Bayesian Modeling of Inhomogeneous Marked Spatial Point Patterns with Location Dependent Mark Distributions

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1 Inhomogeneous with Location Dependent Mark Distributions Matt Bognar Department of Statistics and Actuarial Science University of Iowa December 4, 2014

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3 A spatial point pattern describes the spatial location of events in a region W location of particles in a fluid or gas location of trees in a forest location of bird nests on an island A mark (or multiple marks) may be associated with each point in the pattern type of particle trunk diameter species of bird

4 Ant Nests (Harkness & Isham 1983) Marked spatial point pattern of n = 62 ant nests in a foot field located in Spain (subset of the original dataset) Messor wasmanni ( ) and Cataglyphis bicolor ( )

5 Complete Spatial Randomness (CSR) Randomly generated 47 and 15 in W in right plot Ant nests on left, CSR on right

6 Ant Nests Observations Higher intensity in lower region Modeling spatial inhomogeneity is no big deal See Ogata & Katsura (1988), Bognar (2005), etc. Within species: Fewer close pairs of points than under CSR This regular spacing is called spatial regularity or spatial inhibition Addressed in various papers Between species: Inter-point distances are not-unlike CSR; there are close pairs and distant pairs of points Higher proportion of Cataglyphis bicolor ( ) in lower region Mark distribution appears to depend upon location Ogata & Katsura (1988) addressed this for Poisson processes (i.e. assuming no interaction between points) Spurred current work Can we model interaction between points too?

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8 Setup Let W be a bounded subset of R 2 or R 3 A spatial point process X is a finite random subset of W A realization of X is called a spatial point pattern (SPP) A SPP is denoted by x = {x 1,..., x n } i.e. x consists of n distinct points in W

9 Density Of A Spatial Point Process The density of a spatial point process X (with respect to the unit rate Poisson process) is f (x) def = g(x) Z g(x) is the unnormalized density (or kernel) Z is the normalization constant (or partition function) Z = n=0 e W n! W g(x) dx 1... dx n W When n = 0, interpret the integral to equal 1 Z is typically intractable Assume the unnormalized density g is integrable, i.e. Z < The density f is normalized in the sense that n=0 e W n! W f (x) dx 1... dx n = 1 W

10 Gibbs Point Process Hammersley-Clifford-Ripley-Kelly Theorem (Ripley & Kelly 1977, Baddeley et al. 2004, 2013): A finite Gibbs point process has probability density (with respect to the unit rate Poisson process) of the form f (x) = g(x) Z def = exp [ U(x)] Z where the potential function (or energy function) U(x) = n i=1 n 1 V (1) (x i ) + n i=1 j=i+1 V (2) (x i, x j ) + V (i) = i th order potential (maps sub-configurations with i points into R { }) Normalizer e W Z = g(x) dx 1... dx n n! W W n=0 is typically intractable

11 Special Cases Poisson point processes V (1) is the highest order potential function Homogeneous: V (1) (x i ) = log(κ) (where κ is intensity) Inhomogeneous: V (1) (x i ) = log(κ(x i )) Z is tractable (hence the popularity of the Poisson process) Can not model interaction between points Pairwise interaction point processes V (2) is the highest order interaction (Ripley 1977) Z is intractable Can model interaction between points Triplets process V (3) is the highest order interaction (Geyer 1999) Z is intractable Can model interaction between points Assume V (i) 0 when i 2 = integrability (i.e. Z < ) (Møller et al. 2006) Restricts us to repulsive or inhibitory spatial point processes Better models exist for modeling clustering

12 Selected Literature Baddeley & Turner (2000), Bognar (2005), Bognar & Cowles (2004), Diggle (2003), Diggle et al. (1987, 1994), Geyer (1999), Harkness & Isham (1983), Heikkinen & Penttinen (1999), Mateu & Montes (2001), Møller & Waagepetersen (2004), Ogata & Tanemura (1981, 1984, 1986, 1989), Penttinen (1984), Stoyan & Stoyan (1998), Takacs (1986) and van Lieshout (2000)

