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

Introduction Definition Parametric 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).

Introduction Definition Parametric 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 Conclusion and references

Introduction Definition Parametric models Approximation Inference Conclusion

Introduction

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

• Poisson

DPP DPP with stronger repulsion

Introduction Definition Parametric models Approximation Inference Conclusion

Introduction

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 Parametric 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. Statistical models and inference have so far been largely

unexplored.

Introduction Definition Parametric 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 Conclusion and references

Introduction Definition Parametric models Approximation Inference Conclusion

Notation

X : spatial point process on Rd

Introduction Definition Parametric models Approximation Inference Conclusion

Notation

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

Introduction Definition Parametric 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 Parametric 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 Parametric 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 Parametric 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, . . . Some conditions on C are necessary for existence (see later), e.g. C must satisfy : for all x1, . . . , xn, det[C](x1, . . . , xn) ≥ 0.

Introduction Definition Parametric models Approximation Inference Conclusion

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

Introduction Definition Parametric models Approximation Inference Conclusion

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

From the definition, if C is continuous,

ρ(n)(x1, . . . , xn) ≈ 0 whenever xi ≈ xj for some i = j, = ⇒ the points of X repel each other.

Introduction Definition Parametric models Approximation Inference Conclusion

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

From the definition, if C is continuous,

ρ(n)(x1, . . . , xn) ≈ 0 whenever xi ≈ xj for some i = j, = ⇒ the points of X repel each other.

The intensity of X is ρ(x) = C(x, x).

Introduction Definition Parametric models Approximation Inference Conclusion

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

From the definition, if C is continuous,

ρ(n)(x1, . . . , xn) ≈ 0 whenever xi ≈ xj for some i = j, = ⇒ the points of X repel each other.

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 Parametric models Approximation Inference Conclusion

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

From the definition, if C is continuous,

ρ(n)(x1, . . . , xn) ≈ 0 whenever xi ≈ xj for some i = j, = ⇒ the points of X repel each other.

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 Parametric models Approximation Inference Conclusion

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

From the definition, if C is continuous,

ρ(n)(x1, . . . , xn) ≈ 0 whenever xi ≈ xj for some i = j, = ⇒ the points of X repel each other.

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 Parametric models Approximation Inference Conclusion

Existence

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

Introduction Definition Parametric models Approximation Inference Conclusion

Existence

As noted before, 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 Parametric models Approximation Inference Conclusion

Existence

As noted before, 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 Parametric 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 Parametric 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 Parametric 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 Parametric 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 Parametric 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 Parametric 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 Parametric 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, which

becomes a determinantal projection process. This is done by a sequential algorithm.

Introduction Definition Parametric models Approximation Inference Conclusion

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 Parametric 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 Conclusion and references

Introduction Definition Parametric models Approximation Inference Conclusion

Stationary models

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

Introduction Definition Parametric 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 Parametric 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 Parametric 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 Parametric models Approximation Inference Conclusion

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

Introduction Definition Parametric models Approximation Inference Conclusion

Several parametric families of covariance function 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 Parametric models Approximation Inference Conclusion

Several parametric families of covariance function 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 Parametric models Approximation Inference Conclusion

Several parametric families of covariance function 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 Parametric 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 Parametric 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 Conclusion and references

Introduction Definition Parametric 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 Parametric 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 Parametric 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 Parametric 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 Parametric 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 Parametric 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 Parametric 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

• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ● ●
• ● ●
• ● ● ● ● ● ● ● ●

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

• ●
• ● ● ● ● ● ● ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ● ●
• ●
• ● ● ● ● ● ● ● ● ● ●
• ● ●
• ● ● ● ● ● ● ● ●

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 Parametric 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 Conclusion and references

Introduction Definition Parametric 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 Parametric 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 Parametric 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 Parametric models Approximation Inference Conclusion

Parameter estimation

When C(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 e.g. minimum contrast estimation ˆ ψ = argminψ rmax

• K(r) −
• Kψ(r)
• 2

dr

Introduction Definition Parametric 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 Conclusion and references

Introduction Definition Parametric models Approximation Inference Conclusion

Conclusion

• DPP’s provide flexible parametric models of repulsive point

processes.

Introduction Definition Parametric models Approximation Inference Conclusion

Conclusion

• DPP’s provide flexible parametric models of repulsive point

processes.

• DPP’s possess appealing properties:

Easily and quickly simulated Closed form expressions for all orders of moments. Closed form expression for the density of a DPP on any

bounded set.

Parametric models are available Inference is feasible, including likelihood inference.

Introduction Definition Parametric models Approximation Inference Conclusion

Conclusion

• DPP’s provide flexible parametric models of repulsive point

processes.

• DPP’s possess appealing properties:

Easily and quickly simulated Closed form expressions for all orders of moments. Closed form expression for the density of a DPP on any

bounded set.

Parametric models are available Inference is feasible, including likelihood inference.

• More (in our paper) : extension to non-stationary DPPs ;

case-studies with real dataset where DPPs prove to be useful models.

Introduction Definition Parametric models Approximation Inference Conclusion

References

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.

Determinantal point process models and statistical inference. submitted (arxiv:1205.4818). Macchi, O. (1975). The coincidence approach to stochastic point processes. Advances in Applied Probability 7, 83–122. 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. Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55, 923–975.