rmh.default

0th

Percentile

Simulate Point Process Models using the Metropolis-Hastings Algorithm.

Generates a random point pattern, simulated from a chosen point process model, using the Metropolis-Hastings algorithm.

Usage
rmh.default(model, start, control, verbose=TRUE, ...)
Arguments
model
Data specifying the point process model that is to be simulated.
start
Data determining the initial state of the algorithm.
control
Data controlling the iterative behaviour and termination of the algorithm.
verbose
Logical flag indicating whether to print progress reports.
...
Further arguments which will be passed to any trend functions in model.
Details

This function generates simulated realisations from any of a range of spatial point processes, using the Metropolis-Hastings algorithm. It is the default method for the generic function rmh.

This function executes a Metropolis-Hastings algorithm with birth, death and shift proposals as described in Geyer and Moller (1994).

The argument model specifies the point process model to be simulated. It is either a list, or an object of class "rmhmodel", with the following components:

[object Object],[object Object],[object Object],[object Object],[object Object] For full details of these parameters, see rmhmodel.

The argument start determines the initial state of the Metropolis-Hastings algorithm. It is either a list, or an object of class "rmhstart", with the following components:

[object Object],[object Object],[object Object] For full details of these parameters, see rmhstart.

The third argument control controls the simulation procedure, iterative behaviour, and termination of the Metropolis-Hastings algorithm. It is either a list, or an object of class "rmhcontrol", with components: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object] For full details of these parameters, see rmhcontrol.

It is possible to simulate the model conditionally upon the number of points, or in the case of multitype processes, upon the number of points of each type. To condition upon the total number of points, set control$p (the probability of a shift) equal to 1. The number of points is then determined by the starting state, which may be specified either by setting start$n.start to be a scalar, or by setting the initial pattern start$x.start.

To condition upon the number of points of each type, set control$p equal to 1 and control$fixall to be TRUE. The number of points is then determined by the starting state, which may be specified either by setting start$n.start to be an integer vector, or by setting the initial pattern start$x.start. When we simulate conditionally, no expansion of the window is possible.

Value

  • A point pattern (an object of class "ppp", see ppp.object).

    The returned value has an attribute info containing modified versions of the arguments model, start, and control which together specify the exact simulation procedure.

References

Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283 -- 322.

Diggle, P. J. (2003) Statistical Analysis of Spatial Point Patterns (2nd ed.) Arnold, London.

Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 -- 212.

Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction point processes. Biometrika 74, 763 -- 770.

Geyer, C.J. and M{ller, J. (1994) Simulation procedures and likelihood inference for spatial point processes. Scandinavian Journal of Statistics 21, 359--373.

Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80. Pages 79--140. }

Warnings
{

There is never a guarantee that the Metropolis-Hastings algorithm has converged to its limiting distribution.

If start$x.start is specified then expand is set equal to 1 and simulation takes place in x.start$window. Any specified value for expand is simply ignored.

The presence of both a component w of model and a non-null value for x.start$window makes sense ONLY if w is contained in x.start$window.

For multitype processes make sure that, even if there is to be no trend corresponding to a particular type, there is still a component (a NULL component) for that type, in the list. }

rmh, rmh.ppm, ppp, ppm, Strauss, Softcore, StraussHard, MultiStrauss, MultiStraussHard, DiggleGratton

Extensions
{ The argument model$cif matches the name of a Fortran subroutine which calculates the conditional intensity function for the model. It is intended that more options will be added in the future. The very brave user could try to add her own. Note that in addition to writing Fortran code for the new conditional intensity function, the user would have to modify the code in the files cif.f and rmh.default.R appropriately. (And of course re-install the spatstat package so as to update the dynamically loadable shared object spatstat.so.)

