dppm: Fit Determinantal Point Process Model

Description

Fit a determinantal point process model to a point pattern.

Usage

dppm(formula, family, data=NULL,
       ...,
       startpar = NULL,
       method = c("mincon", "clik2", "palm"),
       weightfun=NULL,
       control=list(),
       algorithm="Nelder-Mead",
       statistic="K",
       statargs=list(),
       rmax = NULL,
       covfunargs=NULL,
       use.gam=FALSE,
       nd=NULL, eps=NULL)

Arguments

formula

A formula in the Rlanguage specifying the data (on the left side) and the form of the model to be fitted (on the right side). For a stationary model it suffices to provide a point pattern without a formula. See Details.

family

Information specifying the family of point processes to be used in the model. Typically one of the family functions dppGauss, dppMatern, <

data

The values of spatial covariates (other than the Cartesian coordinates) required by the model. A named list of pixel images, functions, windows, tessellations or numeric constants.

...

Additional arguments. See Details.

startpar

Named vector of starting parameter values for the optimization.

method

The fitting method. Either "mincon" for minimum contrast, "clik2" for second order composite likelihood, or "palm" for Palm likelihood. Partially matched.

weightfun

Optional weighting function $w$ in the composite likelihood or Palm likelihood. A function in the Rlanguage. See Details.

control

List of control parameters passed to the optimization function optim.

algorithm

Character string determining the mathematical optimisation algorithm to be used by optim. See the argument method of optim.

statistic

Name of the summary statistic to be used for minimum contrast estimation: either "K" or "pcf".

statargs

Optional list of arguments to be used when calculating the statistic. See Details.

rmax

Maximum value of interpoint distance to use in the composite likelihood.

covfunargs,use.gam,nd,eps

Arguments passed to ppm when fitting the intensity.

Value

An object of class "dppm" representing the fitted model. There are methods for printing, plotting, predicting and simulating objects of this class.

concept

point process model
determinantal point process

Details

This function fits a determinantal point process model to a point pattern dataset as described in Lavancier et al. (2015).

The model to be fitted is specified by the arguments formula and family. The argument formula should normally be a formula in the Rlanguage. The left hand side of the formula specifies the point pattern dataset to which the model should be fitted. This should be a single argument which may be a point pattern (object of class "ppp") or a quadrature scheme (object of class "quad"). The right hand side of the formula is called the trend and specifies the form of the logarithm of the intensity of the process. Alternatively the argument formula may be a point pattern or quadrature scheme, and the trend formula is taken to be ~1.

The argument family specifies the family of point processes to be used in the model. It is typically one of the family functions dppGauss, dppMatern, dppCauchy, dppBessel or dppPowerExp. Alternatively it may be a character string giving the name of a family function, or the result of calling one of the family functions. A family function belongs to class "detpointprocfamilyfun". The result of calling a family function is a point process family, which belongs to class "detpointprocfamily". The algorithm first estimates the intensity function of the point process using ppm. If the trend formula is ~1 (the default if a point pattern or quadrature scheme is given rather than a "formula") then the model is homogeneous. The algorithm begins by estimating the intensity as the number of points divided by the area of the window. Otherwise, the model is inhomogeneous. The algorithm begins by fitting a Poisson process with log intensity of the form specified by the formula trend. (See ppm for further explanation).

The interaction parameters of the model are then fitted either by minimum contrast estimation, or by maximum composite likelihood.

[object Object],[object Object],[object Object] In all three methods, the optimisation is performed by the generic optimisation algorithm optim. The behaviour of this algorithm can be modified using the argument control. Useful control arguments include trace, maxit and abstol (documented in the help for optim).

Finally, it is also possible to fix any parameters desired before the optimisation by specifying them as name=value in the call to the family function. See Examples.

References

Lavancier, F. latex{Mller{Moller}, J. and Rubak, E. (2015) Determinantal point process models and statistical inference Journal of the Royal Statistical Society, Series B 77, 853--977.

Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association 101, 1502--1512.

Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43--57.

Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252--258. } jpines <- residualspaper$Fig1

dppm(jpines ~ 1, dppGauss)

dppm(jpines ~ 1, dppGauss, method="c") dppm(jpines ~ 1, dppGauss, method="p")

# Fixing the intensity to lambda=2 rather than the Poisson MLE 2.04: dppm(jpines ~ 1, dppGauss(lambda=2))

if(interactive()) { # The following is quite slow (using K-function) dppm(jpines ~ x, dppMatern) }

# much faster using pair correlation function dppm(jpines ~ x, dppMatern, statistic="pcf", statargs=list(stoyan=0.2))

# Fixing the Matern shape parameter to nu=2 rather than estimating it: dppm(jpines ~ x, dppMatern(nu=2)) [object Object],[object Object] spatial models