Fit a determinantal point process model to a point pattern.
dppm(formula, family, data=NULL,
...,
startpar = NULL,
method = c("mincon", "clik2", "palm", "adapcl"),
weightfun = NULL,
control = list(),
algorithm,
statistic = "K",
statargs = list(),
rmax = NULL,
epsilon = 0.01,
covfunargs = NULL,
use.gam = FALSE,
nd = NULL, eps = NULL)A formula in the R language
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.
Information specifying the family of point processes
to be used in the model.
Typically one of the family functions
dppGauss, dppMatern,
dppCauchy, dppBessel
or dppPowerExp.
Alternatively a character string giving the name
of a family function, or the result of calling one of the
family functions. See Details.
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.
Named vector of starting parameter values for the optimization.
The fitting method. Either
"mincon" for minimum contrast,
"clik2" for second order composite likelihood,
"adapcl" for adaptive second order composite likelihood,
or "palm" for Palm likelihood.
Partially matched.
Optional weighting function \(w\)
in the composite likelihoods or Palm likelihood.
A function in the R language.
See Details.
List of control parameters passed to the optimization function
optim.
Character string determining the mathematical algorithm
to be used to solve the fitting problem.
If method="mincon", "clik2" or "palm" this argument
is passed to the generic optimization function
optim (renamed as the argument method)
with default "Nelder-Mead".
If method="adapcl") the argument is passed to the
equation solver nleqslv,
with default "Bryden".
Name of the summary statistic to be used
for minimum contrast estimation: either "K" or "pcf".
Optional list of arguments to be used when calculating
the statistic. See Details.
Maximum value of interpoint distance to use in the composite likelihood.
Tuning parameter for the adaptive composite likelihood.
Arguments passed to ppm when fitting the intensity.
An object of class "dppm" representing the fitted model.
There are methods for printing, plotting, predicting and simulating
objects of this class.
The following details allow greater control over the fitting procedure.
For the first three fitting methods
(method="mincon", "clik2" and "palm"),
the optimisation is performed by the generic
optimisation algorithm optim.
The behaviour of this algorithm can be modified using the
arguments control and algorithm.
Useful control arguments include
trace, maxit and abstol
(documented in the help for optim).
For method="adapcl", the estimating equation is solved
using the nonlinear equation solver nleqslv
from the package nleqslv.
Arguments available for controlling the solver are
documented in the help for
nleqslv; they include control,
globStrat, startparm for the initial estimates and
algorithm for the method.
The package nleqslv must be installed in order to use this option.
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
R language. 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 a composite likelihood method (maximum composite likelihood, maximum Palm likelihood, or by solving the adaptive composite likelihood estimating equation).
If method = "mincon" (the default) interaction parameters of
the model will be fitted
by minimum contrast estimation, that is, by matching the theoretical
\(K\)-function of the model to the empirical \(K\)-function
of the data, as explained in mincontrast.
For a homogeneous model ( trend = ~1 )
the empirical \(K\)-function of the data is computed
using Kest,
and the interaction parameters of the model are estimated by
the method of minimum contrast.
For an inhomogeneous model,
the inhomogeneous \(K\) function is estimated
by Kinhom using the fitted intensity.
Then the interaction parameters of the model
are estimated by the method of minimum contrast using the
inhomogeneous \(K\) function. This two-step estimation
procedure is heavily inspired by Waagepetersen (2007).
If statistic="pcf" then instead of using the
\(K\)-function, the algorithm will use
the pair correlation function pcf for homogeneous
models and the inhomogeneous pair correlation function
pcfinhom for inhomogeneous models.
In this case, the smoothing parameters of the pair correlation
can be controlled using the argument statargs,
as shown in the Examples.
Additional arguments … will be passed to
clusterfit to control the minimum contrast fitting
algorithm.
