Description Usage Arguments Details Value Author(s) References See Also Examples

Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum Contrast using the pair correlation function.

1 2 3 4 5 |

`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 |

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 + 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^2*. 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 * exp(-r/alpha).
*

To determine the template model, the string `"modelname"`

will be
prefixed by `"RM"`

and the code will search for
a function of this name in the RandomFields package.
For a list of available models see
`RMmodel`

in the
RandomFields package. For example the
Matern covariance with exponent *nu = 0.3* is specified
by `covmodel=list(model="matern", nu=0.3)`

corresponding
to the function `RMmatern`

in the RandomFields package.

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 |

with modifications by Shen Guochun and Rasmus Waagepetersen [email protected] and \ege.

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.

`lgcp.estK`

for alternative method 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.

`RMmodel`

in the
RandomFields package, for covariance function models.

`pcf`

for the pair correlation function.

1 2 3 4 5 6 7 | ```
data(redwood)
u <- lgcp.estpcf(redwood, c(var=1, scale=0.1))
u
plot(u)
if(require(RandomFields)) {
lgcp.estpcf(redwood, covmodel=list(model="matern", nu=0.3))
}
``` |

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