Fit Point Process Model by Maximum Pseudolikelihood
Fits a point process model to an observed point pattern by the method of maximum pseudolikelihood.
mpl(Q, trend=~1, interaction=NULL, data, correction="border", rbord=0, use.gam=FALSE)
- A data point pattern (of class
"ppp") to which the model will be fitted, or preferably a quadrature scheme (of class
"quad") containing this pattern.
- An Rformula object specifying the spatial trend to be fitted.
The default formula,
~1, indicates the model is stationary and no trend is to be fitted.
- An object of class
"interact"describing the point process interaction structure, or
NULLindicating that a Poisson process (stationary or nonstationary) should be fitted.
- An optional data frame of spatial covariates (evaluated at the
locations given in the quadrature scheme
- The name of the edge correction to be used. The default
"border"indicating the border correction. Other possibilities may include
correction = "border"this argument specifies the distance by which the window should be eroded for the border correction.
- Logical flag; if
TRUEthen computations are performed using
This function fits a point process model
to an observed point pattern by the method of maximum pseudolikelihood
The model may include
spatial trend, interpoint interaction, and dependence on covariates.
The algorithm is an implementation of the method of
Baddeley and Turner (2000), which approximates the pseudolikelihood
by a special type of quadrature sum generalising the Berman-Turner (1992)
Q should be either a point pattern
or a quadrature scheme. If it is a point pattern, it is converted
into a quadrature scheme.
A quadrature scheme is an object of class
which specifies both the data point pattern and the dummy points
for the quadrature scheme, as well as the quadrature weights
associated with these points.
Q is simply a point pattern
then it is interpreted as specifying the
data points only; a set of dummy points specified
default.dummy() is added, and the default weighting rule is
invoked to compute the quadrature weights.
The usage of
mpl() is closely analogous to the Splus/Rfunctions
The analogy is:
The point process model to be fitted is specified by the
which are respectively analogous to
family arguments of glm().
Systematic effects (spatial trend and/or dependence on
spatial covariates) are specified by the argument
trend. This is an Splus/Rformula object, which may be expressed
in terms of the Cartesian coordinates
or the variables in the data frame
data (if supplied), or both.
It specifies the logarithm of the first order potential
of the process.
The formula should not use the names
which are reserved for internal use.
trend is absent or equal to the default,
the model to be fitted is stationary (or at least, its first order
potential is constant).
Stochastic interactions between random points of the point process
are defined by the argument
interaction. This is an object of
"interact" which is initialised in a very similar way to the
usage of family objects in
See the examples below.
interaction is missing or
NULL, then the model to be fitted
has no interpoint interactions, that is, it is a Poisson process
(stationary or nonstationary according to
trend). In this case
the method of maximum pseudolikelihood
coincides with maximum likelihood.
data, if supplied, must be a data frame with
as many rows as there are points in
The $i$th row of
data should contain the values of
spatial variables which have been observed
at the $i$th point of
Q. In this case
Q must be a
quadrature scheme, not merely a point pattern.
Thus, it is not sufficient to have observed
a spatial variable only at the points of the data point pattern;
the variable must also have been observed at certain other
locations in the window.
The variable names
are reserved for the Cartesian
coordinates and the mark values,
and these should not be used for variables in
correction is the name of an edge correction method.
correction="border" specifies the border correction,
in which the quadrature window (the domain of integration of the
pseudolikelihood) is obtained by trimming off a margin of width
from the observation window of the data pattern.
Not all edge corrections are implemented (or implementable)
for arbitrary windows.
Other options depend on the argument
interaction, but these generally
"periodic" (the periodic or toroidal edge correction
in which opposite edges of a rectangular window are identified)
(the translation correction, see Baddeley 1998 and Baddeley and Turner
For pairwise interaction there is also Ripley's isotropic correction,
The fitted point process model returned by this function can be printed
(by the print method
to inspect the fitted parameter values.
If a nonparametric spatial trend was fitted, this can be extracted using
the predict method
This algorithm approximates the log pseudolikelihood
by a sum over a finite set of quadrature points.
Finer quadrature schemes (i.e. those with more
quadrature points) generally yield a better approximation, at the
expense of higher computational load.
Complete control over the quadrature scheme is possible.
quadscheme for an overview.
Note that the method of maximum pseudolikelihood is believed to be inefficient and biased for point processes with strong interpoint interactions. In such cases, it is advisable to use iterative maximum likelihood methods such as Monte Carlo Maximum Likelihood (Geyer, 1999) provided the appropriate simulation algorithm exists. The maximum pseudolikelihood parameter estimate often serves as a good initial starting point for these iterative methods. Maximum pseudolikelihood may also be used profitably for model selection in the initial phases of modelling.
