Fit Point Process Model Involving Irregular Trend Parameters
Experimental extension to ppm
which finds optimal values of the irregular trend parameters in a
point process model.
ippm(Q, ...,
iScore=NULL,
start=list(),
covfunargs=start,
nlm.args=list(stepmax=1/2),
silent=FALSE,
warn.unused=TRUE)Q,... |
Arguments passed to |
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 |
nlm.args |
Optional list of arguments passed to |
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 |
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,
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), (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.
A fitted point process model (object of class "ppm")
which also belongs to the special class "ippm".
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
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),
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