lgcp.estK
Fit a Log-Gaussian Cox Point Process by Minimum Contrast
Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum Contrast.
Usage
lgcp.estK(X, startpar=c(sigma2=1,alpha=1),
covmodel=list(model="exponential"),
lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
Arguments
- 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.
- covmodel
- Specification of the covariance model for the log-Gaussian field. See Details.
- 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.
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
[object Object],[object Object]
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 (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 $C(r)$. 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 $K$-function of the LGCP is $$K(r) = \int_0^r 2\pi s \exp(C(s)) \, {\rm d}s.$$ The intensity of the LGCP is $$\lambda = \exp(\mu + \frac{C(0)}{2}).$$ The covariance function $C(r)$ is parametrised in the form $$C(r) = \sigma^2 c(r/\alpha)$$ where $\sigma^2$ and $\alpha$ are parameters controlling the strength and the scale of autocorrelation, respectively, and $c(r)$ is a known covariance function determining the shape of the covariance. The strength and scale parameters $\sigma^2$ and $\alpha$ will be estimated by the algorithm. The template covariance function $c(r)$ must be specified as explained below. In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters $\sigma^2$ and $\alpha$. Then the remaining parameter $\mu$ is inferred from the estimated intensity $\lambda$.
The template covariance function $c(r)$ is specified
using the argument covmodel
. It may be any of the
covariance functions recognised by the command
Covariance
in the
covmodel
should be of the form
list(model="modelname", ...)
where
modelname
is the string name of one of the covariance models
recognised by the command
Covariance
in the
...
are arguments of the
form tag=value
giving the values of parameters controlling the
shape of these models. For example the exponential covariance is
specified by covmodel=list(model="exponential")
while the
Matern covariance with exponent $\nu=0.3$ is specified
by covmodel=list(model="matern", nu=0.3)
.
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.
Value
- 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.
Note
This function is considerably slower than lgcp.estpcf
because of the computation time required for the integral
in the $K$-function.
References
Moller, J, Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451--482. Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252--258.
See Also
lgcp.estpcf
for alternative method of fitting LGCP.
matclust.estK
,
thomas.estK
for other models.
mincontrast
for the generic minimum contrast
fitting algorithm, including important parameters that affect
the accuracy of the fit.
Covariance
in the
Kest
for the $K$ function.
Examples
data(redwood)
u <- lgcp.estK(redwood, c(sigma2=0.1, alpha=1))
u
plot(u)
<testonly>if(require(RandomFields)) {
lgcp.estK(redwood, covmodel=list(model="matern", nu=0.3),
control=list(maxit=5))
}</testonly>
if(require(RandomFields)) {
lgcp.estK(redwood, covmodel=list(model="matern", nu=0.3))
}