lgcp.estK {spatstat} | R Documentation |
Fits a log-Gaussian Cox point process model (with exponential covariance function) to a point pattern dataset by the Method of Minimum Contrast.
lgcp.estK(X, startpar=list(sigma2=1,alpha=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
X |
Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the log-Gaussian Cox process model. |
lambda |
Optional. An estimate of the intensity of the point process. |
q,p |
Optional. Exponents for the contrast criterion. |
rmin, rmax |
Optional. The interval of r values for the contrast criterion. |
... |
Optional arguments passed to optim
to control the optimisation algorithm. See Details.
|
This algorithm fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum Contrast, using the K function.
The argument X
can be either
"ppp"
representing a point pattern dataset.
The K function of the point pattern will be computed
using Kest
, and the method of minimum contrast
will be applied to this.
"fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the K function,
and this object should have been obtained by a call to
Kest
or one of its relatives.
The algorithm fits a log-Gaussian Cox point process (LGCP)
model to X
, by finding the parameters of the LGCP model
which give the closest match between the
theoretical K function of the LGCP model
and the observed K function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The model fitted is a stationary, isotropic log-Gaussian Cox process with exponential covariance (Moller and Waagepetersen, 2003, pp. 72-76). To define this process we start with a stationary Gaussian random field Z in the two-dimensional plane, with constant mean mu and covariance function
c(r) = sigma^2 * exp(-r/alpha)
where sigma^2 and alpha are parameters. Given Z, we generate a Poisson point process Y with intensity function lambda(u) = exp(Z(u)) at location u. Then Y is a log-Gaussian Cox process.
The theoretical K-function of the LGCP is
K(r) = integral from 0 to r of (2 * pi * s * exp(sigma^2 * exp(-s/alpha))) ds.
The theoretical intensity of the LGCP is
lambda= exp(mu + sigma^2/2).
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters sigma^2 and alpha^2. Then the remaining parameter mu is inferred from the estimated intensity lambda.
If the argument lambda
is provided, then this is used
as the value of lambda. Otherwise, if X
is a
point pattern, then lambda
will be estimated from X
.
If X
is a summary statistic and lambda
is missing,
then the intensity lambda cannot be estimated, and
the parameter mu will be returned as NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class "fv" )
containing the observed values of the summary statistic
(observed ) and the theoretical values of the summary
statistic computed from the fitted model parameters.
|
Rasmus Waagepetersen rw@math.auc.dk. Adapted for spatstat by Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2006). An estimation function approach to inference for inhomogeneous Neyman-Scott processes. Submitted.
lgcp.estK
,
matclust.estK
,
mincontrast
,
Kest
data(redwood) u <- lgcp.estK(redwood, c(sigma2=0.1, alpha=1)) u plot(u)