Introduction ● ● ● ● ● ● ● ● ● Jesper Møller ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3 Definition, existence ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● and basic properties ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Stationary DPPs and ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● approximations ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Parametric models ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Simulation ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Stationary data ● ● ● ● ● ● ● ● ● ● ● ● example ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Non-stat. example ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● DPPs on the sphere ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (on going research ● ● ● ● ● ● ● ● ● ● ● project) DPP with Poisson DPP Concluding remarks stronger inhibition ◮ Determinantal point processes (DPP) are inhibitive/regular/repulsive point processes. ◮ Introduced by O. Macchi in 1975 to model fermions in quantum mechanics. ◮ Several theoretical studies appeared in the 2000’s. Dept. of Mathematical Sciences ◮ Statistical models and inference have so far been largely unexplored. Aalborg University Denmark 32
Gibbs point processes — the usual class of point pro- cesses used for modelling inhibition Example: Strauss hard-core process Jesper Møller 4 Definition, existence 1 c ( r , R , β, γ ) β n � γ 1 {� xi − xj �≤ R } 1 {� x i − x j � > r } , and basic properties f ( { x 1 , . . . , x n } ) = { x 1 , . . . , x n } ⊂ S , Stationary DPPs and i < j approximations Parametric models where S ⊂ R d is compact; n = 0 , 1 , . . . ; 0 ≤ r < R , β > 0, 0 ≤ γ ≤ 1 are Simulation parameters; the density is w.r.t. the unit rate Poisson process. Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Gibbs point processes — the usual class of point pro- cesses used for modelling inhibition Example: Strauss hard-core process Jesper Møller 4 Definition, existence 1 c ( r , R , β, γ ) β n � γ 1 {� xi − xj �≤ R } 1 {� x i − x j � > r } , and basic properties f ( { x 1 , . . . , x n } ) = { x 1 , . . . , x n } ⊂ S , Stationary DPPs and i < j approximations Parametric models where S ⊂ R d is compact; n = 0 , 1 , . . . ; 0 ≤ r < R , β > 0, 0 ≤ γ ≤ 1 are Simulation parameters; the density is w.r.t. the unit rate Poisson process. Stationary data example ◮ The normalizing constant c ( r , R , β, γ ) is intractable. Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Gibbs point processes — the usual class of point pro- cesses used for modelling inhibition Example: Strauss hard-core process Jesper Møller 4 Definition, existence 1 c ( r , R , β, γ ) β n � γ 1 {� xi − xj �≤ R } 1 {� x i − x j � > r } , and basic properties f ( { x 1 , . . . , x n } ) = { x 1 , . . . , x n } ⊂ S , Stationary DPPs and i < j approximations Parametric models where S ⊂ R d is compact; n = 0 , 1 , . . . ; 0 ≤ r < R , β > 0, 0 ≤ γ ≤ 1 are Simulation parameters; the density is w.r.t. the unit rate Poisson process. Stationary data example ◮ The normalizing constant c ( r , R , β, γ ) is intractable. Non-stat. example DPPs on the sphere (on going research ◮ Interpretations? We don’t know the intensity or any other moment properties. project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Gibbs point processes — the usual class of point pro- cesses used for modelling inhibition Example: Strauss hard-core process Jesper Møller 4 Definition, existence 1 c ( r , R , β, γ ) β n � γ 1 {� xi − xj �≤ R } 1 {� x i − x j � > r } , and basic properties f ( { x 1 , . . . , x n } ) = { x 1 , . . . , x n } ⊂ S , Stationary DPPs and i < j approximations Parametric models where S ⊂ R d is compact; n = 0 , 1 , . . . ; 0 ≤ r < R , β > 0, 0 ≤ γ ≤ 1 are Simulation parameters; the density is w.r.t. the unit rate Poisson process. Stationary data example ◮ The normalizing constant c ( r , R , β, γ ) is intractable. Non-stat. example DPPs on the sphere (on going research ◮ Interpretations? We don’t know the intensity or any other moment properties. project) Concluding remarks ◮ We don’t know the distribution of the number of points. Dept. of Mathematical Sciences Aalborg University Denmark 32
Gibbs point processes — the usual class of point pro- cesses used for modelling inhibition Example: Strauss hard-core process Jesper Møller 4 Definition, existence 1 c ( r , R , β, γ ) β n � γ 1 {� xi − xj �≤ R } 1 {� x i − x j � > r } , and basic properties f ( { x 1 , . . . , x n } ) = { x 1 , . . . , x n } ⊂ S , Stationary DPPs and i < j approximations Parametric models where S ⊂ R d is compact; n = 0 , 1 , . . . ; 0 ≤ r < R , β > 0, 0 ≤ γ ≤ 1 are Simulation parameters; the density is w.r.t. the unit rate Poisson process. Stationary data example ◮ The normalizing constant c ( r , R , β, γ ) is intractable. Non-stat. example DPPs on the sphere (on going research ◮ Interpretations? We don’t know the intensity or any other moment properties. project) Concluding remarks ◮ We don’t know the distribution of the number of points. ◮ (Long) MCMC runs are needed... Dept. of Mathematical Sciences Aalborg University Denmark 32
Gibbs point processes — the usual class of point pro- cesses used for modelling inhibition Example: Strauss hard-core process Jesper Møller 4 Definition, existence 1 c ( r , R , β, γ ) β n � γ 1 {� xi − xj �≤ R } 1 {� x i − x j � > r } , and basic properties f ( { x 1 , . . . , x n } ) = { x 1 , . . . , x n } ⊂ S , Stationary DPPs and i < j approximations Parametric models where S ⊂ R d is compact; n = 0 , 1 , . . . ; 0 ≤ r < R , β > 0, 0 ≤ γ ≤ 1 are Simulation parameters; the density is w.r.t. the unit rate Poisson process. Stationary data example ◮ The normalizing constant c ( r , R , β, γ ) is intractable. Non-stat. example DPPs on the sphere (on going research ◮ Interpretations? We don’t know the intensity or any other moment properties. project) Concluding remarks ◮ We don’t know the distribution of the number of points. ◮ (Long) MCMC runs are needed... ◮ We have ignored edge effects: the restriction to B ⊂ S ( B � = S ) is not a Strauss hard-core process. Dept. of Mathematical Sciences Aalborg University Denmark 32
Gibbs point processes — the usual class of point pro- cesses used for modelling inhibition Example: Strauss hard-core process Jesper Møller 4 Definition, existence 1 c ( r , R , β, γ ) β n � γ 1 {� xi − xj �≤ R } 1 {� x i − x j � > r } , and basic properties f ( { x 1 , . . . , x n } ) = { x 1 , . . . , x n } ⊂ S , Stationary DPPs and i < j approximations Parametric models where S ⊂ R d is compact; n = 0 , 1 , . . . ; 0 ≤ r < R , β > 0, 0 ≤ γ ≤ 1 are Simulation parameters; the density is w.r.t. the unit rate Poisson process. Stationary data example ◮ The normalizing constant c ( r , R , β, γ ) is intractable. Non-stat. example DPPs on the sphere (on going research ◮ Interpretations? We don’t know the intensity or any other moment properties. project) Concluding remarks ◮ We don’t know the distribution of the number of points. ◮ (Long) MCMC runs are needed... ◮ We have ignored edge effects: the restriction to B ⊂ S ( B � = S ) is not a Strauss hard-core process. ◮ On R d a ‘local specification’ is needed and the issue of phase transition has to Dept. of Mathematical Sciences be clarified. Aalborg University Denmark 32
