Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
Determinantal point process models and
statistical inference
Frédéric Lavancier,
Laboratoire de Mathématiques Jean Leray, Nantes (France)
Joint work with Jesper Møller (Aalborg University, Danemark)
and Ege Rubak (Aalborg University, Danemark).
Introduction
Definition
Parametric models
Approximation
Inference
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
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
Introduction
Determinantal point processes (DPPs): a class of repulsive (or
regular, or inhibitive) point processes.
Some examples :
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Poisson
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DPP
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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 :
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Poisson
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DPP
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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
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
Conclusion
Introduction
Definition
Parametric models
Notation
X : spatial point process on Rd
Approximation
Inference
Conclusion
Introduction
Definition
Parametric models
Approximation
Notation
X : spatial point process on Rd
For any borel set B ⊆ Rd , XB = X ∩ B.
Inference
Conclusion
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
First properties (if X ∼ DP P (C) exists)
Inference
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
First properties (if X ∼ DP P (C) exists)
From the definition, if C is continuous,
ρ(n) (x1 , . . . , xn ) ≈ 0
whenever
xi ≈ xj
=⇒ the points of X repel each other.
for some i 6= j,
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
First properties (if X ∼ DP P (C) exists)
From the definition, if C is continuous,
ρ(n) (x1 , . . . , xn ) ≈ 0
whenever
xi ≈ xj
=⇒ the points of X repel each other.
The intensity of X is ρ(x) = C(x, x).
for some i 6= j,
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
First properties (if X ∼ DP P (C) exists)
From the definition, if C is continuous,
ρ(n) (x1 , . . . , xn ) ≈ 0
whenever
xi ≈ xj
for some i 6= 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)
det[C](x, y)
C(x, y)C(y, x)
=
=1−
ρ(x)ρ(y)
C(x, x)C(y, y)
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 ∼ DP P (C) exists)
From the definition, if C is continuous,
ρ(n) (x1 , . . . , xn ) ≈ 0
whenever
xi ≈ xj
for some i 6= 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)
det[C](x, y)
C(x, y)C(y, x)
=
=1−
ρ(x)ρ(y)
C(x, x)C(y, y)
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 ∼ DP P (C) exists)
From the definition, if C is continuous,
ρ(n) (x1 , . . . , xn ) ≈ 0
whenever
xi ≈ xj
for some i 6= 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)
det[C](x, y)
C(x, y)C(y, x)
=
=1−
ρ(x)ρ(y)
C(x, x)C(y, y)
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,
∞
X
CS (x, y) =
λSk φSk (x)φSk (y),
(x, y) ∈ S × S,
k=1
where λSk ≥ 0 and
R
S
φSk (x)φSl (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,
∞
X
CS (x, y) =
λSk φSk (x)φSk (y),
(x, y) ∈ S × S,
k=1
where λSk ≥ 0 and
R
S
φSk (x)φSl (x) dx = 1{k=l} .
Theorem (Macchi, 1975)
Under (C1), existence of DPP(C) is equivalent to :
λSk ≤ 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.
P
S S
S
Recall that CS (x, y) = ∞
k=1 λk φk (x)φk (y).
Theorem (Macchi (1975))
If λSk < 1 ∀k, then XS Poisson(S, 1), with density
f ({x1 , . . . , xn }) = exp(|S| − D) det[C̃](x1 , . . . , xn ),
P
S
where D = − ∞
k=1 log(1 − λk ) and C̃ : S × S → C is given by
C̃(x, y) =
∞
X
λSk
φSk (x)φSk (y)
S
1 − λk
k=1
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
Simulation
Let X ∼ DPP(C).
We want to simulate XS for S ⊂ Rd compact.
P
S S
S
Recall that XS ∼ DPP(CS ) with CS (x, y) = ∞
k=1 λk φk (x)φk (y).
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
Simulation
Let X ∼ DPP(C).
We want to simulate XS for S ⊂ Rd compact.
P
S S
S
Recall that XS ∼ DPP(CS ) with CS (x, y) = ∞
k=1 λk φk (x)φk (y).
Theorem (Hough et al. (2006))
For k ∈ N, let Bk be independent Bernoulli r.v. with mean λSk .
Define
K(x, y) =
∞
X
Bk φSk (x)φSk (y),
k=1
d
Then DPP(CS ) = DPP(K).
(x, y) ∈ S × S.
Introduction
Definition
Parametric models
Approximation
Inference
So simulating XS is equivalent to simulate DPP(K) with
K(x, y) =
∞
X
k=1
Bk φSk (x)φSk (y),
(x, y) ∈ S × S.
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
So simulating XS is equivalent to simulate DPP(K) with
K(x, y) =
∞
X
Bk φSk (x)φSk (y),
(x, y) ∈ S × S.
k=1
1
Simulate M := sup{k ≥ 0 : Bk 6= 0} (by the inversion
method).
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
So simulating XS is equivalent to simulate DPP(K) with
K(x, y) =
∞
X
Bk φSk (x)φSk (y),
(x, y) ∈ S × S.
k=1
1
Simulate M := sup{k ≥ 0 : Bk 6= 0} (by the inversion
method).
2
Given M = m, generate B1 , . . . , Bm−1
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
So simulating XS is equivalent to simulate DPP(K) with
K(x, y) =
∞
X
Bk φSk (x)φSk (y),
(x, y) ∈ S × S.
k=1
1
Simulate M := sup{k ≥ 0 : Bk 6= 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
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
Conclusion
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
Z
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
Z
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
Z
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
p
2
C0 (x) = ρ
arccos(kxk/α) − kxk/α 1 − (kxk/α)2 1kxk<α .
