rmhmodel.default {spatstat.random} | R Documentation |
Build Point Process Model for Metropolis-Hastings Simulation.
Description
Builds a description of a point process model for use in simulating the model by the Metropolis-Hastings algorithm.
Usage
## Default S3 method:
rmhmodel(...,
cif=NULL, par=NULL, w=NULL, trend=NULL, types=NULL)
Arguments
... |
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. |
Details
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
'areaint'
Area-interaction process.
'badgey'
Baddeley-Geyer (hybrid Geyer) process.
'dgs'
Diggle, Gates and Stibbard (1987) process
'diggra'
Diggle and Gratton (1984) process
'fiksel'
Fiksel double exponential process (Fiksel, 1984).
'geyer'
Saturation process (Geyer, 1999).
'hardcore'
Hard core process
'lennard'
Lennard-Jones process
'lookup'
General isotropic pairwise interaction process, with the interaction function specified via a “lookup table”.
'multihard'
Multitype hardcore process
'penttinen'
The Penttinen process
'strauss'
The Strauss process
'straush'
The Strauss process with hard core
'sftcr'
The Softcore process
'straussm'
The multitype Strauss process
'straushm'
Multitype Strauss process with hard core
'triplets'
Triplets process (Geyer, 1999).
It is also possible to specify a hybrid of these interactions
in the sense of Baddeley et al (2013).
In this case, cif
is a character vector containing names from
the list above. For example, cif=c('strauss', 'geyer')
would
specify a hybrid of the Strauss and Geyer models.
The argument par
supplies parameter values appropriate to
the conditional intensity function being invoked.
For the interactions listed above, these parameters are:
- areaint:
-
(Area-interaction process.) A named list with components
beta,eta,r
which are respectively the “base” intensity, the scaled interaction parameter and the interaction radius. - badgey:
-
(Baddeley-Geyer process.) A named list with components
beta
(the “base” intensity),gamma
(a vector of non-negative interaction parameters),r
(a vector of interaction radii, of the same length asgamma
, in increasing order), andsat
(the saturation parameter(s); this may be a scalar, or a vector of the same length asgamma
andr
; all values should be at least 1). Note that because of the presence of “saturation” thegamma
values are permitted to be larger than 1. - dgs:
-
(Diggle, Gates, and Stibbard process. See Diggle, Gates, and Stibbard (1987)) A named list with components
beta
andrho
. This process has pairwise interaction function equal toe(t) = \sin^2\left(\frac{\pi t}{2\rho}\right)
for
t < \rho
, and equal to 1 fort \ge \rho
. - diggra:
-
(Diggle-Gratton process. See Diggle and Gratton (1984) and Diggle, Gates and Stibbard (1987).) A named list with components
beta
,kappa
,delta
andrho
. This process has pairwise interaction functione(t)
equal to 0 fort < \delta
, equal to\left(\frac{t-\delta}{\rho-\delta}\right)^\kappa
for
\delta \le t < \rho
, and equal to 1 fort \ge \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. - fiksel:
-
(Fiksel double exponential process, see Fiksel (1984)) A named list with components
beta
,r
,hc
,kappa
anda
. This process has pairwise interaction functione(t)
equal to 0 fort < hc
, equal to\exp(a \exp(- \kappa t))
for
hc \le t < r
, and equal to 1 fort \ge r
. - geyer:
-
(Geyer's saturation process. See Geyer (1999).) A named list with components
beta
,gamma
,r
, andsat
. The componentsbeta
,gamma
,r
are as for the Strauss model, andsat
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 thresholds
, and parameters\beta
and\gamma
, is the point process in which each pointx_i
in the patternX
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” ofx_i
in the patternX
.If the saturation threshold
s
is infinite, the Geyer process reduces to a Strauss process with interaction parameter\gamma^2
rather than\gamma
. - hardcore:
-
(Hard core process.) A named list with components
beta
andhc
wherebeta
is the base intensity andhc
is the hard core distance. This process has pairwise interaction functione(t)
equal to 1 ift > hc
and 0 ift <= hc
. - lennard:
-
(Lennard-Jones process.) A named list with components
sigma
andepsilon
, wheresigma
is the characteristic diameter andepsilon
is the well depth. SeeLennardJones
for explanation. - multihard:
-
(Multitype hard core process.) A named list with components
beta
andhradii
, wherebeta
is a vector of base intensities for each type of point, andhradii
is a matrix of hard core radii between each pair of types. - penttinen:
-
(Penttinen process.) A named list with components
beta,gamma,r
which are respectively the “base” intensity, the pairwise interaction parameter, and the disc radius. Note thatgamma
must be less than or equal to 1. SeePenttinen
for explanation. (Note that there is also an algorithm for perfect simulation of the Penttinen process,rPenttinen
) - strauss:
-
(Strauss process.) A named list with components
beta,gamma,r
which are respectively the “base” intensity, the pairwise interaction parameter and the interaction radius. Note thatgamma
must be less than or equal to 1. (Note that there is also an algorithm for perfect simulation of the Strauss process,rStrauss
) - straush:
-
(Strauss process with hardcore.) A named list with entries
beta,gamma,r,hc
wherebeta
,gamma
, andr
are as for the Strauss process, andhc
is the hardcore radius. Of coursehc
must be less thanr
. - sftcr:
-
(Softcore process.) A named list with components
beta,sigma,kappa
. Againbeta
is a “base” intensity. The pairwise interaction between two pointsu \neq v
is\exp \left \{ - \left ( \frac{\sigma}{||u-v||} \right )^{2/\kappa} \right \}
Note that it is necessary that
0 < \kappa < 1
. - straussm:
-
(Multitype Strauss process.) A named list with components
-
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 typei
and typej
. -
radii
: A symmetric matrix of interaction radii, with entriesr_{ij}
pertaining to the interaction between typei
and typej
.
