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")

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. Model-fitting is currently performed by the method of maximum pseudolikelihood (Besag, 1975). Other options will be added in future versions. 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) approximation. The argument 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 "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. The usage of ppm() is closely analogous to the Splus/Rfunctions glm and gam. The analogy is: ll{ 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 Splus/Rformula 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). 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. See the examples below. 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 method of maximum pseudolikelihood coincides with maximum likelihood.

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.

The argument covariates, if supplied, should be either a data frame or a list of images. If it is a data frame, it must have as many rows as there are points in Q. The $i$th row of covariates should contain the values of spatial variables which have been observed at the $i$th point of Q. In this case the argument Q must be a quadrature scheme, not merely a point pattern.

Alternatively, covariates may be a list of images, each image giving the values of a spatial covariate at a fine grid of locations. The argument covariates should then be a named list, with the names corresponding to the names of covariates in the model formula trend. Each entry in the list must be an image object (of class "im", see im.object). The software will look up the pixel values of each image at the quadrature points. In this case the argument Q may be either a quadrature scheme or a point pattern.

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. 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 "periodic" (the periodic or toroidal edge correction in which opposite edges of a rectangular window are identified) and "translate" (the translation correction, see Baddeley 1998 and Baddeley and Turner 2000). For pairwise interaction there is also Ripley's isotropic correction, identified by "isotropic" or "Ripley". 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.

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. See 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.

The warning message ``Algorithm did not converge in: glm.fit( ...'' is sometimes obtained. It means that the glm fitting algorithm did not converge in a reasonable number of iterations. In most cases this can be fixed by taking a finer quadrature scheme. See quadscheme.

Warnings

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 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. 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

Aliases

• ppm

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.

 # 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>