R: Fit Point Process Model to Point Pattern Data

ppm.ppp {spatstat.core} | R Documentation

## Fit Point Process Model to Point Pattern Data

### Description

Fits a point process model to an observed point pattern.

### Usage

       ## S3 method for class 'ppp'
    ppm(Q, trend=~1, interaction=Poisson(),
           ...,
           covariates=data,
           data=NULL,
           covfunargs = list(),
           subset,
           clipwin,
           correction="border",
           rbord=reach(interaction),
           use.gam=FALSE,
           method="mpl",
           forcefit=FALSE,
           emend=project,
           project=FALSE,
           prior.mean = NULL,
           prior.var = NULL,
           nd = NULL,
           eps = NULL,
           gcontrol=list(),
           nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh),
           verb=TRUE,
           callstring=NULL)

       ## S3 method for class 'quad'
    ppm(Q, trend=~1, interaction=Poisson(),
           ...,
           covariates=data,
           data=NULL,
           covfunargs = list(),
           subset,
           clipwin,
           correction="border",
           rbord=reach(interaction),
           use.gam=FALSE,
           method="mpl",
           forcefit=FALSE,
           emend=project,
           project=FALSE,
           prior.mean = NULL,
           prior.var = NULL,
           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 R formula 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 a function that makes such an object, or `NULL` indicating that a Poisson process (stationary or nonstationary) should be fitted.
`...` | Ignored.
`data,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, tessellations or single numbers. See Details.
`subset` | Optional. An expression (which may involve the names of the Cartesian coordinates `x` and `y` and the names of entries in `data`) defining a subset of the spatial domain, to which the likelihood or pseudolikelihood should be restricted. See Details. The result of evaluating the expression should be either a logical vector, or a window (object of class `"owin"`) or a logical-valued pixel image (object of class `"im"`).
`clipwin` | Optional. A spatial window (object of class `"owin"`) to which data will be restricted, before model-fitting is performed. 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"`, `"periodic"`, `"translate"` and `"none"`, depending on the `interaction`.
`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, `"logi"` for the Logistic Likelihood method, `"VBlogi"` for the Variational Bayes Logistic Likelihood method, 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.
`emend,project` | (These are equivalent: `project` is an older name for `emend`.) Logical value. Setting `emend=TRUE` will ensure that the fitted model is always a valid point process by applying `emend.ppm`.
`prior.mean` | Optional vector of prior means for canonical parameters (for `method="VBlogi"`). See Details.
`prior.var` | Optional prior variance covariance matrix for canonical parameters (for `method="VBlogi"`). See Details.
`nd` | Optional. Integer or pair of integers. The dimension of the grid of dummy points (`nd * nd` or `nd[1] * nd[2]`) used to evaluate the integral in the pseudolikelihood. Incompatible with `eps`.
`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 to `gam.control` if `use.gam=TRUE`) controlling the model-fitting algorithm.
`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"`)
`callstring` | Internal use only.

### Details

NOTE: This help page describes the old syntax of the function `ppm`, described in many older documents. This old syntax is still supported. However, if you are learning about `ppm` for the first time, we recommend you use the new syntax described in the help file for `ppm`.

This function fits a point process model to an observed point pattern. The model may include spatial trend, interpoint interaction, and dependence on covariates.

basic use:

  In basic use, `Q` is a point pattern dataset (an object of class `"ppp"`) to which we wish to fit a model.

  The syntax of `ppm()` is closely analogous to the R functions `glm` and `gam`. The analogy is:

  glm | ppm
  `formula` | `trend`
  `family` | `interaction`

  The point process model to be fitted is specified by the arguments `trend` and `interaction` which are respectively analogous to the `formula` and `family` arguments of glm().

  Systematic effects (spatial trend and/or dependence on spatial covariates) are specified by the argument `trend`. This is an R formula object, which may be expressed in terms of the Cartesian coordinates `x`, `y`, the marks `marks`, or the variables in `covariates` (if supplied), or both. It specifies the logarithm of the first order potential of the process. The formula should not use any names beginning with `.mpl` as these are reserved for internal use. If `trend` is absent or equal to the default, `~1`, then the model to be fitted is stationary (or at least, its first order potential is constant).

  The symbol `.` in the trend expression stands for all the covariates supplied in the argument `data`. For example the formula `~ .` indicates an additive model with a main effect for each covariate in `data`.