π
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
p
2
C0 (x) = ρ
arccos(kxk/α) − kxk/α 1 − (kxk/α)2 1kxk<α .
π
DPP(C0 ) exists iff ϕ ≤ 1 ⇔ ρα2 ≤ 4/π.
⇒ Tradeoff between the intensity ρ and the range of repulsion α.
Whittle-Matérn (includes Exponential and Gaussian) :
C0 (x) = ρ
21−ν
kx/αkν Kν (kx/αk),
Γ(ν)
DPP(C0 ) exists iff ρ ≤
Γ(ν)
√
Γ(ν+d/2)(2 πα)d
x ∈ Rd ,
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
p
2
C0 (x) = ρ
arccos(kxk/α) − kxk/α 1 − (kxk/α)2 1kxk<α .
π
DPP(C0 ) exists iff ϕ ≤ 1 ⇔ ρα2 ≤ 4/π.
⇒ Tradeoff between the intensity ρ and the range of repulsion α.
Whittle-Matérn (includes Exponential and Gaussian) :
C0 (x) = ρ
21−ν
kx/αkν Kν (kx/αk),
Γ(ν)
DPP(C0 ) exists iff ρ ≤
x ∈ Rd ,
Γ(ν)
√
Γ(ν+d/2)(2 πα)d
Generalized Cauchy
C0 (x) =
ρ
ν+d/2
(1 + kx/αk2 )
DPP(C0 ) exists iff ρ ≤
Γ(ν+d/2)
√
Γ(ν)( πα)d
,
x ∈ Rd .
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-Matérn, for different values of ν
0.0
0.2
0.4
g(r)
0.6
0.8
1.0
In green : C0 is generalized Cauchy, for different values of ν
0.0
0.2
0.4
0.6
r = |x − y|
0.8
1.0
Introduction
Definition
Parametric models
Approximation
Inference
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
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
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) =
∞
X
λSk φSk (x)φSk (y),
(x, y) ∈ S × S,
k=1
but in general λSk and φSk are not expressible on closed form.
Conclusion
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) =
∞
X
λSk φSk (x)φSk (y),
(x, y) ∈ S × S,
k=1
but in general λSk and φSk are not expressible on closed form.
• Consider w.l.g. the unit box S = [− 12 , 12 ]d and the Fourier expansion
C0 (y − x) =
X
k∈Zd
ck e2πik·(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) =
∞
X
λSk φSk (x)φSk (y),
(x, y) ∈ S × S,
k=1
but in general λSk and φSk are not expressible on closed form.
• Consider w.l.g. the unit box S = [− 12 , 12 ]d and the Fourier expansion
C0 (y − x) =
X
ck e2πik·(y−x) ,
k∈Zd
The Fourier coefficients are
Z
ck =
C0 (u)e−2πik·u du
S
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) =
∞
X
λSk φSk (x)φSk (y),
(x, y) ∈ S × S,
k=1
but in general λSk and φSk are not expressible on closed form.
• Consider w.l.g. the unit box S = [− 12 , 12 ]d and the Fourier expansion
C0 (y − x) =
X
ck e2πik·(y−x) ,
y − x ∈ S.
k∈Zd
The Fourier coefficients are
Z
Z
ck =
C0 (u)e−2πik·u du ≈
S
C0 (u)e−2πik·u du = ϕ(k)
Rd
which is a good approximation if C0 (u) ≈ 0 for |u| > 12 .
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) =
∞
X
λSk φSk (x)φSk (y),
(x, y) ∈ S × S,
k=1
but in general λSk and φSk are not expressible on closed form.
• Consider w.l.g. the unit box S = [− 12 , 12 ]d and the Fourier expansion
C0 (y − x) =
X
ck e2πik·(y−x) ,
y − x ∈ S.
k∈Zd
The Fourier coefficients are
Z
Z
ck =
C0 (u)e−2πik·u du ≈
S
C0 (u)e−2πik·u du = ϕ(k)
Rd
which is a good approximation if C0 (u) ≈ 0 for |u| > 12 .
• 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
X
Capp,0 (y − x) =
ϕ(k)e2πi(y−x)·k , x, y ∈ S,
k∈Zd
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
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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
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ν = 1.00, ρmax = 275
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ν = 4.00, ρmax = 298
ν = ∞, ρmax = 293
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◦ Circles : pair correlation from the approximated kernel
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Generalized Cauchy
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Introduction
Definition
Parametric models
Approximation
Inference
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
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
Consider a stationary and isotropic parametric DPP(C0 ), with
C0 (x − y) = ρRψ (kx − yk),
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 (kx − yk),
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 (kx − yk)2 .
Introduction
Definition
Parametric models
Approximation
Inference
Conclusion
Consider a stationary and isotropic parametric DPP(C0 ), with
C0 (x − y) = ρRψ (kx − yk),
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 (kx − yk)2 .
Ripley’s K-function is given in terms of Rψ :
If e.g. d = 2,
Z
Z
2
2
1 − Rψ (kxk) dx = πr −2π
Kψ (r) :=
kxk≤r
0
r
tRψ (t)2 dt.
Introduction
Definition
Parametric models
Approximation
Inference
Parameter estimation
When
C(x, y) = ρRψ (kx − yk)
estimate
1
using the kernel approximation for the likelihood
(ρ, ψ) by MLE
or 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
2
Z rmax q
q
b
ψ̂ = argminψ
K(r) − Kψ (r) dr
0
Conclusion
Introduction
Definition
Parametric models
Approximation
Inference
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
Conclusion
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
References
Hough, J. B., M. Krishnapur, Y. Peres, and B. Viràg (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.
Conclusion