R: Fit Point Process Model Involving Irregular Trend Parameters

ippm {spatstat.core} | R Documentation

## Fit Point Process Model Involving Irregular Trend Parameters

### Description

Experimental extension to `ppm` which finds optimal values of the irregular trend parameters in a point process model.

### Usage

    ippm(Q, ...,
              iScore=NULL,
              start=list(),
              covfunargs=start,
              nlm.args=list(stepmax=1/2),
              silent=FALSE,
              warn.unused=TRUE)

### Arguments

`Q,...` | Arguments passed to `ppm` to fit the point process model.
`iScore` | Optional. A named list of R functions that compute the partial derivatives of the logarithm of the trend, with respect to each irregular parameter. See Details.
`start` | Named list containing initial values of the irregular parameters over which to optimise.
`covfunargs` | Argument passed to `ppm`. A named list containing values for all irregular parameters required by the covariates in the model. Must include all the parameters named in `start`.
`nlm.args` | Optional list of arguments passed to `nlm` to control the optimization algorithm.
`silent` | Logical. Whether to print warnings if the optimization algorithm fails to converge.
`warn.unused` | Logical. Whether to print a warning if some of the parameters in `start` are not used in the model.

### Details

This function is an experimental extension to the point process model fitting command `ppm`. The extension allows the trend of the model to include irregular parameters, which will be maximised by a Newton-type iterative method, using `nlm`.

For the sake of explanation, consider a Poisson point process with intensity function λ(u) at location u. Assume that

λ(u) = exp(α + β * Z(u)) * f(u, γ)

where α,β,γ are parameters to be estimated, Z(u) is a spatial covariate function, and f is some known function. Then the parameters α,β are called regular because they appear in a loglinear form; the parameter γ is called irregular.

To fit this model using `ippm`, we specify the intensity using the `trend` formula in the same way as usual for `ppm`. The trend formula is a representation of the log intensity. In the above example the log intensity is

log(λ(u)) = α + β * Z(u) + log(f(u, γ))

So the model above would be encoded with the trend formula `~Z + offset(log(f))`. Note that the irregular part of the model is an offset term, which means that it is included in the log trend as it is, without being multiplied by another regular parameter.

The optimisation runs faster if we specify the derivative of log(f(u,γ)) with respect to γ. We call this the irregular score. To specify this, the user must write an R function that computes the irregular score for any value of γ at any location `(x,y)`.

Thus, to code such a problem,

1. The argument `trend` should define the log intensity, with the irregular part as an offset;

2. The argument `start` should be a list containing initial values of each of the irregular parameters;

3. The argument `iScore`, if provided, must be a list (with one entry for each entry of `start`) of functions with arguments `x,y,...`, that evaluate the partial derivatives of log(f(u,gamma)) with respect to each irregular parameter.

The coded example below illustrates the model with two irregular parameters gamma,delta and irregular term

f((x,y), (γ, δ)) = 1 + \exp(γ - δ * x^3)

Arguments `...` passed to `ppm` may also include `interaction`. In this case the model is not a Poisson point process but a more general Gibbs point process; the trend formula `trend` determines the first-order trend of the model (the first order component of the conditional intensity), not the intensity.

### Value

A fitted point process model (object of class `"ppm"`) which also belongs to the special class `"ippm"`.

### Author(s)

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

### See Also

`ppm`, `profilepl`

### Examples

      nd <- 32

      gamma0 <- 3
      delta0 <- 5
      POW <- 3
      # Terms in intensity
      Z <- function(x,y) { -2*y }
      f <- function(x,y,gamma,delta) { 1 + exp(gamma - delta * x^POW) }
      # True intensity
      lamb <- function(x,y,gamma,delta) { 200 * exp(Z(x,y)) * f(x,y,gamma,delta) }
      # Simulate realisation
      lmax <- max(lamb(0,0,gamma0,delta0), lamb(1,1,gamma0,delta0))
      set.seed(42)
      X <- rpoispp(lamb, lmax=lmax, win=owin(), gamma=gamma0, delta=delta0)
      # Partial derivatives of log f
      DlogfDgamma <- function(x,y, gamma, delta) {
        topbit <- exp(gamma - delta * x^POW)
        topbit/(1 + topbit)
      }
      DlogfDdelta <- function(x,y, gamma, delta) {
        topbit <- exp(gamma - delta * x^POW)
        - (x^POW) * topbit/(1 + topbit)
      }
      # irregular score
      Dlogf <- list(gamma=DlogfDgamma, delta=DlogfDdelta)
      # fit model
      ippm(X ~Z + offset(log(f)),
           covariates=list(Z=Z, f=f),
           iScore=Dlogf,
           start=list(gamma=1, delta=1),
           nlm.args=list(stepmax=1),
           nd=nd)

[Package spatstat.core version 2.0-0 Index]