ippm: 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.

Value

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

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 $\lambda(u)$ 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.

The optimisation runs faster if we specify the derivative of $\log f(u,\gamma)$ with respect to $\gamma$. 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 $\gamma$ at any location (x,y).

Thus, to code such a problem,

The argument trend should define the log intensity, with the irregular part as an offset;
The argument start should be a list containing initial values of each of the irregular parameters;
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 $$ f((x,y), (\gamma, \delta)) = 1 + \exp(\gamma - \delta x^3) $$

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.

Examples

Run this code

# NOT RUN {
  nd <- 32
  
# }
# NOT RUN {
  
  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)
# }