  Stochastic interactions between random points of the point process are defined by the argument `interaction`. This is an object of class `"interact"` which is initialised in a very similar way to the usage of family objects in `glm` and `gam`. The models currently available are: `AreaInter`, `BadGey`, `Concom`, `DiggleGatesStibbard`, `DiggleGratton`, `Fiksel`, `Geyer`, `Hardcore`, `HierHard`, `HierStrauss`, `HierStraussHard`, `Hybrid`, `LennardJones`, `MultiHard`, `MultiStrauss`, `MultiStraussHard`, `OrdThresh`, `Ord`, `Pairwise`, `PairPiece`, `Penttinen`, `Poisson`, `Saturated`, `SatPiece`, `Softcore`, `Strauss`, `StraussHard` and `Triplets`. See the examples below. It is also possible to combine several interactions using `Hybrid`.

  If `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 methods of maximum pseudolikelihood and maximum logistic likelihood coincide with maximum likelihood.

  The fitted point process model returned by this function can be printed (by the print method `print.ppm`) to inspect the fitted parameter values. If a nonparametric spatial trend was fitted, this can be extracted using the predict method `predict.ppm`.

Models with covariates:

  To fit a model involving spatial covariates other than the Cartesian coordinates x and y, the values of the covariates should be supplied in the argument `covariates`. Note that it is not sufficient to have observed the covariate only at the points of the data point pattern; the covariate must also have been observed at other locations in the window.

  Typically the argument `covariates` is a list, with names corresponding to variables in the `trend` formula. Each entry in the list is either

  a pixel image,

    giving the values of a spatial covariate at a fine grid of locations. It should be an object of class `"im"`, see `im.object`.

  a function,

    which can be evaluated at any location `(x,y)` to obtain the value of the spatial covariate. It should be a `function(x, y)` or `function(x, y, ...)` in the R language. The first two arguments of the function should be the Cartesian coordinates x and y. The function may have additional arguments; if the function does not have default values for these additional arguments, then the user must supply values for them, in `covfunargs`. See the Examples.

  a window,

    interpreted as a logical variable which is `TRUE` inside the window and `FALSE` outside it. This should be an object of class `"owin"`.

  a tessellation,

    interpreted as a factor covariate. For each spatial location, the factor value indicates which tile of the tessellation it belongs to. This should be an object of class `"tess"`.

  a single number,

    indicating a covariate that is constant in this dataset.

  The software will look up the values of each covariate at the required locations (quadrature points).

  Note that, for covariate functions, only the name of the function appears in the trend formula. A covariate function is treated as if it were a single variable. The function arguments do not appear in the trend formula. See the Examples.

  If `covariates` is a list, the list entries should have names corresponding to the names of covariates in the model formula `trend`. The variable names `x`, `y` and `marks` are reserved for the Cartesian coordinates and the mark values, and these should not be used for variables in `covariates`.

  If `covariates` is a data frame, `Q` must be a quadrature scheme (see under Quadrature Schemes below). Then `covariates` must have as many rows as there are points in `Q`. The ith row of `covariates` should contain the values of spatial variables which have been observed at the ith point of `Q`.

Quadrature schemes:

  In advanced use, `Q` may be a ‘quadrature scheme’. This was originally just a technicality but it has turned out to have practical uses, as we explain below.

  Quadrature schemes are required for our implementation of the method of maximum pseudolikelihood. The definition of the pseudolikelihood involves an integral over the spatial window containing the data. In practice this integral must be approximated by a finite sum over a set of quadrature points. We use the technique of Baddeley and Turner (2000), a generalisation of the Berman-Turner (1992) device. In this technique the quadrature points for the numerical approximation include all the data points (points of the observed point pattern) as well as additional ‘dummy’ points.

  Quadrature schemes are also required for the method of maximum logistic likelihood, which combines the data points with additional ‘dummy’ points.

  A quadrature scheme is an object of class `"quad"` (see `quad.object`) 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. If `Q` is simply a point pattern (of class `"ppp"`, see `ppp.object`) then it is interpreted as specifying the data points only; a set of dummy points specified by `default.dummy()` is added, and the default weighting rule is invoked to compute the quadrature weights.

  Finer quadrature schemes (i.e. those with more dummy points) generally yield a better approximation, at the expense of higher computational load.

  An easy way to fit models using a finer quadrature scheme is to let `Q` be the original point pattern data, and use the argument `nd` to determine the number of dummy points in the quadrature scheme.

  Complete control over the quadrature scheme is possible. See `quadscheme` for an overview. Use `quadscheme(X, D, method="dirichlet")` to compute quadrature weights based on the Dirichlet tessellation, or `quadscheme(X, D, method="grid")` to compute quadrature weights by counting points in grid squares, where `X` and `D` are the patterns of data points and dummy points respectively. Alternatively use `pixelquad` to make a quadrature scheme with a dummy point at every pixel in a pixel image.

