Jesper Møller ) and Kateřina Helisová )

Transcription

1 Jesper Møller ) and ) ) Aalborg University (Denmark) ) Czech Technical University/Charles University in Prague 5 th May 2008

2 Outline 1. Describing model 2. Simulation 3. Power tessellation of a union of discs 4. Markov property of the model 5. Data 6. Statistical inference 7. Nowadays work

3 Point processes Definition Consider N the system of locally finite subsets of R d with the σ-algebra N = σ({x N : (x A) = m} : A B, m N 0 ). A point process X defined on R d is a measurable mapping from some probability space (Ω, F, P ) to (N, N ). Definition A locally finite, diffusion measure µ on B satisfying µ(a) = EX(A) for all A B is called the intensity measure. Definition If there exists a function ρ(x) for x R d such that µ(a) = ρ(x)dx, then ρ(x) is called the intensity function. A Definition A point process is called finite point process if µ(r d ) <.

4 Poisson point process Definition The Poisson process Y is the process which satisfies: for any finite collection {A n } of disjoint sets in R d, the numbers of points in these sets, Y (A n ), are independent random variables, for each A R d such that µ(a) <, Y (A) has Poisson distribution with parameter µ(a), i.e. P [Y (A) = k] = µ(a)k e k, where µ k! is the intensity measure.

5 Point process given by the density with respect to Poisson process Let Y be the Poisson process with an intensity measure µ. For F N, denote Π(F ) = P (Y F ). Definition A point process X is given by density f with respect to the Poisson process Y if P (X F ) = f(x)π(dx). F

6 Denoting b = b(z, r) a disc with centre in z R 2 and radius r (0, ), we have a process of discs b i = b(z i, r i ). Then, we identify b with the point x = (z, r) in R 2 (0, ) and the process of discs b i = b(z i, r i ) with a point process on R 2 (0, ). The reference point process: A Poisson process Y (so that the reference Boolean model is the random set given by the union of discs in Y ) with intensity measure ρ(z) dz Q(dr) on S (0, ), where S R 2 is bounded. : The process X absolutely continuous with respect to the reference Poisson process Y, and given by density f(x) for a finite configurations x = {x 1,..., x n }.

7 Exponential family model General form of the density: f θ (x) = exp (θ T (U x )) /c θ Set T = (A, L, χ, N ic, N h, N bv ), where A = A(U x )...the area L = L(U x )...the perimeter χ = χ(u x )...the Euler-Poincaré characteristic N h = N h (U x )...the number of holes N ic = N ic (U x )...the number of isolated cells N bv = N bv (U x )...the number of boundary vertices, i.e. the density is of the form f θ (x) = 1 c θ exp (θ 1 A(U x ) + θ 2 L(U x ) + θ 3 χ(u x ) + θ 4 N h (U x )+ +θ 5 N ic (U x ) + θ 6 N bv (U x )).

8 Some simulated results A power tessellation of a realization of the reference Poisson process with Q the uniform distribution on the interval [0, 2], ρ(u) = 0.2 on a rectangular region S = [0, 30] [0, 30], and ρ(u) = 0 outside S (left) and A-interaction model with parameters θ 1 = 0.1 (middle), resp. θ 1 = 0.1 (right).

9 Some simulated results (A, L, N cc )-interaction process, where N cc (U x ) is the number of connected components, with parameters (0.6, 1, 1) (left), (0.6, 1, 2) (middle) and (0.6, 1, 5) (right).

10 Papangelou conditional intensity Definition For finite x S (0, ) and v S (0, ) \ x, Papangelou conditional intensity is defined as λ θ (x, v) = f θ (x {v})/f θ (x). Denoting W = A, L,... and defining W (x, v) = W (x v) W (x), we get λ θ (x, v) = exp (θ 1 A(x, v) + θ 2 L(x, v) + + θ 6 N bv (x, v)).

