximation Inference Conclusion

Conclusion

• DPPs provide flexible parametric models of repulsive point

processes.

• DPPs possess appealing properties:

Easily and quickly simulated Closed form expressions for all moments. Closed form expression for the density on any bounded set. Parametric models are available Inference is feasible, including likelihood inference.

• Parametric models, inference methods and so on are available

in R (developed by E. Rubak) in the package spatstat.

Introduction Definition Stationary models Approximation Inference Conclusion

References

Macchi, O. (1975). The coincidence approach to stochastic point processes. Advances in Applied Probability 7, 83–122. Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55, 923–975. Shirai, T. and Y. Takahashi (2003). Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. Journal of Functional Analysis 2, 414–463. Hough, J. B., M. Krishnapur, Y. Peres, and B. Vir` ag (2006). Determinantal processes and independence. Probability Surveys 3, 206–229.

• F. Lavancier, J. Møller, and E. Rubak (2015).

Determinantal point process models and statistical inference. JRSS B 77, 853–877.

• C. Biscio and F. Lavancier.

Quantifying repulsiveness of determinantal point processes. to appear in Bernoulli.