dppm | R Documentation |
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)
formula |
A |
family |
Information specifying the family of point processes
to be used in the model.
Typically one of the family functions
|
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
|
weightfun |
Optional weighting function w
in the composite likelihoods or Palm likelihood.
A |
control |
List of control parameters passed to the optimization function
|
algorithm |
Character string determining the mathematical algorithm
to be used to solve the fitting problem.
If |
statistic |
Name of the summary statistic to be used
for minimum contrast estimation: either |
statargs |
Optional list of arguments to be used when calculating
the |
rmax |
Maximum value of interpoint distance to use in the composite likelihood. |
epsilon |
Tuning parameter for the adaptive composite likelihood. |
covfunargs,use.gam,nd,eps |
Arguments passed to |
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
∑[i,j] w(d[i,j]) log(ρ(d[i,j]; θ)) - (∑[i,j] w(d[i,j])) log(integral[D,D] w(||u-v||) ρ(||u-v||; θ) du dv)
where the sums are taken over all pairs of data points
x[i], x[j] separated by a distance
d[i,j] = ||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 ρ(d;θ) is the
pair correlation function of the model with
interaction parameters θ.
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 ≤ 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)
∑[i,j] w(x[i], x[j]) log(λ[P](x[j] | x[i]; θ) - integral[D] w(x[i], u) λ[P](u | x[i]; θ) du
with the same notation as above. Here λ[P](u|v;θ) 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
∑[u,v] w(epsilon |g(0; theta)-1|/(g(||u-v||; theta)-1)) g'(||u-v||; theta) / g(||u-v||; theta) - integral[D,D] w(epsilon |g(0; theta)-1|/(g(||u-v||; theta)-1)) 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|| ≤ 1)
exp(1/(d^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.
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.
. Adaptive composite likelihood method contributed by Chiara Fend and modified by Adrian Baddeley.
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
jpines <- residualspaper$Fig1 dppm(jpines ~ 1, dppGauss) dppm(jpines ~ 1, dppGauss, method="c") dppm(jpines ~ 1, dppGauss, method="p") dppm(jpines ~ 1, dppGauss, method="a") if(interactive()) { # Fixing the intensity at lambda=2 rather than the Poisson MLE 2.04: dppm(jpines ~ 1, dppGauss(lambda=2)) # 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 at nu=2 rather than estimating it: dppm(jpines ~ x, dppMatern(nu=2))
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