ON THE PROBABILITY APPROXIMATION OF SPATIAL

Transcription

1 ON THE PROBABILITY APPROIMATION OF SPATIAL POINT PROCESSES Giovanni Luca Torrisi Consiglio Nazionale delle Ricerche Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

2 POINT PROCESSES locally compact metric space (serving as the state space of the points), B() Borel σ-field on. Γ set of locally finite point configurations of : Γ :={x : (x K ) < relatively compact K B()} where x K := x K. Here for any subset x, (x) is the cardinality of x, setting (x) := if x is not finite. We endow Γ with the vague topology and denote by B(Γ ) the Borel σ-field on Γ. A point process is a probability measure on (Γ, B(Γ )). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

3 POINT PROCESSES WITH CONDITIONAL INTENSITY: DEFINITION We assume that the probability measure µ on (Γ, B(Γ )) has conditional (or Papangelou) intensity π, i.e. π : Γ [0, + ] is a measurable function such that Γ x x φ(x, x \ {x}) µ(dx) = Γ φ(x, x) π(x, x)σ(dx)µ(dx) for functions φ(x, x) which are non-negative, Papangelou (1974), Georgii (1976), Nguyen and Zessin (1979). Here σ is a diffuse and locally finite measure on (, B()). µ is a Poisson process with intensity measure σ if and only if π 1. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

4 POINT PROCESSES WITH CONDITIONAL INTENSITY: INTERPRETATION The Papangelou intensity π x (x) has the following interpretation: π x (x) σ(dx) is the infinitesimal probability of finding a point of the process in the region dx around x, given that the point process agrees with the configuration x outside dx. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

5 MAIN EAMPLE: GIBBS POINT PROCESSES WITH PAIR POTENTIAL A pair potential is a function ϕ : R d R {+ } such that ϕ(x) = ϕ( x). We define the relative energy of interaction between a particle at x R d \ x and the configuration x Γ R d by E(x, x) := { y x ϕ(x y) if + otherwise. y x ϕ(x y) < µ on Γ R d is called Gibbs point process with activity z > 0 and pair potential ϕ if it has Papangelou intensity of the form π x (x) := z exp( E(x, x)) with reference measure dx. G s (z, ϕ) if ϕ is superstable, lower-regular and 1 e ϕ L 1 (R d, dx), Ruelle (1970). µ is called inhibitory if ϕ 0 and finite range if 1 e ϕ has compact support. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

6 INNOVATION We define the (raw) innovation as δ x (φ) := φ(x) x x φ(x)π x (x) σ(dx) for any measurable function φ : R for which δ x (φ) < µ-a.s.. The innovation δ(φ) is well-defined for all φ L 1 (, E[π x ]σ(dx)). Here E denotes the mean with respect to µ. The innovation is used in spatial statistics e.g. to check the validity of a point process model fitted to data, Baddeley, Møller and Pakes (2008). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

7 FINITE DIFFERENCE OPERATOR We define the finite difference operator by D x F(x) := F (x {x}) F (x), x, x Γ. Here F : Γ R is a measurable function. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

8 WASSERSTEIN DISTANCE Let F L 1 (Γ, µ) and Z a standard normal random variable defined on (Ω, F, P). We denote by E P the mean with respect to P. The Wasserstein distance between F and Z is d W (F, Z ) := sup E[h(F )] E P [h(z )], h Lip(1) where Lip(1) is the class of real-valued Lipschitz functions with Lipschitz constant less than or equal to 1. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

9 GAUSSIAN APPROIMATION: GENERAL BOUND If µ has conditional intensity π and then d W (δ(φ), Z ) 2/π 1 2 φ L 1,2 (, E[π x ] σ(dx)), φ(x) 2 E[π x ] σ(dx) + φ(x)φ(y) 2 E[π x π y ] σ 2 (dxdy) 2 φ(x)φ(y) E[ D x π y π x ] σ 2 (dxdy) 2 + φ 3 L 3 (,E[π x ] σ(dx)) + 2/π + 2 φ(x) 2 φ(y) E[ D x π y π x ] σ 2 (dxdy) 2 + φ(x)φ(y)φ(w) E[ D x π y D x π w π x ] σ 3 (dxdydw). 3 Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

