R: Set Control Parameters for Metropolis-Hastings Algorithm.

rmhcontrol {spatstat} | R Documentation

## Set Control Parameters for Metropolis-Hastings Algorithm.

### Description

Sets up a list of parameters controlling the iterative behaviour of the Metropolis-Hastings algorithm.

### Usage

       rmhcontrol(...)

       ## Default S3 method:
    rmhcontrol(..., p=0.9, q=0.5, nrep=5e5,
                          expand=NULL, periodic=NULL, ptypes=NULL,
                          x.cond=NULL, fixall=FALSE, nverb=0,
                          nsave=NULL, nburn=nsave, track=FALSE,
                          pstage=c("block", "start"))

### Arguments

`...` | Arguments passed to methods.
`p` | Probability of proposing a shift (as against a birth/death).
`q` | Conditional probability of proposing a death given that a birth or death will be proposed.
`nrep` | Total number of steps (proposals) of Metropolis-Hastings algorithm that should be run.
`expand` | Simulation window or expansion rule. Either a window (object of class `"owin"`) or a numerical expansion factor, specifying that simulations are to be performed in a domain other than the original data window, then clipped to the original data window. This argument is passed to `rmhexpand`. A numerical expansion factor can be in several formats: see `rmhexpand`.
`periodic` | Logical value (or `NULL`) indicating whether to simulate “periodically”, i.e. identifying opposite edges of the rectangular simulation window. A `NULL` value means “undecided.”
`ptypes` | For multitype point processes, the distribution of the mark attached to a new random point (when a birth is proposed)
`x.cond` | Conditioning points for conditional simulation.
`fixall` | (Logical) for multitype point processes, whether to fix the number of points of each type.
`nverb` | Progress reports will be printed every `nverb` iterations
`nsave,nburn` | If these values are specified, then intermediate states of the simulation algorithm will be saved every `nsave` iterations, after an initial burn-in period of `nburn` iterations.
`track` | Logical flag indicating whether to save the transition history of the simulations.
`pstage` | Character string specifying when to generate proposal points. Either `"start"` or `"block"`.

### Details

The Metropolis-Hastings algorithm, implemented as `rmh`, generates simulated realisations of point process models. The function `rmhcontrol` sets up a list of parameters which control the iterative behaviour and termination of the Metropolis-Hastings algorithm, for use in a subsequent call to `rmh`. It also checks that the parameters are valid.

(A separate function `rmhstart` determines the initial state of the algorithm, and `rmhmodel` determines the model to be simulated.)

The parameters are as follows:

p

  The probability of proposing a “shift” (as opposed to a birth or death) in the Metropolis-Hastings algorithm.

  If p = 1 then the algorithm only alters existing points, so the number of points never changes, i.e. we are simulating conditionally upon the number of points. The number of points is determined by the initial state (specified by `rmhstart`).

  If p=1 and `fixall=TRUE` and the model is a multitype point process model, then the algorithm only shifts the locations of existing points and does not alter their marks (types). This is equivalent to simulating conditionally upon the number of points of each type. These numbers are again specified by the initial state.

  If p = 1 then no expansion of the simulation window is allowed (see `expand` below).

  The default value of `p` can be changed by setting the parameter `rmh.p` in `spatstat.options`.

q

  The conditional probability of proposing a death (rather than a birth) given that a shift is not proposed. This is of course ignored if `p` is equal to 1.

  The default value of `q` can be changed by setting the parameter `rmh.q` in `spatstat.options`.

nrep

  The number of repetitions or iterations to be made by the Metropolis-Hastings algorithm. It should be large.

  The default value of `nrep` can be changed by setting the parameter `rmh.nrep` in `spatstat.options`.

expand

  Either a number or a window (object of class `"owin"`). Indicates that the process is to be simulated on a domain other than the original data window `w`, then clipped to `w` when the algorithm has finished. This would often be done in order to approximate the simulation of a stationary process (Geyer, 1999) or more generally a process existing in the whole plane, rather than just in the window `w`.

  If `expand` is a window object, it is taken as the larger domain in which simulation is performed.

  If `expand` is numeric, it is interpreted as an expansion factor or expansion distance for determining the simulation domain from the data window. It should be a named scalar, such as `expand=c(area=2)`, `expand=c(distance=0.1)`, `expand=c(length=1.2)`. See `rmhexpand()` for more details. If the name is omitted, it defaults to `area`.

  Expansion is not permitted if the number of points has been fixed by setting `p = 1` or if the starting configuration has been specified via the argument `x.start` in `rmhstart`.

  If `expand` is `NULL`, this is interpreted to mean “not yet decided”. An expansion rule will be determined at a later stage, using appropriate defaults. See `rmhexpand`.

periodic

  A logical value (or `NULL`) determining whether to simulate “periodically”. If `periodic` is `TRUE`, and if the simulation window is a rectangle, then the simulation algorithm effectively identifies opposite edges of the rectangle. Points near the right-hand edge of the rectangle are deemed to be close to points near the left-hand edge. Periodic simulation usually gives a better approximation to a stationary point process. For periodic simulation, the simulation window must be a rectangle. (The simulation window is determined by `expand` as described above.)

