# 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
A named list of objects specifying the point process model that is to be simulated, having (some of) the following components: [object Object],[object Object],[object Object],[object Object],[object Object] See Details for details.
start
List of parameters determining the initial state of the algorithm: [object Object],[object Object],[object Object] The parameters n.start and x.start are incompatible; precisely one of them should be specif
control
List of parameters controlling the iterative behaviour and termination of the algorithm: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object] See Details for d
verbose
A logical scalar; if verbose is TRUE then warnings are printed out whenever the storage space alloted to the underlying Fortran code, to hold the generated points, gets increased. If it is FALSE then this process proceeds silently.
...
Further arguments, e.g. to be passed to trend functions.
##### 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 a list with the following components:

[object Object],[object Object] The argument start determines the initial state of the Metropolis-Hastings algorithm. Possible components are [object Object],[object Object],[object Object] The parameters n.start and x.start are incompatible.

The third argument control controls the simulation procedure, iterative behaviour, and termination of the Metropolis-Hastings algorithm. It is a list with components: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

##### Value

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

The returned value has an attribute info consisting of arguments supplied to the function (or default values of arguments which were not explicitly supplied). These are given so that it is possible to reconstruct exactly the manner in which the pattern was generated. The components of info are model, start, and control which in turn are lists:

model=list(cif, par, trend) start=list(n.start,x.start,iseed)

control=list(p=p,q=q,nrep=nrep,expand,periodic, ptypes=ptypes,fixall=fixall)

Note that only one of x.start and x.start appear in in the start list.

##### 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. }

Note
{ It is possible to simulate 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 p (the probability of a shift) equal to 1, and specify n.start to be a scalar (as usual). To condition upon the number of points of each type, set p equal to 1, fixall equal to TRUE, and specify n.start to be a vector of length $nt$ where $nt$ is the number of types.

In these circumstances

• The value ofexpandmust be equal to 1; it defaults to 1, and it is an error to specify a value larger than 1.
• The resulting simulated pattern will have precisely the number of points (of each type) specified byn.start.
}
Warnings in Respect of lookup''
{

The syntax of rmh.default in respect of the lookup cif has changed from the previous release of spatstat (versions 1.4-7 to 1.5-1). Read the Details carefully. In particular it is now required that the first entry of the r component of par be strictly positive. (This is the opposite of what was required in the previous release, which was that this first entry had to be 0.)

It is also now required that the entries of r be sorted into ascending order. (In the previous release it was assumed that the entries of r and h were in corresponding order and the two vectors were sorted commensurately. It was decided that this is dangerous sand unnecessary.)

Note that if you specify the lookup pairwise interaction function via stepfun() the arguments x and y which are passed to stepfun() are slightly different from r and h: length(y) is equal to 1+length(x); the final entry of y must be equal to 1 --- i.e. this value is explicitly supplied by the user rather than getting tacked on internally.

The step function returned by stepfun() must be right continuous (this is the default behaviour of stepfun()) otherwise an error is given. }

Other Warnings
{

There is never a guarantee that the Metropolis-Hastings algorithm has converged to the steady state.

If 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$window. However no checking is done for this. For multitype processes make sure that, even if there is to be no trend corresponding to a particular type, there is still a component (a NULL component) for that type, in the list. No checking is done to verify non-negativity of any specified trend or trends. } rmh, rmh.ppm, ppp, ppm, Strauss, Softcore, StraussHard, MultiStrauss, MultiStraussHard, DiggleGratton Extensions { The argument model$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. } require(spatstat) 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), control=list(nrep=nr,nverb=nv,iseed=c(42,17,69))) # 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), control=list(nrep=nr,nverb=nv,iseed=c(42,17,69))) # 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),tmax=list(1.5,1.65)) 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,tmax=1.5) 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 1.5-4, License: GPL version 2 or newer

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