  A practical advantage of quadrature schemes arises when we want to fit a model involving covariates (e.g. soil pH). Suppose we have only been able to observe the covariates at a small number of locations. Suppose `cov.dat` is a data frame containing the values of the covariates at the data points (i.e.\ `cov.dat[i,]` contains the observations for the `i`th data point) and `cov.dum` is another data frame (with the same columns as `cov.dat`) containing the covariate values at another set of points whose locations are given by the point pattern `Y`. Then setting `Q = quadscheme(X,Y)` combines the data points and dummy points into a quadrature scheme, and `covariates = rbind(cov.dat, cov.dum)` combines the covariate data frames. We can then fit the model by calling `ppm(Q, ..., covariates)`.

Model-fitting technique:

  There are several choices for the technique used to fit the model.

  method="mpl"

    (the default): the model will be fitted by maximising the pseudolikelihood (Besag, 1975) using the Berman-Turner computational approximation (Berman and Turner, 1992; Baddeley and Turner, 2000). Maximum pseudolikelihood is equivalent to maximum likelihood if the model is a Poisson process. Maximum pseudolikelihood is biased if the interpoint interaction is very strong, unless there is a large number of dummy points. The default settings for `method='mpl'` specify a moderately large number of dummy points, striking a compromise between speed and accuracy.

  method="logi":

    the model will be fitted by maximising the logistic likelihood (Baddeley et al, 2014). This technique is roughly equivalent in speed to maximum pseudolikelihood, but is believed to be less biased. Because it is less biased, the default settings for `method='logi'` specify a relatively small number of dummy points, so that this method is the fastest, in practice.

  method="VBlogi":

    the model will be fitted in a Bayesian setup by maximising the posterior probability density for the canonical model parameters. This uses the variational Bayes approximation to the posterior derived from the logistic likelihood as described in Rajala (2014). The prior is assumed to be multivariate Gaussian with mean vector `prior.mean` and variance-covariance matrix `prior.var`.

  method="ho":

    the model will be fitted by applying the approximate maximum likelihood method of Huang and Ogata (1999). See below. The Huang-Ogata method is slower than the other options, but has better statistical properties.

  Note that `method='logi'`, `method='VBlogi'` and `method='ho'` involve randomisation, so that the results are subject to random variation.

Huang-Ogata method:

  If `method="ho"` then the model will be fitted using the Huang-Ogata (1999) approximate maximum likelihood method. First the model is fitted by maximum pseudolikelihood as described above, yielding an initial estimate of the parameter vector theta0. From this initial model, `nsim` simulated realisations are generated. The score and Fisher information of the model at theta=theta0 are estimated from the simulated realisations. Then one step of the Fisher scoring algorithm is taken, yielding an updated estimate theta1. The corresponding model is returned.

  Simulated realisations are generated using `rmh`. The iterative behaviour of the Metropolis-Hastings algorithm is controlled by the arguments `start` and `control` which are passed to `rmh`.

  As a shortcut, the argument `nrmh` determines the number of Metropolis-Hastings iterations run to produce one simulated realisation (if `control` is absent). Also if `start` is absent or equal to `NULL`, it defaults to `list(n.start=N)` where `N` is the number of points in the data point pattern.

Edge correction

  Edge correction should be applied to the sufficient statistics of the model, to reduce bias. The argument `correction` is the name of an edge correction method. The default `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 `rbord` 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 include `correction="periodic"` (the periodic or toroidal edge correction in which opposite edges of a rectangular window are identified) and `correction="translate"` (the translation correction, see Baddeley 1998 and Baddeley and Turner 2000). For pairwise interaction models there is also Ripley's isotropic correction, identified by `correction="isotropic"` or `"Ripley"`.

Subsetting

  The arguments `subset` and `clipwin` specify that the model should be fitted to a restricted subset of the available data. These arguments are equivalent for Poisson point process models, but different for Gibbs models. If `clipwin` is specified, then all the available data will be restricted to this spatial region, and data outside this region will be discarded, before the model is fitted. If `subset` is specified, then no data are deleted, but the domain of integration of the likelihood or pseudolikelihood is restricted to the `subset`. For Poisson models, these two arguments have the same effect; but for a Gibbs model, interactions between points inside and outside the `subset` are taken into account, while interactions between points inside and outside the `clipwin` are ignored.

### Value

An object of class `"ppm"` describing a fitted point process model.

See `ppm.object` for details of the format of this object and methods available for manipulating it.

### Interaction parameters

Apart from the Poisson model, every point process model fitted by `ppm` has parameters that determine the strength and range of ‘interaction’ or dependence between points. These parameters are of two types:

regular parameters:

  A parameter phi is called regular if the log likelihood is a linear function of theta where theta = theta(psi) is some transformation of psi. [Then theta is called the canonical parameter.]

irregular parameters

  Other parameters are called irregular.

Typically, regular parameters determine the ‘strength’ of the interaction, while irregular parameters determine the ‘range’ of the interaction. For example, the Strauss process has a regular parameter gamma controlling the strength of interpoint inhibition, and an irregular parameter r determining the range of interaction.