Notation ◮ X : spatial point process on R d Jesper Møller 5 Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Notation ◮ X : spatial point process on R d Jesper Møller 5 Definition, existence ◮ For any Borel set B ⊆ R d , X B = X ∩ B . and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Notation ◮ X : spatial point process on R d Jesper Møller 5 Definition, existence ◮ For any Borel set B ⊆ R d , X B = X ∩ B . and basic properties Stationary DPPs and approximations ◮ For any integer n > 0, denote ρ ( n ) the n ’th order joint intensity of X : Parametric models Simulation � � ρ ( n ) ( x 1 , . . . , x n ) d x 1 · · · d x n Stationary data E [# X B 1 · · · # X B n ] = · · · example B 1 B n Non-stat. example DPPs on the sphere for disjoint Borel sets B 1 , . . . , B n ⊆ R d . (on going research project) Concluding remarks ◮ Intuitively, ρ ( n ) ( x 1 , . . . , x n ) d x 1 · · · d x n is the probability that for each i = 1 , . . . , n , X has a point in a region around x i of volume d x i . Dept. of Mathematical Sciences Aalborg University Denmark 32
Notation ◮ X : spatial point process on R d Jesper Møller 5 Definition, existence ◮ For any Borel set B ⊆ R d , X B = X ∩ B . and basic properties Stationary DPPs and approximations ◮ For any integer n > 0, denote ρ ( n ) the n ’th order joint intensity of X : Parametric models Simulation � � ρ ( n ) ( x 1 , . . . , x n ) d x 1 · · · d x n Stationary data E [# X B 1 · · · # X B n ] = · · · example B 1 B n Non-stat. example DPPs on the sphere for disjoint Borel sets B 1 , . . . , B n ⊆ R d . (on going research project) Concluding remarks ◮ Intuitively, ρ ( n ) ( x 1 , . . . , x n ) d x 1 · · · d x n is the probability that for each i = 1 , . . . , n , X has a point in a region around x i of volume d x i . ◮ In particular ρ = ρ ( 1 ) is the intensity function . Dept. of Mathematical Sciences Aalborg University Denmark 32
Definition of a DPP Jesper Møller Definition Let C be a function R d × R d → C . 6 Definition, existence and basic properties X is a determinantal point process with kernel C , denoted X ∼ DPP ( C ) , if Stationary DPPs and approximations ρ ( n ) ( x 1 , . . . , x n ) = det { C ( x i , x j } i , j = 1 ,..., n , Parametric models n = 1 , 2 , . . . Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Definition of a DPP Jesper Møller Definition Let C be a function R d × R d → C . 6 Definition, existence and basic properties X is a determinantal point process with kernel C , denoted X ∼ DPP ( C ) , if Stationary DPPs and approximations ρ ( n ) ( x 1 , . . . , x n ) = det { C ( x i , x j } i , j = 1 ,..., n , Parametric models n = 1 , 2 , . . . Simulation Stationary data example ◮ The Poisson process with intensity ρ ( x ) is the special case where Non-stat. example C ( x , x ) = ρ ( x ) and C ( x , y ) = 0 if x � = y . DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Definition of a DPP Jesper Møller Definition Let C be a function R d × R d → C . 6 Definition, existence and basic properties X is a determinantal point process with kernel C , denoted X ∼ DPP ( C ) , if Stationary DPPs and approximations ρ ( n ) ( x 1 , . . . , x n ) = det { C ( x i , x j } i , j = 1 ,..., n , Parametric models n = 1 , 2 , . . . Simulation Stationary data example ◮ The Poisson process with intensity ρ ( x ) is the special case where Non-stat. example C ( x , x ) = ρ ( x ) and C ( x , y ) = 0 if x � = y . DPPs on the sphere (on going research ◮ For existence, conditions on the kernel C are mandatory. project) Concluding remarks E.g. det { C ( x i , x j } i , j = 1 ,..., n ≥ 0. Dept. of Mathematical Sciences Aalborg University Denmark 32
Definition of a DPP Jesper Møller Definition Let C be a function R d × R d → C . 6 Definition, existence and basic properties X is a determinantal point process with kernel C , denoted X ∼ DPP ( C ) , if Stationary DPPs and approximations ρ ( n ) ( x 1 , . . . , x n ) = det { C ( x i , x j } i , j = 1 ,..., n , Parametric models n = 1 , 2 , . . . Simulation Stationary data example ◮ The Poisson process with intensity ρ ( x ) is the special case where Non-stat. example C ( x , x ) = ρ ( x ) and C ( x , y ) = 0 if x � = y . DPPs on the sphere (on going research ◮ For existence, conditions on the kernel C are mandatory. project) Concluding remarks E.g. det { C ( x i , x j } i , j = 1 ,..., n ≥ 0. For ease of exposition assume (C1) C is a continuous (complex) covariance function . Dept. of Mathematical Sciences Aalborg University Denmark 32
Basic properties (if X ∼ DPP ( C ) exists) ◮ The intensity of X is ρ ( x ) = C ( x , x ) . Jesper Møller 7 Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Basic properties (if X ∼ DPP ( C ) exists) ◮ The intensity of X is ρ ( x ) = C ( x , x ) . Jesper Møller ◮ Inhibition , since 7 Definition, existence and basic properties ρ ( n ) ( x 1 , . . . , x n ) ≤ ρ ( x 1 ) · · · ρ ( x n ) Stationary DPPs and approximations with equality iff X is a Poisson process with intensity function ρ . Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Basic properties (if X ∼ DPP ( C ) exists) ◮ The intensity of X is ρ ( x ) = C ( x , x ) . Jesper Møller ◮ Inhibition , since 7 Definition, existence and basic properties ρ ( n ) ( x 1 , . . . , x n ) ≤ ρ ( x 1 ) · · · ρ ( x n ) Stationary DPPs and approximations with equality iff X is a Poisson process with intensity function ρ . Parametric models ◮ The pair correlation function is Simulation Stationary data example g ( x , y ) := ρ ( 2 ) ( x , y ) ρ ( x ) ρ ( y ) = 1 − C ( x , y ) C ( y , x ) C ( x , x ) C ( y , y ) = 1 − | R ( x , y ) | 2 ≤ 1 Non-stat. example DPPs on the sphere (on going research project) where R is the correlation function corresponding to C . Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Basic properties (if X ∼ DPP ( C ) exists) ◮ The intensity of X is ρ ( x ) = C ( x , x ) . Jesper Møller ◮ Inhibition , since 7 Definition, existence and basic properties ρ ( n ) ( x 1 , . . . , x n ) ≤ ρ ( x 1 ) · · · ρ ( x n ) Stationary DPPs and approximations with equality iff X is a Poisson process with intensity function ρ . Parametric models ◮ The pair correlation function is Simulation Stationary data example g ( x , y ) := ρ ( 2 ) ( x , y ) ρ ( x ) ρ ( y ) = 1 − C ( x , y ) C ( y , x ) C ( x , x ) C ( y , y ) = 1 − | R ( x , y ) | 2 ≤ 1 Non-stat. example DPPs on the sphere (on going research project) where R is the correlation function corresponding to C . Concluding remarks ◮ Any smooth transformation or independent thinning of X is still a DPP with an explicitly given kernel. Dept. of Mathematical Sciences Aalborg University Denmark 32