Note that the lookup conditional intensity function permits the simulation (in theory, to any desired degree of approximation) of any pairwise interaction process for which the interaction depends only on the distance between the pair of points. } nr <- 1e5 nv <- 5000 nr <- 10 nv <- 5 set.seed(961018) # Strauss process. mod01 <- list(cif="strauss",par=c(beta=2,gamma=0.2,r=0.7), w=c(0,10,0,10)) X1.strauss <- rmh(model=mod01,start=list(n.start=80), control=list(nrep=nr,nverb=nv)) # Strauss process, conditioning on n = 80: X2.strauss <- rmh(model=mod01,start=list(n.start=80), control=list(p=1,nrep=nr,nverb=nv)) # Strauss process equal to pure hardcore: mod02 <- list(cif="strauss",par=c(beta=2,gamma=0,r=0.7),w=c(0,10,0,10)) X3.strauss <- rmh(model=mod02,start=list(n.start=60,iseed=c(42,17,69)), control=list(nrep=nr,nverb=nv)) # Strauss process in a polygonal window. x <- c(0.55,0.68,0.75,0.58,0.39,0.37,0.19,0.26,0.42) y <- c(0.20,0.27,0.68,0.99,0.80,0.61,0.45,0.28,0.33) mod03 <- list(cif="strauss",par=c(beta=2000,gamma=0.6,r=0.07), w=owin(poly=list(x=x,y=y))) X4.strauss <- rmh(model=mod03,start=list(n.start=90), control=list(nrep=nr,nverb=nv)) # Strauss process in a polygonal window, conditioning on n = 80. X5.strauss <- rmh(model=mod03,start=list(n.start=90), control=list(p=1,nrep=nr,nverb=nv)) # Strauss process, starting off from X4.strauss, but with the # polygonal window replace by a rectangular one. At the end, # the generated pattern is clipped to the original polygonal window. xxx <- X4.strauss xxx$window <- as.owin(c(0,1,0,1)) X6.strauss <- rmh(model=mod03,start=list(x.start=xxx), control=list(nrep=nr,nverb=nv)) # Strauss with hardcore: mod04 <- list(cif="straush",par=c(beta=2,gamma=0.2,r=0.7,hc=0.3), w=c(0,10,0,10)) X1.straush <- rmh(model=mod04,start=list(n.start=70), control=list(nrep=nr,nverb=nv)) # Another Strauss with hardcore (with a perhaps surprising result): mod05 <- list(cif="straush",par=c(beta=80,gamma=0.36,r=45,hc=2.5), w=c(0,250,0,250)) X2.straush <- rmh(model=mod05,start=list(n.start=250), control=list(nrep=nr,nverb=nv)) # Pure hardcore (identical to X3.strauss). mod06 <- list(cif="straush",par=c(beta=2,gamma=1,r=1,hc=0.7), w=c(0,10,0,10)) X3.straush <- rmh(model=mod06,start=list(n.start=60, iseed=c(42,17,69)), control=list(nrep=nr,nverb=nv)) # Soft core: par3 <- c(0.8,0.1,0.5) w <- c(0,10,0,10) mod07 <- list(cif="sftcr",par=c(beta=0.8,sigma=0.1,kappa=0.5), w=c(0,10,0,10)) X.sftcr <- rmh(model=mod07,start=list(n.start=70), control=list(nrep=nr,nverb=nv)) # Multitype Strauss: beta <- c(0.027,0.008) gmma <- matrix(c(0.43,0.98,0.98,0.36),2,2) r <- matrix(c(45,45,45,45),2,2) mod08 <- list(cif="straussm",par=list(beta=beta,gamma=gmma,radii=r), w=c(0,250,0,250)) X1.straussm <- rmh(model=mod08,start=list(n.start=80), control=list(ptypes=c(0.75,0.25),nrep=nr,nverb=nv)) # Multitype Strauss conditioning upon the total number # of points being 80: X2.straussm <- rmh(model=mod08,start=list(n.start=80), control=list(p=1,ptypes=c(0.75,0.25),nrep=nr, nverb=nv)) # Conditioning upon the number of points of type 1 being 60 # and the number of points of type 2 being 20: X3.straussm <- rmh(model=mod08,start=list(n.start=c(60,20)), control=list(fixall=TRUE,p=1,ptypes=c(0.75,0.25), nrep=nr,nverb=nv)) # Multitype Strauss hardcore: