ppm
Fit Point Process Model to Data
Fits a point process model to an observed point pattern
- Keywords
- spatial
Usage
ppm(Q, trend=~1, interaction=NULL, covariates=NULL,
correction="border", rbord=0, use.gam=FALSE, method="mpl",
forcefit=FALSE, nsim=100, nrmh=1e5,
start=NULL,
control=list(nrep=nrmh),
verb=TRUE)
Arguments
- Q
- A data point pattern (of class
"ppp"
) to which the model will be fitted, or a quadrature scheme (of class"quad"
) containing this pattern. - trend
- 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. - interaction
- An object of class
"interact"
describing the point process interaction structure, orNULL
indicating that a Poisson process (stationary or nonstationary) should be fitted. - covariates
- The values of any spatial covariates (other than the Cartesian coordinates) required by the model. Either a data frame, or a list of images. See Details.
- correction
- The name of the edge correction to be used. The default
is
"border"
indicating the border correction. Other possibilities may include"Ripley"
,"isotropic"
,"translate"
and"none"
- rbord
- If
correction = "border"
this argument specifies the distance by which the window should be eroded for the border correction. - use.gam
- Logical flag; if
TRUE
then computations are performed usinggam
instead ofglm
. - method
- The method used to fit the model. Options are
"mpl"
for the method of Maximum PseudoLikelihood, and"ho"
for the Huang-Ogata approximate maximum likelihood method. - forcefit
- Logical flag for internal use.
If
forcefit=FALSE
, some trivial models will be fitted by a shortcut. Ifforcefit=TRUE
, the generic fitting method will always be used. - nsim
- Number of simulated realisations
to generate (for
method="ho"
) - nrmh
- Number of Metropolis-Hastings iterations
for each simulated realisation (for
method="ho"
) - start,control
- Arguments passed to
rmh
controlling the behaviour of the Metropolis-Hastings algorithm (formethod="ho"
) - verb
- Logical flag indicating whether to print progress reports
(for
method="ho"
)
Details
This function fits a point process model to an observed point pattern. The model may include spatial trend, interpoint interaction, and dependence on covariates.
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Value
- An object of class
"ppm"
describing a fitted point process model. The fitted parameters can be obtained just by printing this object. Fitted spatial trends can be extracted using thepredict
method for this object (seepredict.ppm
).See
ppm.object
for details of the format of this object.
Warnings
The implementation of the Huang-Ogata method is experimental.
See the comments above about the possible inefficiency
and bias of the maximum pseudolikelihood estimator.
The accuracy of the Berman-Turner 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 ppm()$theta[2]
less than or equal to 0
.
The current version of ppm
maximises the pseudolikelihood
without constraining the parameters, and does not apply any checks for
sanity after fitting the model.
The trend
formula should not use any variable names
beginning with the prefixes .mpl
or Interaction
as these names are reserved
for internal use. The data frame covariates
should have as many rows
as there are points in Q
. It should not contain
variables called x
, y
or marks
as these names are reserved for the Cartesian coordinates
and the marks.
If the model formula involves one of the functions
poly()
, bs()
or ns()
(e.g. applied to spatial coordinates x
and y
),
the fitted coefficients can be misleading.
The resulting fit is not to the raw spatial variates
(x
, x^2
, x*y
, etc.)
but to a transformation of these variates. The transformation is implemented
by poly()
in order to achieve better numerical stability.
However the
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, logbeta
.
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.
Worse still, the result of predict.ppm
can be
completely wrong if the trend formula contains one of the
functions poly()
, bs()
or ns()
. This is a weakness of the underlying
function predict.glm
.
If you wish to fit a polynomial trend,
we offer an alternative to poly()
,
namely polynom()
, which avoids the
difficulty induced by transformations. It is completely analogous
to 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 rbord
.
References
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.
Huang, F. and Ogata, Y. Improvements of the maximum pseudo-likelihood estimators in various spatial statistical models. Journal of Computational and Graphical Statistics 8 (1999) 510-530. 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.
See Also
ppp
,
quadscheme
,
ppm.object
,
Poisson
,
Strauss
,
StraussHard
,
MultiStrauss
,
MultiStraussHard
,
Softcore
,
DiggleGratton
,
Pairwise
,
PairPiece
,
Geyer
,
LennardJones
,
Saturated
,
OrdThresh
,
Ord
Examples
data(nztrees)
ppm(nztrees)
# fit the stationary Poisson process
# to point pattern 'nztrees'
Q <- quadscheme(nztrees)
ppm(Q)
# equivalent.
ppm(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
ppm(nztrees, ~ polynom(x,2))
# fit the nonstationary Poisson process
# with intensity function lambda(x,y) = exp(a + bx + cx^2)
library(splines)
ppm(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
ppm(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
ppm(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.
# Huang-Ogata fit:
ppm(nztrees, ~1, Strauss(r=10), rbord=10, method="ho")
<testonly>ppm(nztrees, ~1, Strauss(r=10), rbord=10, method="ho", nsim=10)</testonly>
# COVARIATES
#
X <- rpoispp(42)
weirdfunction <- function(x,y){ 10 * x^2 + runif(length(x))}
Zimage <- as.im(weirdfunction, unit.square())
#
# (a) covariate values in pixel image
ppm(X, ~ y + Z, covariates=list(Z=Zimage))
#
# (b) covariate values in data frame
Q <- quadscheme(X)
xQ <- x.quad(Q)
yQ <- y.quad(Q)
Zvalues <- weirdfunction(xQ,yQ)
ppm(Q, ~ y + Z, covariates=data.frame(Z=Zvalues))
# Note Q not X
## MULTITYPE POINT PROCESSES ###
data(lansing)
# Multitype point pattern --- trees marked by species
<testonly># equivalent functionality - smaller dataset
data(betacells)</testonly>
# fit stationary marked Poisson process
# with different intensity for each species
ppm(lansing, ~ marks, Poisson())
<testonly>ppm(betacells, ~ marks, Poisson())</testonly>
# fit nonstationary marked Poisson process
# with different log-cubic trend for each species
ppm(lansing, ~ marks * polynom(x,y,3), Poisson())
<testonly>ppm(betacells, ~ marks * polynom(x,y,2), Poisson())</testonly>