Basic properties (if X ∼ DPP ( C ) exists) ◮ The intensity of X is ρ ( x ) = C ( x , x ) . Jesper Møller ◮ Inhibition , since 7 Definition, existence and basic properties ρ ( n ) ( x 1 , . . . , x n ) ≤ ρ ( x 1 ) · · · ρ ( x n ) Stationary DPPs and approximations with equality iff X is a Poisson process with intensity function ρ . Parametric models ◮ The pair correlation function is Simulation Stationary data example g ( x , y ) := ρ ( 2 ) ( x , y ) ρ ( x ) ρ ( y ) = 1 − C ( x , y ) C ( y , x ) C ( x , x ) C ( y , y ) = 1 − | R ( x , y ) | 2 ≤ 1 Non-stat. example DPPs on the sphere (on going research project) where R is the correlation function corresponding to C . Concluding remarks ◮ Any smooth transformation or independent thinning of X is still a DPP with an explicitly given kernel. ◮ The restriction to any Borel set B ⊂ S d is a DPP with kernel C B ( x , y ) = C ( x , y ) if x , y ∈ B and C B ( x , y ) = 0 else. Dept. of Mathematical Sciences Aalborg University Denmark 32
Basic properties (if X ∼ DPP ( C ) exists) ◮ The intensity of X is ρ ( x ) = C ( x , x ) . Jesper Møller ◮ Inhibition , since 7 Definition, existence and basic properties ρ ( n ) ( x 1 , . . . , x n ) ≤ ρ ( x 1 ) · · · ρ ( x n ) Stationary DPPs and approximations with equality iff X is a Poisson process with intensity function ρ . Parametric models ◮ The pair correlation function is Simulation Stationary data example g ( x , y ) := ρ ( 2 ) ( x , y ) ρ ( x ) ρ ( y ) = 1 − C ( x , y ) C ( y , x ) C ( x , x ) C ( y , y ) = 1 − | R ( x , y ) | 2 ≤ 1 Non-stat. example DPPs on the sphere (on going research project) where R is the correlation function corresponding to C . Concluding remarks ◮ Any smooth transformation or independent thinning of X is still a DPP with an explicitly given kernel. ◮ The restriction to any Borel set B ⊂ S d is a DPP with kernel C B ( x , y ) = C ( x , y ) if x , y ∈ B and C B ( x , y ) = 0 else. ◮ Given a kernel C , there exists at most one DPP ( C ) . Dept. of Mathematical Sciences Aalborg University Denmark 32
Existence By Mercer’s theorem, for any compact set S ⊂ R d , C restricted to S × S , denoted Jesper Møller C S , has a spectral representation , 8 Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Existence By Mercer’s theorem, for any compact set S ⊂ R d , C restricted to S × S , denoted Jesper Møller C S , has a spectral representation , 8 Definition, existence and basic properties ∞ Stationary DPPs and � λ S k φ S k ( x ) φ S approximations C S ( x , y ) = k ( y ) , ( x , y ) ∈ S × S , Parametric models k = 1 Simulation Stationary data where λ S k ≥ 0 and { φ S k } is a set of orthonormal basis functions for L 2 ( S ) , i.e., example Non-stat. example � DPPs on the sphere φ S k ( x ) φ S l ( x ) d x = 1 { k = l } . (on going research project) S Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Existence By Mercer’s theorem, for any compact set S ⊂ R d , C restricted to S × S , denoted Jesper Møller C S , has a spectral representation , 8 Definition, existence and basic properties ∞ Stationary DPPs and � λ S k φ S k ( x ) φ S approximations C S ( x , y ) = k ( y ) , ( x , y ) ∈ S × S , Parametric models k = 1 Simulation Stationary data where λ S k ≥ 0 and { φ S k } is a set of orthonormal basis functions for L 2 ( S ) , i.e., example Non-stat. example � DPPs on the sphere φ S k ( x ) φ S l ( x ) d x = 1 { k = l } . (on going research project) S Concluding remarks Theorem (Macchi, 1975) Under (C1), existence of DPP ( C ) is equivalent to : k ≤ 1 for all compact S ⊂ R d and all k. λ S Dept. of Mathematical Sciences Aalborg University Denmark 32
Density on a compact set S Let X ∼ DPP ( C ) and S ⊂ R d be any compact set. Jesper Møller 9 Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Density on a compact set S Let X ∼ DPP ( C ) and S ⊂ R d be any compact set. Jesper Møller Theorem (Macchi (1975)) 9 Definition, existence and basic properties If λ S k < 1 ∀ k, then X S ≪ Poisson ( S , 1 ) , with density Stationary DPPs and approximations Parametric models f ( { x 1 , . . . , x n } ) = exp ( | S | − D ) det { C ( x i , x j ) } i , j = 1 ,..., n , Simulation Stationary data where D = − � ∞ k ) and ˜ k = 1 log ( 1 − λ S C : S × S → C is given by example Non-stat. example ∞ λ S DPPs on the sphere ˜ � ˜ ˜ (on going research λ S k φ S k ( x ) φ S λ S k C ( x , y ) = k ( y ) , k = . project) 1 − λ S k k = 1 Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Density on a compact set S Let X ∼ DPP ( C ) and S ⊂ R d be any compact set. Jesper Møller Theorem (Macchi (1975)) 9 Definition, existence and basic properties If λ S k < 1 ∀ k, then X S ≪ Poisson ( S , 1 ) , with density Stationary DPPs and approximations Parametric models f ( { x 1 , . . . , x n } ) = exp ( | S | − D ) det { C ( x i , x j ) } i , j = 1 ,..., n , Simulation Stationary data where D = − � ∞ k ) and ˜ k = 1 log ( 1 − λ S C : S × S → C is given by example Non-stat. example ∞ λ S DPPs on the sphere ˜ � ˜ ˜ (on going research λ S k φ S k ( x ) φ S λ S k C ( x , y ) = k ( y ) , k = . project) 1 − λ S k k = 1 Concluding remarks ◮ Thus to calculate the density/likelihood we need the spectral representation. Dept. of Mathematical Sciences Aalborg University Denmark 32
Density on a compact set S Let X ∼ DPP ( C ) and S ⊂ R d be any compact set. Jesper Møller Theorem (Macchi (1975)) 9 Definition, existence and basic properties If λ S k < 1 ∀ k, then X S ≪ Poisson ( S , 1 ) , with density Stationary DPPs and approximations Parametric models f ( { x 1 , . . . , x n } ) = exp ( | S | − D ) det { C ( x i , x j ) } i , j = 1 ,..., n , Simulation Stationary data where D = − � ∞ k ) and ˜ k = 1 log ( 1 − λ S C : S × S → C is given by example Non-stat. example ∞ λ S DPPs on the sphere ˜ � ˜ ˜ (on going research λ S k φ S k ( x ) φ S λ S k C ( x , y ) = k ( y ) , k = . project) 1 − λ S k k = 1 Concluding remarks ◮ Thus to calculate the density/likelihood we need the spectral representation. ◮ Conversely, existence of X S is ensured by that ˜ λ S λ S k k = < 1 . 1 + ˜ λ S Dept. of Mathematical k Sciences Aalborg University Denmark 32
Simulation Let X ∼ DPP ( C ) . We want to simulate X S for S ⊂ R d compact. Jesper Møller 10 Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation Let X ∼ DPP ( C ) . We want to simulate X S for S ⊂ R d compact. Jesper Møller 10 Definition, existence Theorem (Hough et al. (2006)) and basic properties Stationary DPPs and Let B 1 , B 2 , . . . be independent Bernoulli variables with means λ S 1 , λ S 1 , . . . , and approximations Parametric models ∞ Simulation � B k φ S k ( x ) φ S K ( x , y ) = k ( y ) , ( x , y ) ∈ S × S . Stationary data example k = 1 Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation Let X ∼ DPP ( C ) . We want to simulate X S for S ⊂ R d compact. Jesper Møller 10 Definition, existence Theorem (Hough et al. (2006)) and basic properties Stationary DPPs and Let B 1 , B 2 , . . . be independent Bernoulli variables with means λ S 1 , λ S 1 , . . . , and approximations Parametric models ∞ Simulation � B k φ S k ( x ) φ S K ( x , y ) = k ( y ) , ( x , y ) ∈ S × S . Stationary data example k = 1 Non-stat. example d DPPs on the sphere Then DPP ( C S ) = DPP ( K ) . (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation Let X ∼ DPP ( C ) . We want to simulate X S for S ⊂ R d compact. Jesper Møller 10 Definition, existence Theorem (Hough et al. (2006)) and basic properties Stationary DPPs and Let B 1 , B 2 , . . . be independent Bernoulli