rhc <- matrix(c(9.1,5.0,5.0,2.5),2,2) mod09 <- list(cif="straushm",par=list(beta=beta,gamma=gmma, iradii=r,hradii=rhc),w=c(0,250,0,250)) X.straushm <- rmh(model=mod09,start=list(n.start=80), control=list(ptypes=c(0.75,0.25),nrep=nr,nverb=nv)) # Multitype Strauss hardcore with trends for each type: beta <- c(0.27,0.08) tr3 <- function(x,y){x <- x/250; y <- y/250; exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6) } # log quadratic trend tr4 <- function(x,y){x <- x/250; y <- y/250; exp(-0.6*x+0.5*y)} # log linear trend mod10 <- list(cif="straushm",par=list(beta=beta,gamma=gmma, iradii=r,hradii=rhc),w=c(0,250,0,250), trend=list(tr3,tr4)) X1.straushm.trend <- rmh(model=mod10,start=list(n.start=350), control=list(ptypes=c(0.75,0.25), nrep=nr,nverb=nv)) # Multitype Strauss hardcore with trends for each type, given as images: x <- seq(0,250,length=51) xy <- expand.grid(x=x,y=x) i1 <- im(matrix(tr3(xy$x,xy$y),nrow=51),x,x) i2 <- im(matrix(tr4(xy$x,xy$y),nrow=51),x,x) mod11 <- list(cif="straushm",par=list(beta=beta,gamma=gmma, iradii=r,hradii=rhc),w=c(0,250,0,250), trend=list(i1,i2)) X2.straushm.trend <- rmh(model=mod11,start=list(n.start=350), control=list(ptypes=c(0.75,0.25),expand=1, nrep=nr,nverb=nv)) # Diggle, Gates, and Stibbard: mod12 <- list(cif="dgs",par=c(beta=3600,rho=0.08),w=c(0,1,0,1)) X.dgs <- rmh(model=mod12,start=list(n.start=300), control=list(nrep=nr,nverb=nv)) # Diggle-Gratton: mod13 <- list(cif="diggra", par=c(beta=1800,kappa=3,delta=0.02,rho=0.04), w=square(1)) X.diggra <- rmh(model=mod13,start=list(n.start=300), control=list(nrep=nr,nverb=nv)) # Geyer: mod14 <- list(cif="geyer",par=c(beta=1.25,gamma=1.6,r=0.2,sat=4.5), w=c(0,10,0,10)) X1.geyer <- rmh(model=mod14,start=list(n.start=200), control=list(nrep=nr,nverb=nv)) # Geyer; same as a Strauss process with parameters # (beta=2.25,gamma=0.16,r=0.7): mod15 <- list(cif="geyer",par=c(beta=2.25,gamma=0.4,r=0.7,sat=10000), w=c(0,10,0,10)) X2.geyer <- rmh(model=mod15,start=list(n.start=200), control=list(nrep=nr,nverb=nv)) mod16 <- list(cif="geyer",par=c(beta=8.1,gamma=2.2,r=0.08,sat=3)) data(redwood) X3.geyer <- rmh(model=mod16,start=list(x.start=redwood), control=list(periodic=TRUE,nrep=nr,nverb=nv)) # Geyer, starting from the redwood data set, simulating # on a torus, and conditioning on n: X4.geyer <- rmh(model=mod16,start=list(x.start=redwood), control=list(p=1,periodic=TRUE,nrep=nr,nverb=nv))

# Lookup (interaction function h_2 from page 76, Diggle (2003)): r <- seq(from=0,to=0.2,length=101)[-1] # Drop 0. h <- 20*(r-0.05) h[r<0.05] 0="" <-="" h[r="">0.10] <- 1 mod17 <- list(cif="lookup",par=list(beta=4000,h=h,r=r),w=c(0,1,0,1)) X.lookup <- rmh(model=mod17,start=list(n.start=100), control=list(nrep=nr,nverb=nv)) # Strauss with trend tr <- function(x,y){x <- x/250; y <- y/250; exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6) } beta <- 0.3 gmma <- 0.5 r <- 45 mod17 <- list(cif="strauss",par=c(beta=beta,gamma=gmma,r=r),w=c(0,250,0,250), trend=tr3) X1.strauss.trend <- rmh(model=mod17,start=list(n.start=90), control=list(nrep=nr,nverb=nv)) [object Object],[object Object] spatial

Aliases
  • rmh.default
Documentation reproduced from package spatstat, version 1.7-11, License: GPL version 2 or newer

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