13 Marked

14 Marked Let M denote the mark space Define Γ = W M = {(ω 1, ω 2 ) : ω 1 W, ω 2 M} Let (x, m) = (x 1,..., x n, m 1,..., m n ) Density of a marked Gibbs point process has the form f (x, m) = Energy function U(x, m) = g(x, m) Z def = n V (1) (x i, m i ) i=1 n 1 + i=1 j=i+1 + exp [ U(x, m)] Z n V (2) (x i, x j, m i, m j ) Partition function e W Z = n! n=0 Γ g(x, m) d(x 1, m 1)... d(x n, m n) Γ

15 Goulard et al. (1996) Homogeneous marked pairwise interacting point processes with mark dependent intensity Specification: V (1) (x i, m i ) = log [α(m i )] V (2) (x i, x j, m i, m j ) = φ(x i, x j, m i, m j ) V (m) ( ) 0, m 3 α( ) > 0 is the mark chemical activity function (intensity can depend upon mark) φ( ) > 0 is the mark pair potential function Describes interaction between marked points Interaction can depend upon marks Mark distribution does not change with location Intractable normalization constant Z Did maximum pseudo-likelihood analysis Bognar (2008) used a Bayesian framework Computational; natural point and interval estimation

16 Ogata & Katsura (1988) Inhomogeneous marked Poisson point process with location dependent mark distributions Specification: V (1) (x i, m i ) = log [κ(x i )p(m i x i )] V (m) ( ) 0, m 2 κ(x) > 0 is the intensity at location x p( x) is the mark distribution at location x p(m x) dm = 1 (summation in discrete case) M The normalization constant Z reduces to that of the unmarked inhomogeneous case (it is known in closed form) Performed a Bayesian analysis (using splines) Model does not allow for interaction between points

17 Proposed Model Inhomogeneous marked pairwise interacting point processes with location dependent mark distribution Specification: V (1) (x i, m i ) = log [κ(x i )p(m i x i )] V (2) (x i, x j, m i, m j ) = φ(x i, x j, m i, m j ) V (m) ( ) 0, m 3 Intractable normalization constant Z

18 Notes Maximum pseudo-likelihood techniques are popular since avoid intractable Z See Goulard et al. (1996), Jensen & Møller (1991), Baddeley & Turner (2000) Frequentist framework provides for point estimation, but the complex distributional properties of the estimates makes interval estimation difficult Baddeley et al. (2005) formally explores the use of parametric bootstrap techniques Selected literature: Baddeley & Møller (1989), Degenhardt (1999), Fiksel (1984), Ogata & Tanemura (1985), or Stoyan & Penttinen (2000) Lets use a Bayesian approach

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20 We need to model the: inhomogeneous intensity location dependent parameters of the mark distribution A Voronoi tessellation on W partitions the space into tiles; tile i consists of all points in W closer to generating point i than to any other generating point Associate a constant height with each tile Yields a (discontinuous) surface over W Certainly planes, splines, other tessellation techniques, etc. could be used to model the process intensity as well

21 Example of Voronoi Tessellation with 4 tiles ants[, 2]

22 V (1) Intensity κ( ) κ( ) is modeled by a Voronoi tessellation on W Generating points where C κ i C κ = (C κ 1,..., C κ k κ ) W for i = 1,..., k κ Tile heights H κ = (H κ 1,..., H κ k κ ) where H κ i > 0 for i = 1,..., k κ Define γ κ = (C κ, H κ ) Let 2kκ parameters describing κ( ) κ γ κ(x) denote the height of the intensity tile at location x i.e. if point x resides in tile j, then κγ κ(x) = Hj κ

23 V (1) Mark Distribution p( x) Assume p(m x) is indexed by M parameters η p = (η p 1,..., ηp M ) Assume each parameter η p i can spatially vary Model η p i with a Voronoi tessellation on W Generating points C p i = (C p i1,..., C p ik i ) Tile heights H p γ p i i = (H p i1,..., Hp ik i ) = (C p i, H p i ) for i = 1,..., M 2 M i=1 k i parameters Planes, splines, etc. on W could be used as well

