rmhmodel.default: 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

# 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

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.

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.

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

which are both symmetric matrices of nonnegative numbers. The entries of 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.

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. 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.

Examples

Run this code

# NOT RUN {
   # 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
# }

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