| rmhmodel.default {spatstat} | R Documentation |
Builds a description of a point process model for use in simulating the model by the Metropolis-Hastings algorithm.
## Default S3 method:
rmhmodel(...,
cif=NULL, par=NULL, w=NULL, trend=NULL, types=NULL)
... |
Ignored. |
cif |
Character string specifying the choice of model |
par |
Parameters of the model |
w |
Spatial window in which to simulate |
trend |
Specification of the trend in the model |
types |
A vector of factor levels defining the possible marks, for a multitype process. |
The generic function rmhmodel takes a
description of a point process model in some format, and
converts it into an object of class "rmhmodel"
so that simulations of the model can be generated using
the Metropolis-Hastings algorithm rmh.
This function rmhmodel.default is the default method.
It builds a description of the point process model
from the simple arguments listed.
The argument cif is a character string specifying the choice of
interpoint interaction for the point process. The current options are
'strauss''straush''sftcr''straussm''straushm''dgs''diggra''geyer''lookup'
The argument par supplies parameter values appropriate to
the conditional intensity function being invoked. These are:
beta,gamma,r which are respectively the ``base''
intensity, the pair-wise interaction parameter and the
interaction radius. Note that gamma must be less than
or equal to 1.
(Note that there is also an algorithm for perfect simulation
of the Strauss process, rStrauss)
beta,gamma,r,hc where beta, gamma,
and r are as for the Strauss process, and hc is
the hardcore radius. Of course hc must be less than
r.
beta,sigma,kappa. Again beta is a ``base''
intensity. The pairwise interaction between two points
u != v is
-(sigma/||u-v||)^(2/kappa)
Note that it is necessary that 0 < kappa <1.
beta:
A vector of ``base'' intensities, one for each possible type.
gamma:
A symmetric matrix of interaction parameters,
with gamma_ij pertaining to the interaction between
type i and type j.
radii:
A symmetric matrix of interaction radii, with
entries r_ij pertaining to the interaction between type
i and type j.
beta and gamma
as for straussm and
two ``radii'' components:
iradii: the interaction radii
hradii: the hardcore radii
which are both symmetric matrices of nonnegative numbers.
The entries of hradii must be less than the
corresponding entries
of iradii.
beta and rho. This process has pairwise interaction
function equal to
e(t) = sin^2((pi * t)/(2 * rho))
for t < rho, and equal to 1 for t >= rho.
beta,
kappa, delta and rho. This process has
pairwise interaction function e(t) equal to 0
for t < delta, equal to
((t-delta)/(rho-delta))^kappa
for delta <= t < rho, and equal to 1 for t >= rho. Note that here we use the symbol kappa where Diggle, Gates, and Stibbard use beta since we reserve the symbol beta for an intensity parameter.
beta, gamma, r, and sat.
The components beta, gamma, r are as for
the Strauss model, and sat is the ``saturation''
parameter. The model is Geyer's ``saturation'' point process
model, a modification of the Strauss process in which
we effectively impose an upper limit (sat) on the number of
neighbours which will be counted as close to a given point.
Explicitly, a saturation point process with interaction radius r, saturation threshold s, and parameters beta and gamma, is the point process in which each point x[i] in the pattern X contributes a factor
beta gamma^min(s,t(x[i],X))
to the probability density of the point pattern, where t(x[i],X) denotes the number of ``r-close neighbours'' of x[i] in the pattern X.
If the saturation threshold s is infinite, the Geyer process reduces to a Strauss process with interaction parameter gamma^2 rather than gamma.
beta, r, and h, or just with components
beta and h.
This model is the pairwise interaction process with an isotropic interaction given by any chosen function H. Each pair of points x[i], x[j] in the point pattern contributes a factor H(d(x[i],x[j])) to the probability density, where d denotes distance and H is the pair interaction function.
The component beta is a
(positive) scalar which determines the ``base'' intensity
of the process.
In this implementation, H must be a step function. It is specified by the user in one of two ways.
r is present, then r is assumed to
give the locations of jumps in the function H,
while the vector h gives the corresponding
values of the function.
