spatstat (version 1.25-2)

ippm: Optimise Irregular Trend Parameters in Point Process Model

Description

Experimental extension to ppm. Find optimal values of the irregular trend parameters in a point process model using Fisher scoring algorithm.

Usage

ippm(..., iScore=NULL,
          start=list(),
          covfunargs=start,
          maxiter=20, tol=1e-4, progress=TRUE, stepfactor=1,
          dbug=FALSE)

Arguments

...

Arguments passed to ppm to fit the point process model.

iScore

A named list of Rfunctions that compute the partial derivatives of logf 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

maxiter

Integer. Maximum number of iterations of Fisher scoring algorithm.

tol

Numeric value or vector. The algorithm stops when the difference between two successive estimates of the irregular parameter is less than tol.

progress

Logical. Whether to print progress reports.

stepfactor

Numeric value between 0 and 1 indicating that the change in the parameter between successive iterations is only a specified fraction of the step computed by the Newton-Raphson algorithm.

dbug

Logical. Whether to print debugging output.

Value

• A fitted point process model (object of class "ppm").

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 Fisher scoring method.

For the sake of explanation, consider a Poisson point process with intensity function $\lambda(u)$ at location $u$. Assume that $$\lambda(u) = \exp(\alpha + \beta Z(u)) \, f(u, \gamma)$$ where $\alpha,\beta,\gamma$ are parameters to be estimated, $Z(u)$ is a spatial covariate function, and $f$ is some known function. Then the parameters $\alpha,\beta$ are called regular because they appear in a loglinear form; the parameter $\gamma$ 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 $$\log\lambda(u) = \alpha + \beta Z(u) + \log f(u, \gamma)$$ 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.

To perform Fisher scoring we also need the derivative of $\log f(u,\gamma)$ with respect to $\gamma$. We call this the irregular score. The user must write an Rfunction that computes the irregular score for any value of $\gamma$ at any location (x,y). Thus, to code such a problem,

1. The argumenttrendshould define the log intensity, with the irregular part as an offset;
2. The argumentstartshould be a list containing initial values of each of the irregular parameters;
3. The argumentiScoremust be a list (with one entry for each entry ofstart) of functions with argumentsx,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 $$f((x,y), (\gamma, \delta)) = 1 + \exp(\gamma - \delta x^3)$$

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.

See Also

ppm

Examples

nd <- 32
  <testonly>nd <- 10</testonly>
  
  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),
       tol=0.01, nd=nd)