# ppm

##### Fit Point Process Model to Data

Fits a point process model to an observed point pattern

##### 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, or`NULL`

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 using`gam`

instead of`glm`

. - 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. If`forcefit=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 (for`method="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 the`predict`

method for this object (see`predict.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>
```

*Documentation reproduced from package spatstat, version 1.9-1, License: GPL version 2 or newer*