Specifically, the interaction function
H(t) takes the value h[1]
for distances t in the interval
[0, r[1]); takes the value h[i]
for distances t in the interval
[r[i-1], r[i]) where
i = 2, ..., n;
and takes the value 1 for t >= r[n].
Here n denotes the length of r.
The components r and h
must be numeric vectors of equal length.
The r values must be strictly positive, and
sorted in increasing order.
The entries of h must be non-negative.
If any entry of h is greater than 1,
then the entry h[1] must be 0 (otherwise the specified
process is non-existent).
Greatest efficiency is achieved if the values of
r are equally spaced.
[Note: The usage of r and h
has changed from the previous usage in spatstat
versions 1.4-7 to 1.5-1, in which ascending order was not required,
and in which the first entry of r had to be 0.]
r is absent, then h must be
an object of class "stepfun" specifying
a step function. Such objects are created by
stepfun.
The stepfun object h must be right-continuous
(which is the default using stepfun.)
The values of the step function must all be nonnegative.
The values must all be less than 1
unless the function is identically zero on some initial
interval [0,r). The rightmost value (the value of
h(t) for large t) must be equal to 1.
Greatest efficiency is achieved if the jumps (the ``knots'' of the step function) are equally spaced.
The optional argument trend determines the spatial trend in the model,
if it has one. It should be a function or image
(or a list of such, if the model is multitype)
to provide the value of the trend at an arbitrary point.
trend function at the time of simulation,
via the ... argument to rmh.
The function must be vectorized.
That is, it must be capable of accepting vector valued
x and y arguments. Put another way,
it must be capable of calculating the trend value at a
number of points, simultaneously, and should return the
vector of corresponding trend values.
im.object)
provides the trend values at a grid of
points in the observation window and determines the trend
value at other points as the value at the nearest grid point.
Note that the trend or trends must be non-negative; no checking is done for this.
The optional argument w specifies the window
in which the pattern is to be generated. If specified, it must be in
a form which can be coerced to an object of class owin
by as.owin.
The optional argument types specifies the possible
types in a multitype point process. If the model being simulated
is multitype, and types is not specified, then this vector
defaults to 1:ntypes where ntypes is the number of
types.
An object of class "rmhmodel", which is essentially
a list of parameter values for the model.
There is a print method for this class, which prints
a sensible description of the model chosen.
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.
Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf
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. 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,
rmhcontrol,
rmhstart,
ppm,
Strauss,
Softcore,
StraussHard,
MultiStrauss,
MultiStraussHard,
DiggleGratton,
PairPiece
# Strauss process:
mod01 <- rmhmodel(cif="strauss",par=c(beta=2,gamma=0.2,r=0.7),
w=c(0,10,0,10))
# The above could also be simulated using 'rStrauss'
# Strauss with hardcore:
mod04 <- rmhmodel(cif="straush",par=c(beta=2,gamma=0.2,r=0.7,hc=0.3),
w=owin(c(0,10),c(0,5)))
# Soft core:
par3 <- c(0.8,0.1,0.5)
w <- square(10)
mod07 <- rmhmodel(cif="sftcr",
par=c(beta=0.8,sigma=0.1,kappa=0.5),
w=w)
# 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 <- rmhmodel(cif="straussm",
par=list(beta=beta,gamma=gmma,radii=r),
w=square(250))
# specify types
mod09 <- rmhmodel(cif="straussm",
par=list(beta=beta,gamma=gmma,radii=r),
w=square(250),
types=c("A", "B"))
# Multitype Strauss hardcore with trends for each type:
beta <- c(0.27,0.08)
ri <- matrix(c(45,45,45,45),2,2)
rhc <- matrix(c(9.1,5.0,5.0,2.5),2,2)
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 <- rmhmodel(cif="straushm",par=list(beta=beta,gamma=gmma,
iradii=ri,hradii=rhc),w=c(0,250,0,250),
trend=list(tr3,tr4))
# 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 <- rmhmodel(cif="lookup",par=list(beta=4000,h=h,r=r),w=c(0,1,0,1))