The `ppm` command is only designed to estimate regular parameters of the interaction. It requires the values of any irregular parameters of the interaction to be fixed. For example, to fit a Strauss process model to the `cells` dataset, you could type `ppm(cells, ~1, Strauss(r=0.07))`. Note that the value of the irregular parameter `r` must be given. The result of this command will be a fitted model in which the regular parameter gamma has been estimated.

To determine the irregular parameters, there are several practical techniques, but no general statistical theory available. Useful techniques include maximum profile pseudolikelihood, which is implemented in the command `profilepl`, and Newton-Raphson maximisation, implemented in the experimental command `ippm`.

Some irregular parameters can be estimated directly from data: the hard-core radius in the model `Hardcore` and the matrix of hard-core radii in `MultiHard` can be estimated easily from data. In these cases, `ppm` allows the user to specify the interaction without giving the value of the irregular parameter. The user can give the hard core interaction as `interaction=Hardcore()` or even `interaction=Hardcore`, and the hard core radius will then be estimated from the data.

### Error and Warning Messages

Some common error messages and warning messages are listed below, with explanations.

“System is computationally singular”

  The Fisher information matrix of the fitted model has a determinant close to zero, so that the matrix cannot be inverted, and the software cannot calculate standard errors or confidence intervals. This error is usually reported when the model is printed, because the `print` method calculates standard errors for the fitted parameters. Singularity usually occurs because the spatial coordinates in the original data were very large numbers (e.g. expressed in metres) so that the fitted coefficients were very small numbers. The simple remedy is to rescale the data, for example, to convert from metres to kilometres by `X <- rescale(X, 1000)`, then re-fit the model. Singularity can also occur if the covariate values are very large numbers, or if the covariates are approximately collinear.

“Covariate values were NA or undefined at X% (M out of N) of the quadrature points”

  The covariate data (typically a pixel image) did not provide values of the covariate at some of the spatial locations in the observation window of the point pattern. This means that the spatial domain of the pixel image does not completely cover the observation window of the point pattern. If the percentage is small, this warning can be ignored - typically it happens because of rounding effects which cause the pixel image to be one-pixel-width narrower than the observation window. However if more than a few percent of covariate values are undefined, it would be prudent to check that the pixel images are correct, and are correctly registered in their spatial relation to the observation window.

“Model is unidentifiable”

  It is not possible to estimate all the model parameters from this dataset. The error message gives a further explanation, such as “data pattern is empty”. Choose a simpler model, or check the data.

“N data points are illegal (zero conditional intensity)”

  In a Gibbs model (i.e. with interaction between points), the conditional intensity may be zero at some spatial locations, indicating that the model forbids the presence of a point at these locations. However if the conditional intensity is zero at a data point, this means that the model is inconsistent with the data. Modify the interaction parameters so that the data point is not illegal (e.g. reduce the value of the hard core radius) or choose a different interaction.

### Warnings

The implementation of the Huang-Ogata method is experimental; several bugs were fixed in spatstat 1.19-0.

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`.

By default (if `emend=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 `emend=TRUE`. See also the functions `valid.ppm` and `emend.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`.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### References

Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for spatial Gibbs point processes. Biometrika 101 (2) 377–392.

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.

Rajala T. (2014) A note on Bayesian logistic regression for spatial exponential family Gibbs point processes, Preprint on ArXiv.org. https://arxiv.org/abs/1411.0539

### See Also

`ppm.object` for details of how to print, plot and manipulate a fitted model.

`ppp` and `quadscheme` for constructing data.

Interactions: `AreaInter`, `BadGey`, `Concom`, `DiggleGatesStibbard`, `DiggleGratton`, `Fiksel`, `Geyer`, `Hardcore`, `HierHard`, `HierStrauss`, `HierStraussHard`, `Hybrid`, `LennardJones`, `MultiHard`, `MultiStrauss`, `MultiStraussHard`, `OrdThresh`, `Ord`, `Pairwise`, `PairPiece`, `Penttinen`, `Poisson`, `Saturated`, `SatPiece`, `Softcore`, `Strauss`, `StraussHard` and `Triplets`.

See `profilepl` for advice on fitting nuisance parameters in the interaction, and `ippm` for irregular parameters in the trend.

See `valid.ppm` and `emend.ppm` for ensuring the fitted model is a valid point process.

### Examples

     # fit the stationary Poisson process
     # to point pattern 'nztrees'

     ppm(nztrees)
     ppm(nztrees ~ 1)
     # equivalent.

     Q <- quadscheme(nztrees)
     ppm(Q)
     # equivalent.

     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

     # For other examples, see help(ppm)

[Package spatstat.core version 2.3-0 Index]