11 MCMC algorithm 1. Suppose that in time t, we have a configuration x t = {x 1,..., x n } 2. Proposal in time t + 1: (a) with probability 1/2, the proposal is x t {x n+1 } i. we accept the proposal with probability min{1; h(x t, x n+1 )} and set x t+1 = x t {x n+1 } ii. else we set x t+1 = x t (b) else, the proposal is x t \{x i } i. we accept the proposal with probability min{1; 1/h(x t \{x i }, x i )} and set x t+1 = x t \{x i } ii. else x t+1 = x t where h(x t, x n+1 ) = λ θ (x t, x n+1 ) ρ (n+1) and h(x t \{x i }, x i ) = λ θ (x t \{x i }, x i ) S. ρ n S

12 Power tessellation of a union of discs Assume a union of discs U = I b i in the general position. For each disc b i (i I) with ghost sphere s i, let s + i = {(y 1, y 2, y 3 ) s i : y 3 0} denote the corresponding upper hypersphere. For u b i, let y i (u) denote the unique point on s + i those orthogonal projection on R 2 is u. Define C i = {y i (u) : u b i, u y i (u) u y j (u) for u b j, j I}. Denote B i the orthogonal projection of C i on R 2. Definition The system B of all sets B i is called a power tessellation of a union of discs.

13 Power tessellation of a union of discs Left: A configuration of discs in general position. Middle: The upper hemispheres as seen from above. Right: The power tessellation of the union of discs.

14 Usefulness of power tessellation in MCMC algorithm χ(u x ) = N c N ie + N iv calculation of A(U x ): instead of A(U x ) = A(b i ) A(b i1 b i2 )... ( 1) n+1 i we use {i 1,i 2 } A(U x ) = i analogously calculation of L(U x ) B i {i 1,...,i n } A(b i1 b in )

15 Markov property Recall that Papangelou conditional intensity is defined as λ θ (x, v) = f θ (x {v})/f θ (x). Definition The T -interaction process is said to be Markov with respect to a reflexive relation (called neighbourhood) if λ θ (x, v) depends on x only through {u x : u v}, i.e. the neighbours in x to v. Remark: If the relation depends on the configuration x, the process is called nearest neighbour Markov process. Hammersley-Clifford theorem A function f is the density of a Markov process if and only if there exists a function φ such that f(x) = φ(y) and φ(y) = 1 whenever there exist y y x i, y j y such that y i y j.

16 Markov property - in terms of the connected components The density is of the form f θ (x) = 1 exp (θ 1 A(K) + θ 2 L(K) + θ 3 χ(k) + θ 4 N h (K)+ c θ K K(U x ) +θ 5 N ic (K) + θ 6 N bv (K)) where K(U x ) is the set of connected components of U x. Hammersley-Clifford theorem the T -interaction process is a connected component Markov point process.

17 Data Heather dataset first presented in (Diggle, 1981). The image shows the presence of heather (calluna vulgaris) (indicated by black) in a m region at Jädraås, Sweeden.

18 Diggle s model Diggle modeled the presence of heather by a stationary random-disc Boolean model with the centres given by a stationary Poisson process and the radii i.i.d., independent of the centres, with Weibull distribution. Problem: He observed that his fitted model generates more separate patches than in the data. Our model

19 Statistical inference - problems Denote Y the random set and W the observation window. The statistical inference is complicated by the fact that (i) we do not observe the individual discs but only their union within W, (ii) in practice the grains may only approximately be discs and usually only a digital image is observed and the resolution makes it difficult to identify circular structures, (iii) there occurs a problem with edge effects when Y expands outside W.

20 Modified model Recall f θ (x) = exp (θ T (U x )) /c θ. Original statistic is replaced by T = (A, L, χ, N h, N ic, N bv ) T = (A, L, N cc, N h ), i.e. the density is considered in the form f θ (x) = 1 c θ exp (θ 1 A(U x ) + θ 2 L(U x ) + θ 3 N cc (U x ) + θ 4 N h (U x )).

