Introduction to spatial point processes

Transcription

1 Introduction to spatial point processes Jakob G Rasmussen Department of Mathematics Aalborg University Denmark January 26, /17

2 Before we begin Literature: Book: Jesper Møller og Rasmus P Waagepetersen (2004), Statistical Inference and Simulation for Spatial Point Processes Lecture note: Jakob G Rasmussen (2011), Temporal point processes: the conditional intensity function Student presentations - one per student Student activity - questions/comments are encouraged Materials: Slides and R-code 2/17

3 Inspiration for work on your own There are no exercises - but use the rest of the morning in groups anyway! Go through the slides Do you know what each definition/theorem/formula is used for? Practical use (preliminary analysis, model specification, parameter estimation, model checking) Theoretical use (used in proofs) Check if you understand the proofs Try out and modify the R-code Start out the project work I won t circulate your rooms, but you are welcome to come by my office 3/17

4 Project The primary purpose: the theory Cover everything, or focus on parts you like Include data examples if you like (no time for full statistical analyses) - R has many point process datasets Remember: Quality before quantity 4/17

5 Data Examples: Norwegian spruces & Danish barrows Spruces Barrows Note: specification of observation window very important information about where points do not occur is just as important as information about where the points do occur 5/17

6 Other examples of data One-dimensional point patterns: Positions of car accidents on a highway during a month Times of earthquakes in Japan Two-dimensional point patterns: Positions of cities on a map Positions of farms with mad cow disease in UK Positions of broken wires in an electrical network Three-dimensional point patterns: Positions of stars in the visible part of the universe Positions of copper deposits underground Times and positions and earthquakes in Japan Note: the observation windows are typically subsets of R n, but sometimes the observation windows have complicated shapes - however, usually we look at intervals/rectangles/boxes 6/17

7 Statistical inference for spatial point patterns Objective is to infer structure in spatial distribution of points: interaction between points: regularity or clustering ( random ) inhomogeneity linked to covariates ( systematic ) Spatial point processes are stochastic models for spatial point patterns Clustered Regular Inhomogeneous 7/17

8 A bit of measure theory Borel set: A Borel set B S is any set that can be constructed from open subsets of S using the following: If B B, then S\B B If B i B for i = 1,2,3,, then i=1 B i B Note: S is some arbitrary space, eg R d B is the set of all borel sets B 0 is the set of all bounded borel sets Measure: A measure ν : B [0, ] satisfies: If B i B are disjoint, then ν( i=1 B i) = i=1 ν(b i) Examples of measures: Counting measure: N(B) is the number of points in B Lebesgue measure: B is the area/volume of B Probability measure: Given some distribution on S, Π(B) is the probability of falling in B (in this case Π(S) = 1) 8/17

9 What is a spatial point process? Definitions: 1 a random counting measure N on R d 2 a locally finite random subset X of R d Counting measure: N(A) counts the number of points from X falling in any Borel set A R d Locally finite: #(X A) finite for all bounded Borel sets A R d Equivalent if simple point process (ie no multiple points): N(A) = #(X A) 9/17

10 Simple example of point process: Binomial point process Suppose f is a probability density on S Then X is a binomial point process with n points if X = {x 1,,x n } consists of n iid points x i f Binomial: the number of points in A S is binomially distributed b(n,p) with p = A f(x)dx Example with S = [0,1] [0,1], n = 100 and f(x) = 1 10/17

11 Fundamental example: The Poisson process Assume µ locally finite measure on S R d with intensity ρ (µ(b) = B ρ(u)du) X is a Poisson process with intensity measure µ if for any bounded region B with µ(b) > 0: 1 N(B) po(µ(b)) 2 Given N(B), points in X B iid with density ρ(u), u B (ie given N(B), X B is a binomial point process) Examples on S = [0,1] [0,1]: Homogeneous: ρ = 100 Inhomogeneous: ρ(x, y) 200x 11/17

12 Mean measure and intensity µ is called the mean measure or first order moment measure since EN(B) = µ(b) Infinitesimal interpretation of intensity function: N(A) binary variable (presence or absence of point in A) when A very small Hence ρ(u)a µ(a) = EN(A) P(X has a point in A) 12/17

13 Stationarity and isotropy X on R d is stationary if distribution invariant under translations: X {u +s u X}, s R d X on R d is isotropic if distribution invariant under rotations: X {Ru u X} where R denotes a rotation around the origin Poisson process on R d with constant intensity ρ is both stationary and isotropic 13/17

14 Characterization in terms of void probabilities The void probability is the probability that there are no points in some specified set, ie v(b) = P(N(B) = 0) for B B 0 The distribution of X is uniquely determined by the void probabilities Intuitive proof: consider a very fine subdivision of observation window then at most one point in each cell and probabilities of absence/presence determined by void probabilities Void probabilities are often used in proofs, fx to prove that two point processes are the same 14/17

15 Marked point processes Sometimes mark m u M attached to point u Examples of marked point patterns: Trees: size of tree (diameter at breast height), M = (0, ) Car accidents: type of accident (deaths/no deaths), M = {0,1} Times of earthquakes: position and magnitude of earthquake, M = R 2 (0, ) Marked point process X = {(u,m u ) u Y} where Y point process on S may be viewed as a point process with points in product space S M Multitype point process: Marked point process with finite mark space, eg M = {1,,k} 15/17

16 Point processes in R R-packages for dealing with point processes: Spatial point processes: spatstat Temporal point processes: PtProcess Manuals: wwwspatstatorg/spatstat/doc/spatstatjsspaperpdf cranatr-projectorg/web/packages/ptprocess/ PtProcesspdf Many algorithms implemented for Parameter estimation Simulation Model checking 16/17

17 Overview of the rest of the course Lec 2-3 Poisson processes - basic point process, the starting point of of everything else Lec 4-5 Temporal point processes - point processes on the time line Lec 6-7 Summary Statistics - useful tools for preliminary analysis and model checking Lec 8-9 Cox processes - models for clustered point patterns Lec Markov processes - models for regular point patterns Lec 12 Simulation of point processes - MCMC based simulation Lec Inference - estimation of parameters 17/17