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Models and Simulation Techniques from Stochastic Geometry J I Home Page Title Page Contents JJ II Models and simulation techniques from stochastic geometry J I Page 1 of 163 Wilfrid S. Kendall Department of Statistics, University of Warwick Go Back A course of 5 × 75 minute lectures at the Madison Probability Intern Program, June-July 2003 Full Screen Close Quit Introduction Home Page Stochastic geometry is the study of random patterns, whether of points, line seg- ments, or objects. Title Page Related reading: Stoyan et al. (1995); Contents Barndorff-Nielsen et al. (1999); JJ II Møller and Waagepetersen (2003) and the recent expository essays edited by Møller (2003); J I The Stochastic Geometry and Statistical Applications section of Advances in Applied Probability. Page 2 of 163 Go Back Full Screen Close Quit This screen layout is not very suitable for printing. However, if a printout is desired Home Page then here is a button to do the job! Print Title Page Contents JJ II J I Page 3 of 163 Go Back Full Screen Close Quit Home Page Title Page Contents JJ II J I Page 4 of 163 Go Back Full Screen Close Quit Home Page Title Page Contents Contents JJ II J I 1 A rapid tour of point processes 7 Page 5 of 163 2 Simulation for point processes 45 3 Perfect simulation 75 Go Back 4 Deformable templates and vascular trees 103 Full Screen 5 Multiresolution Ising models 123 Close A Notes on the proofs in Chapter 5 145 Quit Home Page Title Page Contents JJ II J I Page 6 of 163 Go Back Full Screen Close Quit Home Page Title Page Contents Chapter 1 JJ II A rapid tour of point processes J I Page 7 of 163 Contents Go Back 1.1 The Poisson point process .................. 8 1.2 Boolean models ........................ 23 Full Screen 1.3 General definitions and concepts .............. 27 1.4 Markov point processes ................... 33 Close 1.5 Further issues ......................... 40 Quit 1.1. The Poisson point process Home Page The simplest example of a random point pattern is the Poisson point process, which models “completely random” distribution of points in the plane R2, or 3-space R3, Title Page or other spaces. It is a basic building block in stochastic geometry and elsewhere, with many symmetries which typically reduce calculations to computations of area Contents or volume. JJ II J I Page 8 of 163 Go Back Full Screen Close Figure 1.1: Which of these point clouds exhibits interaction? Quit Home Page Definition 1.1 Let X be a random point process in a window W ⊆ Rn defined as a collection of random variables X(A), where A runs through Title Page the Borel subsets of W ⊆ Rn. The point process X is Poisson of intensity λ > 0 if the following are true: Contents 1. X(A) is a Poisson random variable of mean λ Leb(A); 2. X(A1),..., X(An) are independent for disjoint A1,..., An. JJ II Examples: J I stars in the night sky; Page 9 of 163 Go Back patterns used to define collections of random discs; Full Screen ... or space-time structures; Close Brownian excursions indexed by local time(!). Quit • In dimension 1 we can construct X using a Markov chain (“number of points Home Page in [0, t]”) and simulate it using Exponential random variables for inter-point spacings. Higher dimensions are trickier! See Theorem 1.5 below. n Title Page • We can replace λ Leb (and W ⊆ R ) by a nonnegative σ-finite measure µ on a measurable space X , in which case X is an inhomogeneous Poisson point process with intensity measure µ. We should also require µ to be Contents diffuse (P [µ({x}) = 0] = 1 for every point {x}) in order to avoid the point process faux pas of placing than one point onto a single location. JJ II Exercise 1.2 Show that any measurable map will carry one (perhaps J I inhomogeneous) Poisson point process into another, so long as – the image of the intensity measure is a diffuse measure; Page 10 of 163 – and we note that the image Poisson point process is degenerate if the image measure fails to be σ-finite. Go Back Full Screen Exercise 1.3 Show that in particular we can superpose two indepen- Close dent Poisson point processes to obtain a third, since the sum of inde- pendent Poisson random variables is itself Poisson. Quit • It is a theorem that the second, independent scattering, requirement of Defi- Home Page nition 1.1 is redundant! In fact even the first requirement is over-elaborate: Title Page Theorem 1.4 Suppose a random point process X satisfies Contents P [X(A) = 0] = exp (−µ(A)) for all measurable A. Then X is an (inhomogeneous) Poisson point process of intensity mea- JJ II sure µ. J I – This can be proved directly. I find it more satisfying to view the re- Page 11 of 163 sult as a consequence of Choquet capacity theory, since this approach extends to cover a theory of random closed sets (Kendall 1974). Com- pare inclusion-exclusion arguments and “correlation functions” used Go Back in the lattice case in statistical mechanics (Minlos 1999). – We don’t have to check every measurable set:1 it is enough to consider Full Screen sets A running through the finite unions of rectangles! – But we cannot be too constrained in our choice of various A (the family Close of convex A is not enough). 