variables with means λ S 1 , λ S 1 , . . . , and approximations Parametric models ∞ Simulation � B k φ S k ( x ) φ S K ( x , y ) = k ( y ) , ( x , y ) ∈ S × S . Stationary data example k = 1 Non-stat. example d DPPs on the sphere Then DPP ( C S ) = DPP ( K ) . (on going research project) The algorithm starts by producing n points: Concluding remarks ∞ ∞ ∞ � � � λ S λ S k ( 1 − λ S n ∼ B k , E [ n ] = k , Var [ n ] = k ) . k = 1 k = 1 k = 1 Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation Let X ∼ DPP ( C ) . We want to simulate X S for S ⊂ R d compact. Jesper Møller 10 Definition, existence Theorem (Hough et al. (2006)) and basic properties Stationary DPPs and Let B 1 , B 2 , . . . be independent Bernoulli variables with means λ S 1 , λ S 1 , . . . , and approximations Parametric models ∞ Simulation � B k φ S k ( x ) φ S K ( x , y ) = k ( y ) , ( x , y ) ∈ S × S . Stationary data example k = 1 Non-stat. example d DPPs on the sphere Then DPP ( C S ) = DPP ( K ) . (on going research project) The algorithm starts by producing n points: Concluding remarks ∞ ∞ ∞ � � � λ S λ S k ( 1 − λ S n ∼ B k , E [ n ] = k , Var [ n ] = k ) . k = 1 k = 1 k = 1 NB: Since C is continuous, � � λ S k = C ( x , x ) d x < ∞ . Dept. of Mathematical Sciences S Aalborg University Denmark 32
Simulation (cont’d) Effectively we pick out n < ∞ eigenfunctions with probability according to their Jesper Møller eigenvalues and simulate the DPP with finite rank kernel 11 Definition, existence and basic properties n Stationary DPPs and � φ S � φ S approximations k ( x ) φ S k i ( x ) φ S K ( x , y ) = k ( y ) = k i ( y ) , ( x , y ) ∈ S × S . Parametric models k : B k = 1 i = 1 Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation (cont’d) Effectively we pick out n < ∞ eigenfunctions with probability according to their Jesper Møller eigenvalues and simulate the DPP with finite rank kernel 11 Definition, existence and basic properties n Stationary DPPs and � φ S � φ S approximations k ( x ) φ S k i ( x ) φ S K ( x , y ) = k ( y ) = k i ( y ) , ( x , y ) ∈ S × S . Parametric models k : B k = 1 i = 1 Simulation This is a projection kernel , and the corresponding DPP can be simulated: The Stationary data example algorithm basically consists of taking a quite abstract procedure described by Non-stat. example Hough et al. (2006) and translating it into implementable linear algebra. DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation (cont’d) Effectively we pick out n < ∞ eigenfunctions with probability according to their Jesper Møller eigenvalues and simulate the DPP with finite rank kernel 11 Definition, existence and basic properties n Stationary DPPs and � φ S � φ S approximations k ( x ) φ S k i ( x ) φ S K ( x , y ) = k ( y ) = k i ( y ) , ( x , y ) ∈ S × S . Parametric models k : B k = 1 i = 1 Simulation This is a projection kernel , and the corresponding DPP can be simulated: The Stationary data example algorithm basically consists of taking a quite abstract procedure described by Non-stat. example Hough et al. (2006) and translating it into implementable linear algebra. DPPs on the sphere (on going research project) This leads to simulation of the first point, the second given the first point, the third Concluding remarks given the first and second points,... At each step we have been using rejection sampling... Dept. of Mathematical Sciences Aalborg University Denmark 32
Status ◮ A DPP on R d is specified through a continuous (complex) covariance function Jesper Møller C : R d × R d → C . 12 Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Status ◮ A DPP on R d is specified through a continuous (complex) covariance function Jesper Møller C : R d × R d → C . 12 Definition, existence and basic properties ◮ C determines the moment properties of the DPP . Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Status ◮ A DPP on R d is specified through a continuous (complex) covariance function Jesper Møller C : R d × R d → C . 12 Definition, existence and basic properties ◮ C determines the moment properties of the DPP . Stationary DPPs and approximations Parametric models ◮ Given the spectral representation of C on a compact set S we Simulation Stationary data ◮ have a simple existence condition, example Non-stat. example ◮ know the distribution of the number of points falling in S , DPPs on the sphere (on going research ◮ can simulate the process on S , project) Concluding remarks ◮ can calculate the density/likelihood. Dept. of Mathematical Sciences Aalborg University Denmark 32
Status ◮ A DPP on R d is specified through a continuous (complex) covariance function Jesper Møller C : R d × R d → C . 12 Definition, existence and basic properties ◮ C determines the moment properties of the DPP . Stationary DPPs and approximations Parametric models ◮ Given the spectral representation of C on a compact set S we Simulation Stationary data ◮ have a simple existence condition, example Non-stat. example ◮ know the distribution of the number of points falling in S , DPPs on the sphere (on going research ◮ can simulate the process on S , project) Concluding remarks ◮ can calculate the density/likelihood. Typically we don’t know the spectral representation! Dept. of Mathematical Sciences Aalborg University Denmark 32
Stationary kernels x , y ∈ R d . Consider a stationary kernel: C ( x , y ) = C 0 ( x − y ) , Jesper Møller Definition, existence and basic properties 13 Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Stationary kernels x , y ∈ R d . Consider a stationary kernel: C ( x , y ) = C 0 ( x − y ) , Jesper Møller Its Fourier transform (or spectral density) is: Definition, existence and basic properties � C 0 ( t ) e − 2 π i x · t d t , x ∈ R d . 13 Stationary DPPs and ϕ ( x ) = approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Stationary kernels x , y ∈ R d . Consider a stationary kernel: C ( x , y ) = C 0 ( x − y ) , Jesper Møller Its Fourier transform (or spectral density) is: Definition, existence and basic properties � C 0 ( t ) e − 2 π i x · t d t , x ∈ R d . 13 Stationary DPPs and ϕ ( x ) = approximations Parametric models Theorem Simulation Under (C1), if C 0 ∈ L 2 ( R d ) , then existence of DPP ( C 0 ) is equivalent to Stationary data example Non-stat. example ϕ ≤ 1 . DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Stationary kernels x , y ∈ R d . Consider a stationary kernel: C ( x , y ) = C 0 ( x − y ) , Jesper Møller Its Fourier transform (or spectral density) is: Definition, existence and basic properties � C 0 ( t ) e − 2 π i x · t d t , x ∈ R d . 13 Stationary DPPs and ϕ ( x ) = approximations Parametric models Theorem Simulation Under (C1), if C 0 ∈ L 2 ( R d ) , then existence of DPP ( C 0 ) is equivalent to Stationary data example Non-stat. example ϕ ≤ 1 . DPPs on the sphere (on going research project) Concluding remarks → This induces a restriction on the parameter space : That is, there is a trade-off between strong inhibiton and large intensity. In practice, this restriction implies that if the intensity is large the range (effective support) of C 0 must be small. Dept. of Mathematical Sciences Aalborg University Denmark 32
