lgcp.estpcf: Fit a Log-Gaussian Cox Point Process by Minimum Contrast
In spatstat.model: Parametric Statistical Modelling and Inference for the 'spatstat' Family
lgcp.estpcf R Documentation
Fit a Log-Gaussian Cox Point Process by Minimum Contrast
Description
Fits a log-Gaussian Cox point process model
to a point pattern dataset by the Method of Minimum Contrast
using the pair correlation function.
Usage
lgcp.estpcf(X,
startpar=c(var=1,scale=1),
covmodel=list(model="exponential"),
lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
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.
pcfargs
Optional list containing arguments passed to pcf.ppp
to control the smoothing in the estimation of the
pair correlation function.
Details
This algorithm fits a log-Gaussian Cox point process (LGCP) model
to a point pattern dataset by the Method of Minimum Contrast,
using the estimated pair correlation function of the point pattern.
The shape of the covariance of the LGCP must be specified:
the default is the exponential covariance function,
but other covariance models can be selected.
The argument X can be either
- a point pattern:
An object of class "ppp"
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using pcf, and the method of minimum contrast
will be applied to this.
- a summary statistic:
An object of class "fv" containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to
pcf 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 pair correlation function of the LGCP model
and the observed pair correlation 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 theoretical pair correlation function of the LGCP is
g(r) = \exp(C(s))
The intensity of the LGCP is
\lambda = \exp(\mu + \frac{C(0)}{2}).
The covariance function C(r) takes 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. This should be of the form
list(model="modelname", ...) where
modelname is a string identifying the template model
as explained below, and ... are optional arguments of the
form tag=value giving the values of parameters controlling the
shape of the template model.
The default is the exponential covariance
c(r) = e^{-r}
so that the scaled covariance is
C(r) = \sigma^2 e^{-r/\alpha}.
For a list of available models see kppm.
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.
Author(s)
\adrian
with modifications by Shen Guochun and
\rasmus and \ege.
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
kppm and lgcp.estK
for alternative methods of fitting LGCP.
matclust.estpcf,
thomas.estpcf for other models.
mincontrast for the generic minimum contrast
fitting algorithm, including important parameters that affect
the accuracy of the fit.
pcf for the pair correlation function.
Examples
u <- lgcp.estpcf(redwood, c(var=1, scale=0.1))
u
plot(u)
lgcp.estpcf(redwood, covmodel=list(model="matern", nu=0.3))
spatstat.model documentation built on Sept. 30, 2024, 9:26 a.m.
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| lgcp.estpcf | R Documentation |
Fit a Log-Gaussian Cox Point Process by Minimum Contrast
Description
Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum Contrast using the pair correlation function.
Usage
lgcp.estpcf(X,
startpar=c(var=1,scale=1),
covmodel=list(model="exponential"),
lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
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 |
... |
Optional arguments passed to |
pcfargs |
Optional list containing arguments passed to |
Details
This algorithm fits a log-Gaussian Cox point process (LGCP) model to a point pattern dataset by the Method of Minimum Contrast, using the estimated pair correlation function of the point pattern.
The shape of the covariance of the LGCP must be specified: the default is the exponential covariance function, but other covariance models can be selected.
The argument X can be either
- a point pattern:
An object of class
"ppp"representing a point pattern dataset. The pair correlation function of the point pattern will be computed usingpcf, and the method of minimum contrast will be applied to this.- a summary statistic:
An object of class
"fv"containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the pair correlation function, and this object should have been obtained by a call topcfor 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 pair correlation function of the LGCP model
and the observed pair correlation 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 theoretical pair correlation function of the LGCP is
g(r) = \exp(C(s))
The intensity of the LGCP is
\lambda = \exp(\mu + \frac{C(0)}{2}).
The covariance function C(r) takes 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. This should be of the form
list(model="modelname", ...) where
modelname is a string identifying the template model
as explained below, and ... are optional arguments of the
form tag=value giving the values of parameters controlling the
shape of the template model.
The default is the exponential covariance
c(r) = e^{-r}
so that the scaled covariance is
C(r) = \sigma^2 e^{-r/\alpha}.
For a list of available models see kppm.
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 |
Author(s)
\adrianwith modifications by Shen Guochun and \rasmus and \ege.
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
kppm and lgcp.estK
for alternative methods of fitting LGCP.
matclust.estpcf,
thomas.estpcf for other models.
mincontrast for the generic minimum contrast
fitting algorithm, including important parameters that affect
the accuracy of the fit.
pcf for the pair correlation function.
Examples
u <- lgcp.estpcf(redwood, c(var=1, scale=0.1))
u
plot(u)
lgcp.estpcf(redwood, covmodel=list(model="matern", nu=0.3))
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