24 V (1) p( x) for Ant Nests Let m i = { 0 Messor wasmanni ( ) 1 Cataglyphis bicolor ( ) Let π(x) = probability of at location x Model π(x) with a Voronoi tessellation on W Generating points C p = (C p 1,..., C p k p ) Tile heights H p = (H p 1,..., Hp k p ) Let γ p = (C p, H p ) π γ p(x) denote the height of the tessellation tile at location x i.e. if point x resides in tile j, then let πγ p (x) = H p j Location dependent mark distribution m x, γ p Bern(π γ p(x)) 2k p parameters

25 V (1) Specification for Ant Nests V (1) is indexed by γ = (γ κ, γ p ) First order potential function is [ ] V γ (1) (x i, m i ) = log κ γ κ(x i ) π γ p(x i ) m i (1 π γ p(x i )) 1 m i for i = 1,..., n

26 Since spatial interaction will depend upon species, let (generalization of Strauss 1975) V (2) ψ (x i, x j, m i, m j ) = h 00 I(m i = m j = 0)I( x i x j < b 00 ) + h 01 I(m i m j )I( x i x j < b 01 ) + h 11 I(m i = m j = 1)I( x i x j < b 11 ) xi x j = Euclidean distance between x i and x j ψ = (b00, h 00, b 01, h 01, b 11, h 11) Straussian parameters h.. 0 describes the strength of inhibition between pairs of points h.. > 0 indicates spatial regularity h.. = 0 corresponds to no spatial interaction Interaction distances b.. > 0 describes the distance at which two points cease to interact

27 Let θ = (γ, ψ) is f (x, m θ) = Energy function U(x, m θ) = g(x, m θ) Z(θ) + n i=1 def = V (1) γ (x i, m i ) n 1 n i=1 j=i+1 exp[ U(x, m θ)] Z(θ) V (2) ψ (x i, x j, m i, m j ) Normalizer is an intractable function of the parameters Z(θ) = n=0 e W n! Γ g(x, m θ) d(x 1, m 1)... d(x n, m n) Γ

28 Priors: Let kκ = k p = 4 (these need not be equal); the k s are a sort of smoothing parameter parameters V (1) γ V (2) ψ C1 κ,..., C 4 κ iid Unif (W ) H1 κ,..., Hκ iid 4 Unif (0, 100) C p 1,..., C p iid 4 Unif (W ) H p 1,..., Hp iid 4 Unif (0, 1) parameters h 00, h 01, h 11 iid Unif [0, 100) b ij iid Unif (min( xi x j ), 100) for ij = 00, 01, parameters Let p(θ) = p(γ, ψ) denote the joint prior : Bayesian inference will be based upon MCMC simulations from the full posterior distribution p(θ x, m) f (x, m θ)p(θ)

29 Standard Metropolis-Hastings Update Suppose the current parameter vector at iteration t is θ (t) A singe iteration of the MH algorithm proceeds as follows: 1. Generate θ from some proposal distribution, q(θ θ (t) ) 2. Set θ (t+1) = θ with probability [ α(θ (t), θ ) = min 1, otherwise set θ (t+1) = θ (t) ] f (x, m θ )p(θ )q(θ (t) θ ) f (x, m θ (t) )p(θ (t) )q(θ θ (t) ) [ = min 1, g(x, m θ )Z(θ (t) )p(θ )q(θ (t) θ ) g(x, m θ (t) )Z(θ )p(θ (t) )q(θ θ (t) ) The acceptance probability contains a ratio of intractable normalizing constants ] Z(θ (t) )/Z(θ ) thus we can not do a standard MH algorithm

30 Back when Matt had beautiful, luscious hair... Estimated the intractable ratio using importance sampling, plugged-in the estimate into the acceptance probability, and accepted/rejected obtaining a new importance sampling estimate within each iteration was computationally costly The limiting distribution resembles p(θ x, m) provided importance sampling estimates are good However, limiting distribution is not p(θ x, m) A similar method is suggested by Liang & Jin (2013)