Approximation Jesper Møller WLOG consider S = [ − 1 / 2 , 1 / 2 ] d . Definition, existence and basic properties 14 Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Approximation Jesper Møller WLOG consider S = [ − 1 / 2 , 1 / 2 ] d . Approximate X S by X app ∼ DPP S ( C app ) where Definition, existence and basic properties � ϕ ( k ) e 2 π i k · ( x − y ) , C app ( x , y ) = x , y ∈ S . 14 Stationary DPPs and approximations k ∈ Z d Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Approximation Jesper Møller WLOG consider S = [ − 1 / 2 , 1 / 2 ] d . Approximate X S by X app ∼ DPP S ( C app ) where Definition, existence and basic properties � ϕ ( k ) e 2 π i k · ( x − y ) , C app ( x , y ) = x , y ∈ S . 14 Stationary DPPs and approximations k ∈ Z d Parametric models Simulation If x − y ∈ S this is effectively the Fourier expansion Stationary data example Non-stat. example � α k e 2 π i k · ( x − y ) C ( x , y ) = C 0 ( x − y ) = DPPs on the sphere (on going research k ∈ Z d project) Concluding remarks since for “most” interesting models � � C 0 ( t ) e − 2 π i k · t d t ≈ R d C 0 ( t ) e − 2 π i k · t d t = ϕ ( k ) . α k = S So we claim that C 0 ( t ) ≈ 0 for t �∈ S : in practice, for any reasonable expected number of points, this is implied by the parameter restriction. Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling based on spectral densities Idea: instead of modelling C 0 , model λ S k and φ S k in Jesper Møller Definition, existence ∞ and basic properties � λ S k φ S k ( x ) φ S C 0 ( y − x ) = k ( y ) . 15 Stationary DPPs and approximations k = 1 Parametric models Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling based on spectral densities Idea: instead of modelling C 0 , model λ S k and φ S k in Jesper Møller Definition, existence ∞ and basic properties � λ S k φ S k ( x ) φ S C 0 ( y − x ) = k ( y ) . 15 Stationary DPPs and approximations k = 1 Parametric models Simulation Following the previous approximation on the unit square: Stationary data example ◮ Choose the Fourier basis: φ S k ( x ) = e − 2 π i k · x . Non-stat. example ◮ Choose λ S k = ϕ ( k ) , where ϕ is a spectral density with ϕ ≤ 1. DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling based on spectral densities Idea: instead of modelling C 0 , model λ S k and φ S k in Jesper Møller Definition, existence ∞ and basic properties � λ S k φ S k ( x ) φ S C 0 ( y − x ) = k ( y ) . 15 Stationary DPPs and approximations k = 1 Parametric models Simulation Following the previous approximation on the unit square: Stationary data example ◮ Choose the Fourier basis: φ S k ( x ) = e − 2 π i k · x . Non-stat. example ◮ Choose λ S k = ϕ ( k ) , where ϕ is a spectral density with ϕ ≤ 1. DPPs on the sphere (on going research project) ◮ Then we have a well-defined DPP on S , which can easily be simulated and the Concluding remarks density/likelihood can be evaluated exactly (up to series truncation). Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling based on spectral densities Idea: instead of modelling C 0 , model λ S k and φ S k in Jesper Møller Definition, existence ∞ and basic properties � λ S k φ S k ( x ) φ S C 0 ( y − x ) = k ( y ) . 15 Stationary DPPs and approximations k = 1 Parametric models Simulation Following the previous approximation on the unit square: Stationary data example ◮ Choose the Fourier basis: φ S k ( x ) = e − 2 π i k · x . Non-stat. example ◮ Choose λ S k = ϕ ( k ) , where ϕ is a spectral density with ϕ ≤ 1. DPPs on the sphere (on going research project) ◮ Then we have a well-defined DPP on S , which can easily be simulated and the Concluding remarks density/likelihood can be evaluated exactly (up to series truncation). ◮ To obtain a DPP on R d start by modelling ϕ ≤ 1. Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling based on spectral densities Idea: instead of modelling C 0 , model λ S k and φ S k in Jesper Møller Definition, existence ∞ and basic properties � λ S k φ S k ( x ) φ S C 0 ( y − x ) = k ( y ) . 15 Stationary DPPs and approximations k = 1 Parametric models Simulation Following the previous approximation on the unit square: Stationary data example ◮ Choose the Fourier basis: φ S k ( x ) = e − 2 π i k · x . Non-stat. example ◮ Choose λ S k = ϕ ( k ) , where ϕ is a spectral density with ϕ ≤ 1. DPPs on the sphere (on going research project) ◮ Then we have a well-defined DPP on S , which can easily be simulated and the Concluding remarks density/likelihood can be evaluated exactly (up to series truncation). ◮ To obtain a DPP on R d start by modelling ϕ ≤ 1. Main drawback: ◮ C 0 (and thus the moment properties) is given as an infinite sum → parameters may be harder to understand/interpret. Dept. of Mathematical Sciences Aalborg University Denmark 32
Intermezzo Jesper Møller Definition, existence and basic properties 16 Stationary DPPs and approximations This concludes the first part of the talk focusing on the probabilistic Parametric models background and approximations for simulation and density Simulation expression. Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Now we start doing statistics, so if you got lost or fell asleep you get a fresh start! Dept. of Mathematical Sciences Aalborg University Denmark 32
Examples of parametric models We will focus on the following parametric models, where ρ > 0 is the intensity, α > 0 Jesper Møller is a scale/range parameter, and ν > 0 is a shape parameter: Definition, existence and basic properties Stationary DPPs and approximations Parametric models 17 Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Examples of parametric models We will focus on the following parametric models, where ρ > 0 is the intensity, α > 0 Jesper Møller is a scale/range parameter, and ν > 0 is a shape parameter: Definition, existence and basic properties ◮ Whittle-Matérn model , which includes the exponential model ( ν = 1 / 2) and the Stationary DPPs and approximations Gaussian model ( ν = ∞ ): Parametric models 17 C 0 ( x ) = ρ 2 1 − ν Simulation Γ( ν ) � x /α � ν K ν ( � x /α � ) , x ∈ R d , Stationary data example Non-stat. example Γ( ν ) DPPs on the sphere The parameter restriction is ρ ≤ Γ( ν + d / 2 )( 2 √ πα ) d . (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Examples of parametric models We will focus on the following parametric models, where ρ > 0 is the intensity, α > 0 Jesper Møller is a scale/range parameter, and ν > 0 is a shape parameter: Definition, existence and basic properties ◮ Whittle-Matérn model , which includes the exponential model ( ν = 1 / 2) and the Stationary DPPs and approximations Gaussian model ( ν = ∞ ): Parametric models 17 C 0 ( x ) = ρ 2 1 − ν Simulation Γ( ν ) � x /α � ν K ν ( � x /α � ) , x ∈ R d , Stationary data example Non-stat. example Γ( ν ) DPPs on the sphere The parameter restriction is ρ ≤ Γ( ν + d / 2 )( 2 √ πα ) d . (on going research project) Concluding remarks ◮ Power exponential spectral model ϕ ( x ) = ρ Γ( d / 2 + 1 ) α d π d / 2 Γ( d /ν + 1 ) exp ( −� α x � ν ) , x ∈ R d . The parameter restriction is ρ ≤ π d / 2 Γ( d /ν + 1 ) . Γ( d / 2 ) α d Dept. of Mathematical Sciences Aalborg University Denmark 32