If method = "clik2" the interaction parameters of the
model will be fitted by maximising the second-order composite likelihood
(Guan, 2006). The log composite likelihood is
$$
\sum_{i,j} w(d_{ij}) \log\rho(d_{ij}; \theta)
- \left( \sum_{i,j} w(d_{ij}) \right)
\log \int_D \int_D w(\|u-v\|) \rho(\|u-v\|; \theta)\, du\, dv
$$
where the sums are taken over all pairs of data points
\(x_i, x_j\) separated by a distance
\(d_{ij} = \| x_i - x_j\|\)
less than rmax,
and the double integral is taken over all pairs of locations
\(u,v\) in the spatial window of the data.
Here \(\rho(d;\theta)\) is the
pair correlation function of the model with
interaction parameters \(\theta\).
The function \(w\) in the composite likelihood
is a weighting function and may be chosen arbitrarily.
It is specified by the argument weightfun.
If this is missing or NULL then the default is
a threshold weight function,
\(w(d) = 1(d \le R)\), where \(R\) is rmax/2.
If method = "palm" the interaction parameters of the
model will be fitted by maximising the Palm loglikelihood
(Tanaka et al, 2008)
$$
\sum_{i,j} w(x_i, x_j) \log \lambda_P(x_j \mid x_i; \theta)
- \int_D w(x_i, u) \lambda_P(u \mid x_i; \theta) {\rm d} u
$$
with the same notation as above. Here
\(\lambda_P(u|v;\theta\) is the Palm intensity of
the model at location \(u\) given there is a point at \(v\).
If method = "cladap" the clustering parameters of the
model will be fitted by solving the adaptive second order
composite likelihood estimating equation (Lavancier et al, 2021).
The estimating function is
$$
\sum_{u, v} w(\epsilon \frac{|g(0; \theta) - 1|}{g(\|u-v\|; \theta)-1})
\frac{\nabla_\theta g(\|u-v\|;\theta)}{g(\|u-v\|;\theta)} -
\int_D \int_D w(\epsilon \frac{M(u,v; \theta)} \nabla_\theta g(\|u-v\|; \theta)
\rho(u) \rho(v)\, du\, dv
$$
where the sum is taken over all distinct pairs of points.
Here \(g(d;\theta)\) is the
pair correlation function with
parameters \(\theta\).
The partial derivative with respect to \(\theta\)
is \(g'(d; \theta)\), and \(\rho(u)\) denotes
the fitted intensity function of the model.
The tuning parameter \(\epsilon\) is
independent of the data. It can be specified by the
argument epsilon and has default value \(0.01\).
The function \(w\) in the estimating function
is a weighting function of bounded support \([-1,1]\).
It is specified by the argument weightfun.
If this is missing or NULL then the default is
\(
w(d) = 1(\|d\| \le 1) \exp(1/(r^2-1))\).
The estimating equation is solved using the
nonlinear equation solver nleqslv
from the package nleqslv. The package nleqslv
must be installed in order to use this option.
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.
Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association 101, 1502--1512.
Lavancier, F., Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference. Journal of the Royal Statistical Society, Series B 77, 853--977.
Lavancier, F., Poinas, A., and Waagepetersen, R. (2021) Adaptive estimating function inference for nonstationary determinantal point processes. Scandinavian Journal of Statistics, 48 (1), 87--107.
Tanaka, U., 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.
methods for dppm objects:
plot.dppm,
fitted.dppm,
predict.dppm,
simulate.dppm,
methods.dppm,
as.ppm.dppm,
Kmodel.dppm,
pcfmodel.dppm.
Minimum contrast fitting algorithm:
higher level interface clusterfit;
low-level algorithm mincontrast.
Deterimantal point process models:
dppGauss,
dppMatern,
dppCauchy,
dppBessel,
dppPowerExp,
Summary statistics:
Kest,
Kinhom,
pcf,
pcfinhom.
See also ppm
# NOT RUN {
jpines <- residualspaper$Fig1
# }
# NOT RUN {
dppm(jpines ~ 1, dppGauss)
dppm(jpines ~ 1, dppGauss, method="c")
dppm(jpines ~ 1, dppGauss, method="p")
dppm(jpines ~ 1, dppGauss, method="a")
# 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))
# }
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