31 Exchange Algorithm (Murray et al. 2006) Exchange algorithm can produce samples from the exact posterior, not an approximation of it Suppose the current parameter vector at iteration t is θ (t) A single iteration of the exchange algorithm proceeds as follows: 1. Generate θ from some proposal distribution, q(θ θ (t) ) 2. Generate a marked spatial point pattern, say (x w, m w ), from f ( θ ) 3. Set θ (t+1) = θ with probability α(θ (t), θ ) [ = min 1, f (x, ] m θ )p(θ )q(θ (t) θ )f (x w, m w θ (t) ) f (x, m θ (t) )p(θ (t) )q(θ θ (t) )f (x w, m w θ ) [ = min 1, g(x, ] m θ )p(θ )q(θ (t) θ )g(x w, m w θ (t) ) g(x, m θ (t) )p(θ (t) )q(θ θ (t) )g(x w, m w θ ) otherwise set θ (t+1) = θ (t) Exchange algorithm can be viewed as a more direct version of the Møller et al. (2006) auxiliary variable algorithm

32 Simulating (x w, m w ) Marked spatial point patterns can be generated using: birth/death algorithm of Geyer & Møller (1994) reversible jump MCMC (RJMCMC) algorithm of Green (1995) (the birth/death algorithm, which preceded RJMCMC, is a special case) Neither algorithm needs the normalization constant Z( ); because we are generating spatial point patterns at a fixed θ, the intractable normalizers Z( ) cancel in the acceptance probability Once the algorithm converges, we could obtain a marked spatial point pattern (x w, m w ) from f ( θ ) and use this to perform a single update in the exchange algorithm Doing updates in this way would yield samples from the exact posterior p(θ x, m)

33 Simulating (x w, m w ): The Catch Both the birth/death and RJMCMC algorithms require burn-in Insufficient burn-in would: 1. yield a marked spatial point pattern (x w, m w ) whose distribution is not f ( θ ) 2. subvert the cancelation of the intractable Z( ) s 3. and yield the wrong limiting distribution

34 Simulating (x w, m w ): Burn-in Period Lets use a very conservative (long) burn-in period Trace plot of the total energy U during the burn-in period of the birth/death algorithm for generating (x w, m w ) from f ( θ ) (starting from a Poisson spatial point pattern) Energy Iteration Used a 5,000 iteration burn-in period for the birth/death algorithm Very computational Every iteration of the exchange algorithm requires running a new birth/death (or RJMCMC) algorithm, with sufficient burn-in, to obtain (x w, m w ) from f ( θ )

35 Simulating (x w, m w ): Notes Tracking the total energy U( ) only provides rough guidance on convergence Insufficient burn-in of the birth/death (RJMCMC) algorithm typically yields an over-dispersed posterior distribution (and incorrect point estimates) Burn-in period of 50 iterations yielded over-dispersed posterior Burn-in period of 500 iterations yielded posterior inferences similar to the 5,000 iteration burn-in period What about perfect (exact) sampling? Appears much too slow to use within the exchange algorithm Appears much more difficult to implement Plus Matt is too lazy to code this Propp & Wilson (1996), van Lieshout & Stoica (2006), Møller (2001), Berthelsen & Møller (2002, 2003)

36 Proposals Within each iteration of the exchange algorithm, a parameter was randomly selected (random scan algorithm, one-at-a-time) Uniform random walk proposals Variance of the proposals was tuned to obtain acceptance rates from 20-50%

37 Used UI Neon Cluster (LT node) Coding in C++ (delicate, tedious, but readable) Largest computational burden: the ( n 2) pairwise Euclidean distances needed to evaluate V (2) ψ Must be computed (or updated) for every iteration of the birth/death (or RJMCMC) algorithm that simulates (x w, m w ) from f ( θ ) (could do a subset of distances if we smartly update; more prone to coding errors) Used OpenMP and the Intel MKL library to parallelize across 16 Intel Xeon processors and an Intel Phi co-processor (with 240 cores) Parallelization reduced computing time only 50% or so Lots of inter-processor communication Small dataset (n = 62) and small simulated datasets (x w, m w ) limit speed gains The exchange algorithm was run for 50,000 iterations (following a 5,000 iteration burn-in period) Animation

38 : κ γ κ(x) Marginal posterior mean of intensity κ γ κ(x) (50 50 grid) Unlike a Poisson process, the intensity is not proportional to the density of points Voronoi tessellation is crude; posterior mean is surprisingly smooth