21 Solving of the edge effects problem Split X into X (a), X (b), X (c) corresponding to discs belonging to connected components of U X which are respectively (a) contained in the window W, (b) intersecting both W and its complement W c, (c) contained in W c. Let x (b) denote a realization of X (b), i.e. x (b) is a finite configuration of discs such that K intersects both W and W c for all K K(U x (b)). It holds that conditional on X (b) = x (b), we have that X (a) and X (c) are independent, and the conditional distribution of X (a) depends only on x (b) through V = W U x (b).

22 Solving edge effects problem W Illustrating possible realizations of X (a) (the full circles), X (b) (the dashed circles), and X (c) (the dotted circles). S

23 Solving edge effects problem Heather dataset without the components intersected by the boundary of the observation window. Remark: For the data, A(V )/A(W ) = when A(Y W )/A(W ) = loss of information.

24 Reference process A realization of the reference Boolean model with the intensity ρ = 2.45 and R following the restriction of N(0.26, ) to the interval [0, 0.50].

25 Estimates of parameters Denote f θ (x) = h θ (x)/c θ (i.e. h θ (x) = exp (θ T (U x )) is the unnormalized density). For an observation x, the log likelihood function is given by l(θ) = log h θ (x) log c θ and the score function u(θ) = dl(θ)/dθ = d log h θ (x)/dθ d log c θ /dθ. In the exponential family case, we have l(θ) = θ T (U x ) log c θ and u(θ) = T (U x ) E θ T (U X ). Problem: c θ and E θ T (U X ) have no explicit expression.

26 Estimates of parameters For fixed θ 0, the log likelihood ratio can be approximated by l(θ) l(θ 0 ) = log(h θ (x)/h θ0 (x)) log(c θ /c θ0 ) l(θ) l(θ 0 ) = log(h θ (x)/h θ0 (x)) log 1 n n 1 m=0 h θ (Y m )/h θ0 (Y m ), where Y m are realizations from f θ0 (x) obtained from MCMC simulations. Recall and denote u(θ) = dl(θ)/dθ j(θ) = du(θ)/dθ.

27 Estimates of parameters Newton-Raphson method for maximising the log likelihood: 1. Set ˆθ (0) = θ 0 ; 2. (k + 1)-th iteration is given by ˆθ (k+1) = ˆθ (k) + u θ0,n(ˆθ (k) )j θ0,n(ˆθ (k) ) 1, where and u θ0,n(ˆθ (k) ) = T (U x ) Eˆθ(k),θ 0,n T (U X) n 1 m=0 = T (U x ) T (U Y )hˆθ(m)(y m m )/h θ0 (Y m ) n 1 hˆθ(k)(y m=0 m )/h θ0 (Y m ) j θ0,n(ˆθ (k) ) = Varˆθ(k),θ 0,n T (U X).

28 Estimates of parameters from the data Using the heather data without the components intersected by the boundary of the observation window, we have T = (A, L, N cc, N h ) = (45.6(m 2 ), 204(m), 32, 2). For the reference Boolean model we obtain T = (A, L, N cc, N h ) = (42.7(m 2 ), 220.8(m), 111, 6).

29 Estimates of parameters from the data Simulation: 1. Simulations from θ 0 = (0, 0, 0, 0) 2. After iterations and 100 steps od N-R method we have ˆθ = ( 1.70, 0.38, 0.73, 0.67) 3. Simulations from θ 0 = ( 1.70, 0.38, 0.73, 0.67). n. Repeating the same procedure seven times, we get ˆθ = ( 4.35, 1.02, 2.23, 0.89) Characteristics of the model with parameters ( 4.35, 1.02, 2.23, 0.89): T = (A, L, N cc, N h ) = (46.82(m 2 ), (m), 32, 2).

30 Simulated process A realization of the T -interaction model with the parameters ( 4.35, 1.02, 2.23, 0.89).

31 Nowadays work Test the model against Boolean model and against some other submodels Check the model Distribution of radii

32 Thank you for your attention!