Parametric models in R (so far: contact Ege Rubak; later on: spatstat) The parametric families are specified in R via the determinantal family functions (of Jesper Møller class detfamily ): detGauss , detMatern , detPowerExp . E.g: Definition, existence and basic properties ◮ model <- detGauss(rho=100, alpha=0.05, d=2) Stationary DPPs and approximations ◮ model <- detMatern(rho=100, alpha=0.03, nu=0.5, d=2) 18 Parametric models Simulation ◮ model <- detPowerExp(rho=100, alpha=0.17, nu=2, d=2) Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Parametric models in R (so far: contact Ege Rubak; later on: spatstat) The parametric families are specified in R via the determinantal family functions (of Jesper Møller class detfamily ): detGauss , detMatern , detPowerExp . E.g: Definition, existence and basic properties ◮ model <- detGauss(rho=100, alpha=0.05, d=2) Stationary DPPs and approximations ◮ model <- detMatern(rho=100, alpha=0.03, nu=0.5, d=2) 18 Parametric models Simulation ◮ model <- detPowerExp(rho=100, alpha=0.17, nu=2, d=2) Stationary data example Non-stat. example Extract the kernel, spectral density, pair correlation function, K -function: DPPs on the sphere ◮ detkernel(model) (on going research project) ◮ detspecden(model) Concluding remarks ◮ pcfmodel(model) ◮ Kmodel(model) Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation in R Simply use the generic function simulate (then R automatically calls the function Jesper Møller simulate.detmodel ): Definition, existence and basic properties ◮ model <- detGauss(rho=100, alpha=0.05, d=2) Stationary DPPs and X <- simulate(model) approximations Parametric models 19 Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation in R Simply use the generic function simulate (then R automatically calls the function Jesper Møller simulate.detmodel ): Definition, existence and basic properties ◮ model <- detGauss(rho=100, alpha=0.05, d=2) Stationary DPPs and X <- simulate(model) approximations Parametric models ◮ Change the window (default is the unit square): 19 Simulation W <- owin(poly=list(x=c(-1,0,1),y=c(0,1,0))) Stationary data example X <- simulate(model, W=W) Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation in R Simply use the generic function simulate (then R automatically calls the function Jesper Møller simulate.detmodel ): Definition, existence and basic properties ◮ model <- detGauss(rho=100, alpha=0.05, d=2) Stationary DPPs and X <- simulate(model) approximations Parametric models ◮ Change the window (default is the unit square): 19 Simulation W <- owin(poly=list(x=c(-1,0,1),y=c(0,1,0))) Stationary data example X <- simulate(model, W=W) Non-stat. example DPPs on the sphere ◮ Several realizations: (on going research project) X <- simulate(model, nsim=4) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Illustration of simulation algorithm Step 1. The first point is sampled uniformly on S (stationary case). Jesper Møller Definition, existence and basic properties Stationary DPPs and approximations Parametric models 20 Simulation Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Illustration of simulation algorithm Step 2. The next point is sampled w.r.t. the following density: Jesper Møller Definition, existence 1 and basic properties Stationary DPPs and approximations Parametric models 0.8 20 Simulation Stationary data example 0.6 Non-stat. example DPPs on the sphere (on going research project) Concluding remarks 0.4 ● 0.2 0 Dept. of Mathematical Sciences Aalborg University Denmark 32
Illustration of simulation algorithm Step 3. The next point is sampled w.r.t. the following density: Jesper Møller Definition, existence 1 and basic properties Stationary DPPs and approximations Parametric models 0.8 20 Simulation Stationary data example 0.6 Non-stat. example DPPs on the sphere (on going research ● project) Concluding remarks 0.4 ● 0.2 0 Dept. of Mathematical Sciences Aalborg University Denmark 32
Illustration of simulation algorithm Step 4. The next point is sampled w.r.t. the following density: Jesper Møller Definition, existence 1 and basic properties ● Stationary DPPs and approximations Parametric models 0.8 20 Simulation Stationary data example 0.6 Non-stat. example DPPs on the sphere (on going research ● project) Concluding remarks 0.4 ● 0.2 0 Dept. of Mathematical Sciences Aalborg University Denmark 32
Illustration of simulation algorithm Step 5. The next point is sampled w.r.t. the following density: Jesper Møller Definition, existence 1 and basic properties ● Stationary DPPs and approximations Parametric models 0.8 20 Simulation Stationary data example 0.6 Non-stat. example DPPs on the sphere (on going research ● project) Concluding remarks 0.4 ● 0.2 ● 0 Dept. of Mathematical Sciences Aalborg University Denmark 32
Illustration of simulation algorithm ...somewhere in the middle... Jesper Møller Definition, existence 0.6 and basic properties ● ● ● ● Stationary DPPs and ● ● approximations ● ● 0.5 Parametric models 20 Simulation ● ● ● Stationary data ● ● ● ● ● ● 0.4 example ● ● Non-stat. example ● ● ● ● ● DPPs on the sphere (on going research 0.3 ● ● project) ● Concluding remarks ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● 0.1 ● ● ● ● ● ● ● ● ● ● ● ● 0 Dept. of Mathematical Sciences Aalborg University Denmark 32
Illustration of simulation algorithm Final point is sampled w.r.t. the following density: Jesper Møller Definition, existence and basic properties ● ● ● ● Stationary DPPs and ● ● 0.15 approximations ● ● ● ● Parametric models ● ● 20 Simulation ● ● ● Stationary data ● ● ● ● ● ● example ● ● ● Non-stat. example ● ● ● ● ● 0.1 ● DPPs on the sphere ● (on going research ● ● ● project) ● ● ● Concluding remarks ● ● ● ● ● ● 0.05 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 Dept. of Mathematical Sciences Aalborg University Denmark 32
Spanish towns dataset Ripley (1988): Strauss hard-core model with 4 parameters: Jesper Møller Definition, existence r =hard-core, R =range of interaction, β =abundance, γ =interaction. and basic properties Stationary DPPs and approximations Parametric models ● ● ● ● ● ● ● ● ● ● ● ● Simulation ● ● ● ● ● ● ● ● ● 21 Stationary data ● ● ● example ● ● ● ● ● Non-stat. example ● ● ● ● ● ● ● DPPs on the sphere ● ● (on going research ● ● ● project) ● ● ● ● ● Concluding remarks ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Dept. of Mathematical Sciences Aalborg University Denmark 32
Spanish towns dataset Ripley (1988): Strauss hard-core model with 4 parameters: Jesper Møller Definition, existence r =hard-core, R =range of interaction, β =abundance, γ =interaction. and basic properties Stationary DPPs and approximations Parametric models ● ● ● ● ● ● ● ● ● ● ● ● Simulation ● ● ● ● ● ● ● ● ● 21 Stationary data ● ● ● example ● ● ● ● ● Non-stat. example ● ● ● ● ● ● ● DPPs on the sphere ● ● (on going research ● ● ● project) ● ● ● ● ● Concluding remarks ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● r = 0 . 83, ˆ Following Illian et al. (2008): ˆ R = 3 . 5. Approximate likelihood method (Huang and Ogata (1999)): ˆ β = 0 . 12 and ˆ γ = 0 . 76. Dept. of Mathematical Sciences Aalborg University Denmark 32
