# 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.

Keywords
spatial
##### 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],[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.

##### 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

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

##### 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. 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{\o}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.

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

• rmh.default
##### Examples
require(spatstat)
nr   <- 1e5
nv  <- 5000
<testonly>nr  <- 10
nv <- 5</testonly>
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,
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))

# 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))
Documentation reproduced from package spatstat, version 1.4-3, License: GPL version 2 or newer

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