# rmhmodel.default

##### Build Point Process Model for Metropolis-Hastings Simulation.

Builds a description of a point process model for use in simulating the model by the Metropolis-Hastings algorithm.

##### Usage

```
# S3 method for default
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 as`gamma`

, in*increasing*order), and`sat`

(the saturation parameter(s); this may be a scalar, or a vector of the same length as`gamma`

and`r`

; all values should be at least 1). Note that because of the presence of ``saturation'' the`gamma`

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`

and`rho`

. This process has pairwise interaction function equal to $$ e(t) = \sin^2\left(\frac{\pi t}{2\rho}\right) $$ for \(t < \rho\), and equal to 1 for \(t \ge \rho\).- diggra:
(Diggle-Gratton process. See Diggle and Gratton (1984) and Diggle, Gates and Stibbard (1987).) A

**named**list with components`beta`

,`kappa`

,`delta`

and`rho`

. This process has pairwise interaction function \(e(t)\) equal to 0 for \(t < \delta\), equal to $$ \left(\frac{t-\delta}{\rho-\delta}\right)^\kappa $$ for \(\delta \le t < \rho\), and equal to 1 for \(t \ge \rho\). 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`

and`a`

. This process has pairwise interaction function \(e(t)\) equal to 0 for \(t < hc\), equal to $$ \exp(a \exp(- \kappa t)) $$ for \(hc \le t < r\), and equal to 1 for \(t \ge r\).- geyer:
(Geyer's saturation process. See Geyer (1999).) A

**named**list with components`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 \(\gamma\).

- hardcore:
(Hard core process.) A

**named**list with components`beta`

and`hc`

where`beta`

is the base intensity and`hc`

is the hard core distance. This process has pairwise interaction function \(e(t)\) equal to 1 if \(t > hc\) and 0 if \(t <= hc\).- lennard:
(Lennard-Jones process.) A

**named**list with components`sigma`

and`epsilon`

, where`sigma`

is the characteristic diameter and`epsilon`

is the well depth. See`LennardJones`

for explanation.- multihard:
(Multitype hard core process.) A

**named**list with components`beta`

and`hradii`

, where`beta`

is a vector of base intensities for each type of point, and`hradii`

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 that`gamma`

must be less than or equal to 1. See`Penttinen`

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 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`

)- straush:
(Strauss process with hardcore.) A

**named**list with entries`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`

.- sftcr:
(Softcore process.) A

**named**list with components`beta,sigma,kappa`

. Again`beta`

is a ``base'' intensity. The pairwise interaction between two points \(u \neq v\) 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 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 \(j\).

- straushm:
(Multitype Strauss process with hardcore.) A

**named**list with components`beta`

and`gamma`

as for`straussm`

and**two**``radii'' components:`iradii`

: the interaction radii`hradii`

: the hardcore radii

`hradii`

must be less than the corresponding entries of`iradii`

.- 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 that`gamma`

must be less than or equal to 1.- lookup:
(Arbitrary pairwise interaction process with isotropic interaction.) A

**named**list with components`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.

**as a vector of values:**If`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 \(i = 2,\ldots, n\); and takes the value 1 for \(t \ge r[n]\). 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.]**as a stepfun object:**If`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.

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 the

`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.- 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.

##### 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.
http://www.jstatsoft.org/v55/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

```
# NOT RUN {
# Strauss process:
mod01 <- rmhmodel(cif="strauss",par=list(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=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)
# 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))
# }
```

*Documentation reproduced from package spatstat, version 1.49-0, License: GPL (>= 2)*