39 : π γ p(x) Marginal posterior mean of π γ p(x) (probability of Cataglyphis bicolor ) under pairwise interaction model

40 : V (2) ψ Parameters mean and 95% credible set for the V (2) ψ parameters Mean 95% Cred. Set Mean 95% Cred. Set b (37.06, 58.76) h (0.256, 0.920) b (8.70, 71.93) h (0.011, 1.245) b (32.04, 49.03) h (1.060, 5.914) Messor wasmanni ( ) exhibit moderate regularity (ĥ00 = 0.556) Cataglyphis bicolor ( ) exhibit strong repulsion (ĥ11 = 2.628) Repulsion of the between-species nest locations is quite weak (not unlike CSR) (ĥ01 = 0.285) Other authors fixed b.. = 45 agree with Takacs & Fiksel (1986) and Särkkä (1993) (used fixed b s)

41 Big Question: Does Modeling the Interaction Matter? Marginal posterior mean of π γ p(x) (probability of Cataglyphis bicolor ) under non-interaction (Poisson) model (V (i) 0 for i 2)

42 Why the difference? Cataglyphis ( ) are strongly inhibitory Messor ( ) are better able to exist near each other Thus, process must push harder to yield a Cataglyphis nest ( ) near the bottom of W Hence, the underlying probability of a Cataglyphis nest ( ), π γ p( ), must be higher near the bottom of W

43 Mini

44 : Dataset was simulated on the square region First order potential: κ(x)=0.001 p(m x)=n(1.5,0.1) κ(x)=0.002 p(m x)=n(1.5,0.1) κ(x)=0.001 p(m x)=n(0.5,0.1) κ(x)=0.002 p(m x)=n(0.5,0.1)

45 : The second order potential function was { V (2) h ψ (x xi x i, x j, m i, m j ) = j s ij b 0 otherwise where s ij = (0.5(m i + m j )) 1 and ψ = (b, h) The interaction distance scales according to the mark For two points with small marks (near 0.5), sij 2 = interaction distance b/2 For two points with large marks (near 1.5), sij 2/3 = interaction distance 3b/2 One large and one small mark will have interaction distance b Used b = 20 and h = 1

46 Simulated Dataset Simulated dataset had 198 points (circle is proportional to mark)

47 : V (1) γ The k = 2 Voronoi tiles were fixed at true locations Uniform priors, RW proposals, 10,000 updates means and 95% credible intervals 500 κ(x)=0.001 κ(x)=0.002 p(m x)=n(1.5,0.1) p(m x)=n(1.5,0.1) 400 κ^(x)= κ^(x)= ( , ) ( , ) µ^= µ^= (1.475,1.530) (1.475,1.530) κ(x)=0.001 κ(x)=0.002 p(m x)=n(0.5,0.1) p(m x)=n(0.5,0.1) κ^(x)= κ^(x)= ( , ) ( , ) µ^= µ^= (0.466,0.498) (0.466,0.498)

48 : V (2) ψ Second order potential parameters: Sim. Val. Post. Mean 95% Cred. Int. b (18.76, 21.12) h (0.596, 1.284)

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50 Discussion It is important to model interaction in marked spatial point patterns to better understand underlying process Bayesian approach: provides inference for all parameters; Frequentist approaches typically fix interaction distance(s) (e.g. b 00, b 01, b 11) provides natural interval estimates (no bootstrap) seems to allow much more complex modeling than Frequentist approaches Can easily get credible sets for intensity and parameters of the mark distribution at any location x W (not shown) Gibbs model is more flexible/general than I thought

51 Open Questions Multiple marks Allow number of tiles k to vary in MCMC sampler Not clear if Exchange algorithm can be modified to handle a dimension change Not clear if RJMCMC can be modified to handle an intractable normalizer Use Pseudo-Marginal algorithm (Andrieu & Roberts 2009) instead of Exchange algorithm Needs more work; mixes poorly in current setup Worked great with diffusions (Stramer & Bognar 2011) Edge effects (literature suggests it is minimal for ants dataset) We assumed interaction parameters ψ didn t change across the space W Want interaction parameters to depend upon location, i.e. ψ(x i, x j )

52 Thank You

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