10 POISSON CASE If µ is a Poisson process with mean measure σ and φ L 1,2 (, σ), then d W (δ(φ), Z ) 1 φ 2 L 2 (,σ) + φ 3 L 3 (,σ), Peccati, Sole, Taqqu and Utzet (2010). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

11 GIBBS CASE If µ is a Gibbs point process with activity z > 0 and pair potential ϕ, and φ L 1,2 (R d, E[π x ] dx), then d W (δ(φ), Z ) 2/π 1 2 φ(x) 2 E[π x ] dx + φ(x)φ(y) 2 E[π x π y ] dxdy R d R 2d + φ 3 L 3 (R d,e[π x ] dx) + 2/π φ(x)φ(y) 1 e ϕ(y x) E[π x π y ] dxdy R 2d + 2 φ(x) 2 φ(y) 1 e ϕ(y x) E[π x π y ] dydx R 2d + φ(x)φ(y)φ(w) 1 e ϕ(y x) 1 e ϕ(w x) E[π x π y π w ] dxdydw. R 3d Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

12 AN EPLICIT BOUND FOR S.I.F.R. GIBBS POINT PROCESSES Suppose µ stationary Gibbs point process with z > 0 and ϕ 0 with compact support and φ L 1,2 (R d, dx). Then, for any p, q, p, q > 1 such that p 1 + q 1 = p 1 + q 1 = 1, d W (δ(φ), Z ) 2/π 1 2c 1 φ 2 L 2 (R d,dx) + zc 2 φ 4 L 2 (R d,dx) + c 2A, where A := φ 3 L 3 (R d,dx) + z 2/π φ 2 L 2 (R d,dx) 1 e ϕ L 1 (R d,dx) + 2z φ 2 L p (R d,dx) φ L q (R d,dx) 1 e ϕ L 1 (R d,dx) and c 1 := + z 2 φ L p p (R d,dx) φ L p q (R d,dx) φ L q (R d,dx) 1 e ϕ 2 L 1 (R d,dx) z 1 + z 1 e ϕ, c 2 := L 1 (R d,dx) z 2 exp( z 1 e ϕ L 1 (R d,dx) ). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

13 EAMPLE Define ϕ(x) := log(1 r d e ρ r (x)), r > 0, e < e, x R d where ρ r (x) := r d ρ(x/r) is the classical mollifier, i.e. ρ(x) := 1{ x 1}e ( x 2 1) 1. Then 1 e ϕ L 1 (R d,dx) = e r d. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

14 EAMPLE Let µ be the Strauss process, i.e. take ϕ(x) := ( log ν)1{ x r}, ν (0, 1), r > 0, x R d. Then 1 e ϕ L 1 (R d,dx) = (1 ν)α dr d, where α d denotes the volume of the unit ball of R d. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

15 A QUANTITATIVE CLT FOR S.I.F.R. GIBBS POINT PROCESSES Assume µ n, n 1, stationary Gibbs point process with z n > 0 and ϕ n 0 with compact support, z n z > 0 and 1 e ϕ n L 1 (R d,dx) 0. In addition, assume that {φ n } n 1 is a sequence of integrable and square integrable functions such that, for some p, q, p, q > 1 with p 1 + q 1 = p 1 + q 1 = 1, φ n 2 L 2 (R d,dx) z 1, φ n L 3 (R d,dx) 0 φ 2 n L p (R d,dx) φ n L q (R d,dx) 1 e ϕn L 1 (R d,dx) 0 φ n L p p (R d,dx) φ n L p q (R d,dx) φ n L q (R d,dx) 1 e ϕn 2 L 1 (R d,dx) 0. Then d W (δ (n) (φ n ), Z ) 2/π + c (n) 2 A n c (n) 1 φ n 2 L 2 (R d,dx) + z nc (n) 2 φ n 4 L 2 (R d,dx) Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

16 EAMPLE {z n } n 1 sequence of positive numbers converging to z > 0, ϕ n, n 1, mollifiers or Strauss with r = n 1, and φ n (x) := 1 Kn (x)(zl(k n )) 1/2, n 1, x R d l Lebesgue measure, K n R d with l(k n ). We have the quantitative CLT d W (δ (n) (φ n ), Z ) 2/π 1 2z 1 c (n) 1 + z 2 z n c (n) 2 + c (n) 2 A n 0 where and c (n) 1 = z n (1 + αz n n d ) 1, c (n) 2 = z n (2 e αz nn d ) 1 A n =z 3/2 l(k n ) α( 2/πz 1 + 2z 3/2 l(k n ) 1 2 )zn n d + α 2 z 3/2 z 2 n l(k n ) 1 2 n 2d. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