-
- straushm:
-
(Multitype Strauss process with hardcore.) A named list with components
beta
andgamma
as forstraussm
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 ofiradii
. -
- triplets:
-
(Triplets process.) A named list with components
beta,gamma,r
which are respectively the “base” intensity, the triplet interaction parameter and the interaction radius. Note thatgamma
must be less than or equal to 1. - lookup:
-
(Arbitrary pairwise interaction process with isotropic interaction.) A named list with components
beta
,r
, andh
, or just with componentsbeta
andh
.This model is the pairwise interaction process with an isotropic interaction given by any chosen function
H
. Each pair of pointsx_i, x_j
in the point pattern contributes a factorH(d(x_i, x_j))
to the probability density, whered
denotes distance andH
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.-
as a vector of values: If
r
is present, thenr
is assumed to give the locations of jumps in the functionH
, while the vectorh
gives the corresponding values of the function.Specifically, the interaction function
H(t)
takes the valueh[1]
for distancest
in the interval[0, r[1])
; takes the valueh[i]
for distancest
in the interval[r[i-1], r[i])
wherei = 2,\ldots, n
; and takes the value 1 fort \ge r[n]
. Heren
denotes the length ofr
.The components
r
andh
must be numeric vectors of equal length. Ther
values must be strictly positive, and sorted in increasing order.The entries of
h
must be non-negative. If any entry ofh
is greater than 1, then the entryh[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
andh
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 ofr
had to be 0.] -
as a stepfun object: If
r
is absent, thenh
must be an object of class"stepfun"
specifying a step function. Such objects are created bystepfun
.The stepfun object
h
must be right-continuous (which is the default usingstepfun
.)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 ofh(t)
for larget
) must be equal to 1.Greatest efficiency is achieved if the jumps (the “knots” of the step function) are equally spaced.
-
For a hybrid model, the argument par
should be a list,
of the same length as cif
, such that par[[i]]
is a list of the parameters required for the interaction
cif[i]
. See the Examples.
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 given as a function:
A trend function may be a function of any number of arguments, but the first two must be the
x,y
coordinates of a point. Auxiliary arguments may be passed to thetrend
function at the time of simulation, via the...
argument tormh
.The function must be vectorized. That is, it must be capable of accepting vector valued
x
andy
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.- trend given as an image:
-
An image (see
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.
Value
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.
Warnings in Respect of “lookup”
For the lookup
cif,
the entries of the r
component of par
must be strictly positive and sorted into ascending order.
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.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner rolfturner@posteo.net
References
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013)
Hybrids of Gibbs point process models and their implementation.
Journal of Statistical Software 55:11, 1–43.
DOI: 10.18637/jss.v055.i11
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.
Fiksel, T. (1984) Estimation of parameterized pair potentials of marked and non-marked Gibbsian point processes. Electronische Informationsverabeitung und Kybernetika 20, 270–278.
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.
See Also
rmh
,
rmhcontrol
,
rmhstart
,
ppm
,
AreaInter
, BadGey
, DiggleGatesStibbard
, DiggleGratton
, Fiksel
, Geyer
, Hardcore
, Hybrid
, LennardJones
, MultiStrauss
, MultiStraussHard
, PairPiece
, Penttinen
, Poisson
, Softcore
, Strauss
, StraussHard
and Triplets
.
Examples
# Strauss process:
mod01 <- rmhmodel(cif="strauss",par=list(beta=2,gamma=0.2,r=0.7),
w=c(0,10,0,10))
mod01
# The above could also be simulated using 'rStrauss'
# Strauss with hardcore:
mod04 <- rmhmodel(cif="straush",par=list(beta=2,gamma=0.2,r=0.7,hc=0.3),
w=owin(c(0,10),c(0,5)))
# Hard core:
mod05 <- rmhmodel(cif="hardcore",par=list(beta=2,hc=0.3),
w=square(5))
# Soft core:
w <- square(10)
mod07 <- rmhmodel(cif="sftcr",
par=list(beta=0.8,sigma=0.1,kappa=0.5),
w=w)
# Penttinen process:
modpen <- rmhmodel(cif="penttinen",par=list(beta=2,gamma=0.6,r=1),
w=c(0,10,0,10))
# Area-interaction process:
mod42 <- rmhmodel(cif="areaint",par=list(beta=2,eta=1.6,r=0.7),
w=c(0,10,0,10))
# Baddeley-Geyer process:
mod99 <- rmhmodel(cif="badgey",par=list(beta=0.3,
gamma=c(0.2,1.8,2.4),r=c(0.035,0.07,0.14),sat=5),
w=unit.square())
# 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 Hardcore:
rhc <- matrix(c(9.1,5.0,5.0,2.5),2,2)
mod08hard <- rmhmodel(cif="multihard",
par=list(beta=beta,hradii=rhc),
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))
# Triplets process:
mod11 <- rmhmodel(cif="triplets",par=list(beta=2,gamma=0.2,r=0.7),
w=c(0,10,0,10))
# 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))
# hybrid model
modhy <- rmhmodel(cif=c('strauss', 'geyer'),
par=list(list(beta=100,gamma=0.5,r=0.05),
list(beta=1, gamma=0.7,r=0.1, sat=2)),
w=square(1))
modhy