Alternative DPP models Gaussian, Whittle-Matérn, and power exponential spectral models fitted using the Jesper Møller function dppm : Definition, existence and basic properties ◮ Default estimation method is “partial likelihood” where we use Stationary DPPs and ρ = n / | W | = 0 . 043 and MLEs for the rest: ˆ approximations Parametric models fit <- dppm(X, detGauss()) Simulation ◮ Full likelihood: 22 Stationary data fit <- dppm(X, detGauss(), method="likelihood") example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Alternative DPP models Gaussian, Whittle-Matérn, and power exponential spectral models fitted using the Jesper Møller function dppm : Definition, existence and basic properties ◮ Default estimation method is “partial likelihood” where we use Stationary DPPs and ρ = n / | W | = 0 . 043 and MLEs for the rest: ˆ approximations Parametric models fit <- dppm(X, detGauss()) Simulation ◮ Full likelihood: 22 Stationary data fit <- dppm(X, detGauss(), method="likelihood") example Non-stat. example Highest likelihood: fitted Whittle-Matérn model. DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Alternative DPP models Gaussian, Whittle-Matérn, and power exponential spectral models fitted using the Jesper Møller function dppm : Definition, existence and basic properties ◮ Default estimation method is “partial likelihood” where we use Stationary DPPs and ρ = n / | W | = 0 . 043 and MLEs for the rest: ˆ approximations Parametric models fit <- dppm(X, detGauss()) Simulation ◮ Full likelihood: 22 Stationary data fit <- dppm(X, detGauss(), method="likelihood") example Non-stat. example Highest likelihood: fitted Whittle-Matérn model. DPPs on the sphere (on going research Simulation based likelihood-ratio test for the simpler Gaussian model vs the project) Whittle-Matérn model: p = 3 % . Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Clockwise from top left: Non-parametric estimate of L ( r ) − r , G ( r ) , J ( r ) , F ( r ) , and simulation based 2 . 5 % and 97 . 5 % pointwise quantiles (based on 400 realizations). Strauss 0.0 0.8 Matern Data 0.6 −0.5 0.4 −1.0 Strauss 0.2 Matern −1.5 Data 0.0 0 2 4 6 8 10 0 1 2 3 6 Strauss Strauss 0.8 Matern Matern 5 Data Data 0.6 4 0.4 3 0.2 2 1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Conclusion of data analysis Whittle-Matérn model: Jesper Møller ◮ has less parameters Definition, existence and basic properties ◮ (arguably) provides a better fit Stationary DPPs and approximations ◮ has a canonical way of estimating parameters ( likelihood ) Parametric models ◮ direct access to the moments (intensity, pair correlation function, ...) Simulation 24 Stationary data example Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Conclusion of data analysis Whittle-Matérn model: Jesper Møller ◮ has less parameters Definition, existence and basic properties ◮ (arguably) provides a better fit Stationary DPPs and approximations ◮ has a canonical way of estimating parameters ( likelihood ) Parametric models ◮ direct access to the moments (intensity, pair correlation function, ...) Simulation 24 Stationary data example For the Strauss hard-core model Non-stat. example ◮ parameter estimation relies to a certain extend on “ad-hoc” methods DPPs on the sphere (on going research ◮ the density and moments can only be obtained by MCMC simulation. project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Mucous membrane dataset Consists of the most abundant type of cell in a bivariate point pattern analysed in Jesper Møller Møller and Waagepetersen (2004). Definition, existence and basic properties Stationary DPPs and approximations ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● Parametric models ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Simulation ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Stationary data ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● example ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 25 Non-stat. example ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● DPPs on the sphere ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● (on going research ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● project) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● Concluding remarks ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Dept. of Mathematical Sciences Aalborg University Denmark 32
Mucous membrane dataset Consists of the most abundant type of cell in a bivariate point pattern analysed in Jesper Møller Møller and Waagepetersen (2004). Definition, existence and basic properties Stationary DPPs and approximations ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● Parametric models ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Simulation ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Stationary data ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● example ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 25 Non-stat. example ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● DPPs on the sphere ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● (on going research ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● project) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● Concluding remarks ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● We use this unmarked point pattern to illustrate how an inhomogenous DPP can be fitted to a real dataset. Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling inhomogeneity Assume second-order intensity-reweighted stationarity (Baddeley, Møller & Jesper Møller Waagepetersen, 2000), i.e., the correlation function is translation invariant: Definition, existence and basic properties C ( x , y ) C ( x , y ) Stationary DPPs and R ( x , y ) = = = R 0 ( x − y ) . approximations � � C ( x , x ) C ( y , y ) ρ ( x ) ρ ( y ) Parametric models Simulation Stationary data example 26 Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling inhomogeneity Assume second-order intensity-reweighted stationarity (Baddeley, Møller & Jesper Møller Waagepetersen, 2000), i.e., the correlation function is translation invariant: Definition, existence and basic properties C ( x , y ) C ( x , y ) Stationary DPPs and R ( x , y ) = = = R 0 ( x − y ) . approximations � � C ( x , x ) C ( y , y ) ρ ( x ) ρ ( y ) Parametric models Simulation Stationary data ◮ Fit a parametric model to ρ depending on relevant covariates (second example 26 Non-stat. example coordinate axis in our case). DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling inhomogeneity Assume second-order intensity-reweighted stationarity (Baddeley, Møller & Jesper Møller Waagepetersen, 2000), i.e., the correlation function is translation invariant: Definition, existence and basic properties C ( x , y ) C ( x , y ) Stationary DPPs and R ( x , y ) = = = R 0 ( x − y ) . approximations � � C ( x , x ) C ( y , y ) ρ ( x ) ρ ( y ) Parametric models Simulation Stationary data ◮ Fit a parametric model to ρ depending on relevant covariates (second example 26 Non-stat. example coordinate axis in our case). DPPs on the sphere ◮ Use the fitted intensity to estimate the inhomogeneous g -function (or (on going research project) K -function). Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling inhomogeneity Assume second-order intensity-reweighted stationarity (Baddeley, Møller & Jesper Møller Waagepetersen, 2000), i.e., the correlation function is translation invariant: Definition, existence and basic properties C ( x , y ) C ( x , y ) Stationary DPPs and R ( x , y ) = = = R 0 ( x − y ) . approximations � � C ( x , x ) C ( y , y ) ρ ( x ) ρ ( y ) Parametric models Simulation Stationary data ◮ Fit a parametric model to ρ depending on relevant covariates (second example 26 Non-stat. example coordinate axis in our case). DPPs on the sphere ◮ Use the fitted intensity to estimate the inhomogeneous g -function (or (on going research project) K -function). Concluding remarks ◮ Fit a parametric model for R 0 via minimum contrast. Dept. of Mathematical Sciences Aalborg University Denmark 32