17 EAMPLE (CONTINUED) If e.g. z n = z and l(k n ) n d, elementary computations show that the bound is asymptotically equivalent to ( ) 1 2αz + n d/2 z π Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

18 PROOF OF THE EPLICIT BOUND FOR S.I.F.R. GIBBS POINT PROCESSES By the general bound for Gibbs point processes we have d W (δ(φ), Z ) 2/π 1 2E[π 0 ] φ 2 L 2 (R d,dx) + ze[π 0] φ 4 L 2 (R d,dx) + E[π 0 ] φ 3 L 3 (R d,dx) + z 2/πE[π 0 ] φ ( φ 1 e ϕ ) L 1 (R d,dx) + 2zE[π 0 ] φ 2 ( φ 1 e ϕ ) L 1 (R d,dx) + z 2 E[π 0 ] φ ( φ 1 e ϕ ) 2 L 1 (R d,dx). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

19 PROOF OF THE EPLICIT BOUND FOR S.I.F.R. GIBBS POINT PROCESSES (CONTINUED) We conclude the proof using Hölder s inequality, f g L p (R d,dx) f L 1 (R d,dx) g L p (R d,dx) and that for s.i.f.r. Gibbs point processes one has the crucial estimates c 1 E[π 0 ] c 2, Stucki and Schuhmacher (2014). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

20 INTEGRATION BY PARTS FORMULAS FOR POINT PROCESSES Peccati et al. (2010) used the IBP formula of the Malliavin calculus on the Poisson space due to Nualart and Vives (1990). In Privault and T. (2013), for the purpose of probability approximation of Poisson functionals, we proposed an IBP formula based on the Clark-Ocone covariance representation. I cite also the IBP formula on the Poisson space due to Albeverio, Kondratiev and Röckner (1998) (construction of diffusion) and the one in Privault and T. (2011), which holds for general finite point processes (density estimation). These latter two formulas involve "gradients" and "divergences" (very) different from those in Nualart and Vives (1990) and are difficult to use for probability approximation. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

21 PROOF OF THE GENERAL BOUND: THE INTEGRATION BY PARTS FORMULA For all F : Γ R and φ : R such that φ(x) E[π x ] σ(dx) < and we have φ(x) E[ D x F π x ] σ(dx) <, φ(x) E[ F π x ] σ(dx) <, [ ] E φ(x)d x F π x σ(dx) = E[F δ(φ)]. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

22 PROOF OF THE GENERAL BOUND: THE INTEGRATION BY PARTS FORMULA (CONTINUED) The following "sample path" identity holds δ x (φdf) = F (x)δ x (φ) δ x (φf ) φ(x)d x F (x)π x (x) σ(dx), where δ x (φdf) := φ(x)d x F (x \ {x}) x x φ(x)d x F (x)π x (x) σ(dx) and the term δ x (φf ) is defined similarly. The claim follows taking the mean. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

23 PROOF OF THE GENERAL BOUND: THE STEIN METHOD It is known that by the Stein equation, for any F L 1 (Γ, µ), one has d W (F, Z ) sup f F W E[f (F ) Ff (F )], where F W denotes the class of twice differentiable functions f such that f 2, f 2/π, f 2. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

24 PROOF OF THE GENERAL BOUND: MALLIAVIN MEETS STEIN Take F = δ(φ), f F W. Following Peccati et al. (2010), we deduce E[f (δ(φ)) δ(φ)f (δ(φ))] [ = E f (δ(φ)) φ x D x f (δ(φ))π x σ(dx)] [ = E f (δ(φ)) φ(x) ( f (δ(φ))d x δ(φ) + R(D x δ(φ)) ) π x σ(dx)] [ 1 ] 2/πE φ(x)d x δ(φ)π x σ(dx) [ ] + E φ(x) D x δ(φ) 2 π x σ(dx). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