See the comments above about the possible inefficiency
and bias of the maximum pseudolikelihood estimator.
The accuracy of the Berman-Turner-Baddeley approximation to
the pseudolikelihood depends on the number of dummy points used
in the quadrature scheme. The number of dummy points should
at least equal the number of data points.
The parameter values of the fitted model
do not necessarily determine a valid point process.
Some of the point process models are only defined when the parameter
values lie in a certain subset. For example the Strauss process only
exists when the interaction parameter $\gamma$
is less than or equal to $1$,
corresponding to a value of
less than or equal to
The current version of
mpl maximises the pseudolikelihood
without constraining the parameters, and does not apply any checks for
sanity after fitting the model.
trend formula should not use the names
which are reserved
for internal use. The data frame
data should have as many rows
as there are points in
Q. It should not contain
as these names are reserved for the Cartesian coordinates
and the marks.
If the model formula involves one of the functions
(e.g. applied to spatial coordinates
the fitted coefficients can be misleading.
The resulting fit is not to the raw spatial variates
but to a transformation of these variates. The transformation is implemented
poly() in order to achieve better numerical stability.
resulting coefficients are appropriate for use with the transformed
variates, not with the raw variates.
This affects the interpretation of the constant
term in the fitted model,
Conventionally, $\beta$ is the background intensity, i.e. the
value taken by the conditional intensity function when all predictors
(including spatial or ``trend'' predictors) are set equal to $0$.
However the coefficient actually produced is the value that the
log conditional intensity takes when all the predictors,
including the transformed
spatial predictors, are set equal to
0, which is not the same thing.
If you wish to fit a polynomial trend,
we offer an alternative to
polynom(), which avoids the
difficulty induced by transformations. It is completely analogous
poly except that it does not orthonormalise.
The resulting coefficient estimates then have
their natural interpretation and can be predicted correctly.
Numerical stability may be compromised.
Values of the maximised pseudolikelihood are not comparable
if they have been obtained with different values of
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283--322. Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31--38. Besag, J. Statistical analysis of non-lattice data. The Statistician 24 (1975) 179-195. Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. On parameter estimation for pairwise interaction processes. International Statistical Review 62 (1994) 99-117. Jensen, J.L. and Moeller, M. Pseudolikelihood for exponential family models of spatial point processes. Annals of Applied Probability 1 (1991) 445--461. Jensen, J.L. and Kuensch, H.R. On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes, Annals of the Institute of Statistical Mathematics 46 (1994) 475-486.
library(spatstat) data(nztrees) mpl(nztrees) # fit the stationary Poisson process # to point pattern 'nztrees' Q <- quadscheme(nztrees) mpl(Q) # equivalent. mpl(nztrees, ~ x) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx) # where x,y are the Cartesian coordinates # and a,b are parameters to be estimated mpl(nztrees, ~ polynom(x,2)) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx + cx^2) library(splines) mpl(nztrees, ~ bs(x,df=3)) # WARNING: do not use predict.ppm() on this result # Fits the nonstationary Poisson process # with intensity function lambda(x,y) = exp(B(x)) # where B is a B-spline with df = 3 mpl(nztrees, ~1, Strauss(r=10), rbord=10) # Fit the stationary Strauss process with interaction range r=10 # using the border method with margin rbord=10 mpl(nztrees, ~ x, Strauss(13), correction="periodic") # Fit the nonstationary Strauss process with interaction range r=13 # and exp(first order potential) = activity = beta(x,y) = exp(a+bx) # using the periodic correction. data(soilsurvey) mpl(soilsurvey, ~ bs(pH,3), Strauss(0.1), rbord=0.1, data=soilchem) # WARNING: do not use predict.ppm() on this result # Fit the nonstationary Strauss process # with intensity modelled as a third order spline function of the # spatial variable "pH" in data frame 'soilchem' ## MULTITYPE POINT PROCESSES ### data(lansing) # Multitype point pattern --- trees marked by species mpl(lansing, ~ marks, Poisson()) # fit stationary marked Poisson process # with different intensity for each species mpl(lansing, ~ marks * polynom(x,y,3), Poisson()) # fit nonstationary marked Poisson process # with different log-cubic trend for each species <testonly># equivalent functionality - smaller dataset data(ganglia) mpl(ganglia, ~ marks * polynom(x,y,2), Poisson())</testonly>