Modelling inhomogeneity Assume second-order intensity-reweighted stationarity (Baddeley, Møller & Jesper Møller Waagepetersen, 2000), i.e., the correlation function is translation invariant: Definition, existence and basic properties C ( x , y ) C ( x , y ) Stationary DPPs and R ( x , y ) = = = R 0 ( x − y ) . approximations � � C ( x , x ) C ( y , y ) ρ ( x ) ρ ( y ) Parametric models Simulation Stationary data ◮ Fit a parametric model to ρ depending on relevant covariates (second example 26 Non-stat. example coordinate axis in our case). DPPs on the sphere ◮ Use the fitted intensity to estimate the inhomogeneous g -function (or (on going research project) K -function). Concluding remarks ◮ Fit a parametric model for R 0 via minimum contrast. ◮ The resulting DPP has kernel ˆ � ρ ( x )ˆ � C ( x , y ) = ˆ R 0 ( x − y ) ρ ( y ) . ˆ Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation of inhomogeneous model NB: If a ‘dominating DPP’ with kernel C dom ( x , y ) is thinned with retention probability Jesper Møller π ( x ) , the resulting process is a new DPP with kernel Definition, existence and basic properties � π ( x ) C dom ( x , y ) � Stationary DPPs and C ( x , y ) = π ( y ) . approximations Parametric models Simulation Stationary data example 27 Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation of inhomogeneous model NB: If a ‘dominating DPP’ with kernel C dom ( x , y ) is thinned with retention probability Jesper Møller π ( x ) , the resulting process is a new DPP with kernel Definition, existence and basic properties � π ( x ) C dom ( x , y ) � Stationary DPPs and C ( x , y ) = π ( y ) . approximations Parametric models ρ ( x ) } and define a stationary DPP X dom with kernel Thus let ˆ ρ max = sup x ∈ S { ˆ Simulation Stationary data example ρ max ˆ C dom ( x , y ) = C dom ( x − y ) = ˆ R 0 ( x − y ) . 0 27 Non-stat. example DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
Simulation of inhomogeneous model NB: If a ‘dominating DPP’ with kernel C dom ( x , y ) is thinned with retention probability Jesper Møller π ( x ) , the resulting process is a new DPP with kernel Definition, existence and basic properties � π ( x ) C dom ( x , y ) � Stationary DPPs and C ( x , y ) = π ( y ) . approximations Parametric models ρ ( x ) } and define a stationary DPP X dom with kernel Thus let ˆ ρ max = sup x ∈ S { ˆ Simulation Stationary data example ρ max ˆ C dom ( x , y ) = C dom ( x − y ) = ˆ R 0 ( x − y ) . 0 27 Non-stat. example DPPs on the sphere Then our fitted model is simulated by thinning X dom with retention probability (on going research project) π ( x ) = ˆ ρ ( x ) / ˆ ρ max , since Concluding remarks � � ρ ( x ) ˆ ρ ( y ) ˆ ρ ( x )ˆ ρ ( y ) = ˆ C dom ( x , y ) � � = ˆ R 0 ( x − y ) ˆ C ( x , y ) . ρ max ˆ ρ max ˆ Dept. of Mathematical Sciences Aalborg University Denmark 32
DPPs on the sphere ◮ On the sphere the spherical harmonics constitute a set of basis functions Jesper Møller (given in terms of associated Legendre polynomials). Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example 28 DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
DPPs on the sphere ◮ On the sphere the spherical harmonics constitute a set of basis functions Jesper Møller (given in terms of associated Legendre polynomials). Definition, existence and basic properties ◮ Thus we only have to make a parametric model for the eigenvalues λ k to have Stationary DPPs and approximations a DPP on the sphere. Parametric models Simulation Stationary data example Non-stat. example 28 DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
DPPs on the sphere ◮ On the sphere the spherical harmonics constitute a set of basis functions Jesper Møller (given in terms of associated Legendre polynomials). Definition, existence and basic properties ◮ Thus we only have to make a parametric model for the eigenvalues λ k to have Stationary DPPs and approximations a DPP on the sphere. Parametric models Simulation ◮ There are covariance functions on the sphere with known eigenvalues. One is Stationary data the Inverse MultiQuadric covariance function. example Non-stat. example 28 DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
DPPs on the sphere ◮ On the sphere the spherical harmonics constitute a set of basis functions Jesper Møller (given in terms of associated Legendre polynomials). Definition, existence and basic properties ◮ Thus we only have to make a parametric model for the eigenvalues λ k to have Stationary DPPs and approximations a DPP on the sphere. Parametric models Simulation ◮ There are covariance functions on the sphere with known eigenvalues. One is Stationary data the Inverse MultiQuadric covariance function. example Non-stat. example ◮ We have implemented it in R: 28 DPPs on the sphere (on going research model <- detIMQ(rho=500,delta=0.998) project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
DPPs on the sphere ◮ On the sphere the spherical harmonics constitute a set of basis functions Jesper Møller (given in terms of associated Legendre polynomials). Definition, existence and basic properties ◮ Thus we only have to make a parametric model for the eigenvalues λ k to have Stationary DPPs and approximations a DPP on the sphere. Parametric models Simulation ◮ There are covariance functions on the sphere with known eigenvalues. One is Stationary data the Inverse MultiQuadric covariance function. example Non-stat. example ◮ We have implemented it in R: 28 DPPs on the sphere (on going research model <- detIMQ(rho=500,delta=0.998) project) Concluding remarks ◮ Simlulations rely on the previously developed code (with some modifications): X <- simulate(model) Dept. of Mathematical Sciences Aalborg University Denmark 32
A simulated DPP consisting of 441 points on planet Earth Jesper Møller Definition, existence and basic properties Stationary DPPs and approximations Parametric models Simulation Stationary data example Non-stat. example 29 DPPs on the sphere (on going research project) Concluding remarks Dept. of Mathematical Sciences Aalborg University Denmark 32
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