25 CONCLUSION OF THE PROOF OF THE GENERAL BOUND Taking the supremum over f, we have d W (δ(φ), Z ) [ 1 ] 2/πE φ(x)d x δ(φ)π x σ(dx) [ ] + E φ(x) D x δ(φ) 2 π x σ(dx). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

26 TOTAL VARIATION DISTANCE Take F : Γ N and let Po(λ) be a Poisson random variable with mean λ > 0 defined on the probability space (Ω, F, P). The total variation distance between F and Po(λ) is defined by d TV (F, Po(λ)) := sup µ(f A) P(Po(λ) A). A N Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

27 POISSON APPROIMATION: GENERAL BOUND Assume φ : N such that φ(x)e[π x ] σ(dx) < and set N x (φ) := x x φ(x). Then d TV (N(φ), Po(λ)) 1 e λ φ(x)(φ(x) 1)E[π x ] σ(dx) λ + 1 e λ λ 2 (φ(x)) 2 (φ(x) 1)E[π x ] σ(dx) ( ) 2 + min 1, φ(x)φ(y)(e[π x π y ] E[π x ]E[π y ]) σ(dx)σ(dy) λe 2 where λ := E[N(φ)] = φ(x)e[π x ] σ(dx). Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

28 POISSON CASE If µ is a Poisson process with mean measure σ and φ L 1 (, σ) and N-valued, then d TV (N(φ), Po(λ)) 1 e λ φ(x)(φ(x) 1) σ(dx) λ where + 1 e λ λ 2 Peccati (2011). (φ(x)) 2 (φ(x) 1) σ(dx), λ := φ(x) σ(dx), Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

29 AN EPLICIT BOUND FOR S.I.F.R. GIBBS POINT PROCESSES µ stationary Gibbs point process with z > 0 and ϕ 0 with compact support. Suppose φ L 1 (R d, dx) and N-valued. Then, for any c [c 1, c 2 ], ( ) d TV (N(φ), Po(c φ 1 )) B(1 e φ 1c 2 1 e φ 1c 2 ) + C ( ) 2 + φ 1 min 1, zc 2 (c 1 ) φ 1 c 1 e 2 + φ 1 (c 2 c 1 ), where B := φ 1 1 φ(φ 1) 1, C := φ 2 1 φ2 (φ 1) 1 and c 1, c 2 are as above. c 1 Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

30 A QUANTITATIVE POISSON LIMIT THEOREM FOR S.I.F.R. GIBBS POINT PROCESSES Assume µ n, n 1, stationary Gibbs with z n > 0 and ϕ n 0 with compact support, z n z > 0, 1 e ϕ n L 1 (R d,dx) 0. Assume {φ n } n 1 N-valued and integrable functions such that Then max{ φ n 1, φ n 2 2, φ n 3 3 } γ (0, ). ( d TV (N (n) (φ n ), Po(zγ)) B n (1 e φ n 1 c (n) 1 e φ n 1 c (n) 2 2 ) + Cn + φ n 1 min ( 1, 2 c (n) 1 φ n 1 e { + max zγ c (n) 1 φ n 1, zγ c (n) 2 φ n 1 c (n) 1 ) z n c (n) 2 (c (n) 1 )2 + φ n 1 (c (n) 2 c (n) 1 ) } 0. ) Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

31 EAMPLE {z n } n 1 sequence of positive numbers converging to z > 0, ϕ n, n 1, mollifiers or Strauss with r = n 1, and φ n (x) := 1 Kn (x), n 1, x R d K n R d with l(k n ) γ > 0, l Lebesgue measure. We have the quantitative Poisson Limit Theorem ( d TV (N (n) 2 (1 Kn ), Po(zγ)) l(k n ) min 1, c (n) 1 l(k n)e + l(k n )(c (n) 2 c (n) 1 ) + max { zγ c (n) 1 l(k n), zγ c (n) 2 l(k n) ) z n c (n) 2 (c (n) 1 )2 } 0. Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

32 EAMPLE (CONTINUED) If e.g. z n = z and l(k n ) γ + n d, elementary computations show that the bound is asymptotically equivalent to γ ( ) αz 3 2 min 1, n d/2. γez Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33

33 MANY, MANY THANKS! Giovanni Luca Torrisi (CNR) Approximation of point processes May / 33