1 Quit Certainly we need check only the BOUNDED measurable sets. • An alternative approach2 lends itself to simulation methods for Poisson point Home Page processes in bounded windows: condition on the total number of points. Title Page Theorem 1.5 Suppose the window W ⊆ Rn has finite area. The fol- lowing constructs X as a Poisson point process of finite intensity λ: Contents – X(W ) is Poisson of mean λ Leb(W ); – Conditional on X(W ) = n, the random variables X(A) are JJ II constructed as point-counts of a pattern of n independent points U1,..., Un uniformly distributed over W . So J I X(A) = # {i : Ui ∈ A} . Page 12 of 163 Go Back Exercise 1.6 Generalize Theorem 1.5 to (inhomogeneous) Poisson Full Screen point processes and to the case of W of infinite area/volume. Close 2The construction described in Theorem 1.5 is the same as the one producing the link via condi- Quit tioning between Poisson and multinomial contingency table models. Home Page Exercise 1.7 Independent thinning. Suppose each point ξ of X, pro- duced as in Theorem 1.5, is deleted independently of all other points Title Page with probability 1 − p(ξ) (so p(ξ) is the retention probability). Show that the resulting thinned random point pattern is an inhomogeneous Poisson point process with intensity measure p(u) Leb( du). Contents JJ II Exercise 1.8 Formulate the generalization of the above exercise J I which uses position-dependent thinning on inhomogeneous Poisson point processes. Page 13 of 163 Go Back Full Screen Close Quit • The process produced by conditioning on X(W ) = n has a claim to being Home Page even more primitive than the Poisson point process. Title Page Definition 1.9 A Binomial point process with n points defined on a window W is produced by scattering n points independently and uni- Contents formly over W . JJ II However you should notice J I – point counts in disjoint subsets are now (perhaps weakly) negatively correlated: the more in one region, the fewer can fall in another; Page 14 of 163 – the total number n of points is fixed, so we cannot extend the notion of a (homogeneous) Binomial point process to windows of infinite mea- sure; Go Back – if you like that sort of thing, the Binomial point process is the simplest representative of a class of point processes (with finite total number of Full Screen points) analogous to canonical ensembles in statistical physics. The Poisson point process relates similarly to grand canonical ensembles Close (grand, because points are indistinguishable and we randomize over the total number of particles). Of course these are trivial cases because there is no interaction between points — but more of that later! Quit Here’s a worry.3 It is easy to prove that the construction of Theorem 1.5 produces Home Page point count random variables which satisfy Definition 1.1 and therefore delivers a Poisson point process. But what about the other way round? Can we travel from Poisson point counts to a random point pattern? Or is there some subtle concealed Title Page random behaviour in Theorem 1.5 going beyond the labelling of points, which cannot be extracted from a point count specification? Contents The answer is that the construction and the point count definition are equivalent. JJ II J I Page 15 of 163 Go Back Full Screen Close 3Perhaps you think this is more properly a neurosis to be dealt with by personal counselling?! Consider that a neurotic concern here is what allows us to introduce measure-theoretic foundations Quit which are most valuable in further theory. Home Page Theorem 1.10 Any Poisson point process given by Definition 1.1 induces a unique probability measure on the space of locally finite point patterns, Title Page when this is furnished with the σ-algebra generated by point counts. Contents Here is a sketch of the proof. Take W = [0, 1)2. For each n = 1, 2, . , make a dyadic dissection of [0, 1)2 into JJ II 4n different sub-rectangles [k2−n, (k + 1)2−n) × [`2−n, (` + 1)2−n).
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