  The value `NULL` means ‘undecided’. The decision is postponed until `rmh` is called. Depending on the point process model to be simulated, `rmh` will then set `periodic=TRUE` if the simulation window is expanded and the expanded simulation window is rectangular; otherwise `periodic=FALSE`.

  Note that `periodic=TRUE` is only permitted when the simulation window (i.e. the expanded window) is rectangular.

ptypes

  A vector of probabilities (summing to 1) to be used in assigning a random type to a new point. Defaults to a vector each of whose entries is 1/nt where nt is the number of types for the process. Convergence of the simulation algorithm should be improved if `ptypes` is close to the relative frequencies of the types which will result from the simulation.

x.cond

  If this argument is given, then conditional simulation will be performed, and `x.cond` specifies the location of the fixed points as well as the type of conditioning. It should be either a point pattern (object of class `"ppp"`) or a `list(x,y)` or a `data.frame`. See the section on Conditional Simulation.

fixall

  A logical scalar specifying whether to condition on the number of points of each type. Meaningful only if a marked process is being simulated, and if p = 1. A warning message is given if `fixall` is set equal to `TRUE` when it is not meaningful.

nverb

  An integer specifying how often “progress reports” (which consist simply of the number of repetitions completed) should be printed out. If nverb is left at 0, the default, the simulation proceeds silently.

nsave,nburn

  If these integers are given, then the current state of the simulation algorithm (i.e. the current random point pattern) will be saved every `nsave` iterations, starting from iteration `nburn`. (Alternatively `nsave` can be a vector, specifying different numbers of iterations between each successive save. This vector will be recycled until the end of the simulations.)

track

  Logical flag indicating whether to save the transition history of the simulations (i.e. information specifying what type of proposal was made, and whether it was accepted or rejected, for each iteration).

pstage

  Character string specifying the stage of the algorithm at which the randomised proposal points should be generated. If `pstage="start"` or if `nsave=0`, the entire sequence of `nrep` random proposal points is generated at the start of the algorithm. This is the original behaviour of the code, and should be used in order to maintain consistency with older versions of spatstat. If `pstage="block"` and `nsave > 0`, then a set of `nsave` random proposal points will be generated before each block of `nsave` iterations. This is much more efficient. The default is `pstage="block"`.

### Value

An object of class `"rmhcontrol"`, which is essentially a list of parameter values for the algorithm.

There is a `print` method for this class, which prints a sensible description of the parameters chosen.

### Conditional Simulation

For a Gibbs point process X, the Metropolis-Hastings algorithm easily accommodates several kinds of conditional simulation:

conditioning on the total number of points:

  We fix the total number of points N(X) to be equal to n. We simulate from the conditional distribution of X given N(X) = n.

conditioning on the number of points of each type:

  In a multitype point process, where Y[[j]] denotes the process of points of type j, we fix the number N(Y[[j]]) of points of type j to be equal to n[j], for j=1,2,...,m. We simulate from the conditional distribution of X given N(Y[[j]]) = n[j] for j=1,2,...,m.

conditioning on the realisation in a subwindow:

  We require that the point process X should, within a specified sub-window V, coincide with a specified point pattern y. We simulate from the conditional distribution of X given (X intersect V) = y.

Palm conditioning:

  We require that the point process X include a specified list of points y. We simulate from the point process with probability density g(x) = c * f(x union y) where f is the probability density of the original process X, and c is a normalising constant.

To achieve each of these types of conditioning we do as follows:

conditioning on the total number of points:

  Set `p=1`. The number of points is determined by the initial state of the simulation: see `rmhstart`.

conditioning on the number of points of each type:

  Set `p=1` and `fixall=TRUE`. The number of points of each type is determined by the initial state of the simulation: see `rmhstart`.

conditioning on the realisation in a subwindow:

  Set `x.cond` to be a point pattern (object of class `"ppp"`). Its window `V=Window(x.cond)` becomes the conditioning subwindow V.

Palm conditioning:

  Set `x.cond` to be a `list(x,y)` or `data.frame` with two columns containing the coordinates of the points, or a `list(x,y,marks)` or `data.frame` with three columns containing the coordinates and marks of the points.

The arguments `x.cond`, `p` and `fixall` can be combined.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### References

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`, `rmhmodel`, `rmhstart`, `rmhexpand`, `spatstat.options`

### Examples

       # parameters given as named arguments
       c1 <- rmhcontrol(p=0.3,periodic=TRUE,nrep=1e6,nverb=1e5)

       # parameters given as a list
       liz <- list(p=0.9, nrep=1e4)
       c2 <- rmhcontrol(liz)

       # parameters given in rmhcontrol object
       c3 <- rmhcontrol(c1)

[Package spatstat version 1.63-3 Index]