ppm
Fit Point Process Model to Data
Fits a point process model to an observed point pattern
Usage
ppm(Q, trend=~1, interaction=Poisson(),
...,
covariates=NULL,
covfunargs = list(),
correction="border",
rbord=reach(interaction),
use.gam=FALSE,
method="mpl",
forcefit=FALSE,
project=FALSE,
nd = NULL,
eps = NULL,
gcontrol=list(),
nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh),
verb=TRUE,
callstring=NULL)
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. - ...
- Ignored.
- covariates
- The values of any spatial covariates (other than the Cartesian coordinates) required by the model. Either a data frame, or a list whose entries are images, functions, windows or single numbers. See Details.
- covfunargs
- A named list containing the values of any additional arguments required by covariate functions.
- 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. - project
- Logical. Setting
project=TRUE
will ensure that the fitted model is always a valid point process by applyingproject.ppm
. - nd
- Optional. Integer or pair of integers.
The dimension of the grid of dummy points (
nd * nd
ornd[1] * nd[2]
) used to evaluate the integral in the pseudolikelihood. Incompatible witheps
. - eps
- Optional.
A positive number, or a vector of two positive numbers, giving the
horizontal and vertical spacing, respectively, of the grid of
dummy points. Incompatible with
nd
. - gcontrol
- Optional. List of parameters passed to
glm.control
(or passed togam.control
ifuse.gam=TRUE
) controlling the model-fitting algo - 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"
) - callstring
- Internal use only.
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. Seeppm.object
for details of the format of this object and methods available for manipulating it.
Warnings
The implementation of the Huang-Ogata method is experimental;
several bugs were fixed in ppm()$theta[2]
less than or equal to 0
.
By default (if project=FALSE
) the algorithm
maximises the pseudolikelihood
without constraining the parameters, and does not apply any checks for
sanity after fitting the model.
This is because the fitted parameter value
could be useful information for data analysis.
To constrain the parameters to ensure that the model is a valid
point process, set project=TRUE
. See also the functions
valid.ppm
and project.ppm
.
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
ppm.object
for details of how to
print, plot and manipulate a fitted model.
ppp
and quadscheme
for constructing data.
Interactions:
Poisson
,
AreaInter
,
BadGey
,
DiggleGatesStibbard
,
DiggleGratton
,
Geyer
,
Fiksel
,
Hardcore
,
LennardJones
,
MultiStrauss
,
MultiStraussHard
,
OrdThresh
,
Ord
,
Pairwise
,
PairPiece
,
Saturated
,
SatPiece
,
Softcore
,
Strauss
and
StraussHard
.
See profilepl
for advice on
fitting nuisance parameters in the interaction,
and ippm
for irregular parameters in the trend.
See valid.ppm
and project.ppm
for
ensuring the fitted model is a valid point process.
Examples
ppm(nztrees)
# fit the stationary Poisson process
# to point pattern 'nztrees'
Q <- quadscheme(nztrees)
ppm(Q)
# equivalent.
ppm(nztrees, nd=128)
<testonly>ppm(nztrees, nd=16)</testonly>
fit1 <- 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
fit1
coef(fit1)
coef(summary(fit1))
ppm(nztrees, ~ polynom(x,2))
<testonly>ppm(nztrees, ~ polynom(x,2), nd=16)</testonly>
# 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)
<testonly>ppm(nztrees, ~1, Strauss(r=10), rbord=10, nd=16)</testonly>
# 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")
<testonly>ppm(nztrees, ~ x, Strauss(13), correction="periodic", nd=16)</testonly>
# 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), method="ho")
<testonly>ppm(nztrees, ~1, Strauss(r=10), method="ho", nd=16, nsim=10)</testonly>
# COVARIATES
#
X <- rpoispp(42)
weirdfunction <- function(x,y){ 10 * x^2 + 5 * sin(10 * y) }
#
# (a) covariate values as function
ppm(X, ~ y + Z, covariates=list(Z=weirdfunction))
#
# (b) covariate values in pixel image
Zimage <- as.im(weirdfunction, unit.square())
ppm(X, ~ y + Z, covariates=list(Z=Zimage))
#
# (c) 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
# COVARIATE FUNCTION WITH EXTRA ARGUMENTS
#
f <- function(x,y,a){ y - a }
ppm(X, ~x + f, covariates=list(f=f), covfunargs=list(a=1/2))
## MULTITYPE POINT PROCESSES ###
# fit stationary marked Poisson process
# with different intensity for each species
ppm(lansing, ~ marks, Poisson())
<testonly>a <- ppm(amacrine, ~ marks, Poisson(), nd=16)</testonly>
# fit nonstationary marked Poisson process
# with different log-cubic trend for each species
ppm(lansing, ~ marks * polynom(x,y,3), Poisson())
<testonly>ppm(amacrine, ~ marks * polynom(x,y,2), Poisson(), nd=16)</testonly>