ppm

Fit Point Process Model to Data

Fits a point process model to an observed point pattern.

Usage

ppm(Q, ...)
   ## S3 method for class 'formula':
ppm(Q, interaction=NULL, \dots, data=NULL)

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.

The function ppm is generic, with methods for the classes formula, ppp and quad. This page describes the method for a formula.

The first argument is a formula in the Rlanguage describing the spatial trend model to be fitted. It has the general form pattern ~ trend where the left hand side pattern is usually the name of a spatial point pattern (object of class "ppp") to which the model should be fitted, or an expression which evaluates to a point pattern; and the right hand side trend is an expression specifying the spatial trend of the model.

Systematic effects (spatial trend and/or dependence on spatial covariates) are specified by the trend expression on the right hand side of the formula. The trend may involve the Cartesian coordinates x, y, the marks marks, the names of entries in the argument data (if supplied), or the names of objects that exist in the Rsession. The trend formula specifies the logarithm of the intensity of a Poisson process, or in general, the logarithm of the first order potential of the Gibbs process. The formula should not use any names beginning with .mpl as these are reserved for internal use. If the formula is pattern~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. The models currently available are: Poisson, AreaInter, BadGey, Concom, DiggleGatesStibbard, DiggleGratton, Fiksel, Geyer, Hardcore, LennardJones, MultiStrauss, MultiStraussHard, OrdThresh, Ord, Pairwise, PairPiece, Saturated, SatPiece, Softcore, Strauss and StraussHard. 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.

To fit a model involving spatial covariates other than the Cartesian coordinates $x$ and $y$, the values of the covariates should either be supplied in the argument data, or should be stored in objects that exist in the Rsession. 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.

If it is given, the argument data is typically a list, with names corresponding to variables in the trend formula. Each entry in the list is either [object Object],[object Object],[object Object],[object Object],[object Object] 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 data is a list, the list entries should have names corresponding to (some of) 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 data.

Alternatively, data may be a data frame giving the values of the covariates at specified locations. Then pattern should be a quadrature scheme (object of class "quad") giving the corresponding locations. See ppm.quad for details.

References

Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2013) A logistic regression estimating function for spatial Gibbs point processes. Research Report, Centre for Stochastic Geometry and Bioimaging, Denmark. www.csgb.dk Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for spatial Gibbs point processes. Biometrika, in press.

Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 283--322. Berman, M. and Turner, T.R. (1992) Approximating point process likelihoods with GLIM. Applied Statistics 41, 31--38. Besag, J. (1975) Statistical analysis of non-lattice data. The Statistician 24, 179-195. Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. (1994) On parameter estimation for pairwise interaction processes. International Statistical Review 62, 99-117.

Huang, F. and Ogata, Y. (1999) Improvements of the maximum pseudo-likelihood estimators in various spatial statistical models. Journal of Computational and Graphical Statistics 8, 510--530. Jensen, J.L. and Moeller, M. (1991) Pseudolikelihood for exponential family models of spatial point processes. Annals of Applied Probability 1, 445--461. Jensen, J.L. and Kuensch, H.R. (1994) On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes, Annals of the Institute of Statistical Mathematics 46, 475--486.

See Also

ppm.ppp and ppm.quad for more details on the fitting technique and edge correction.

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, Hybrid, 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.

Aliases

• ppm
• ppm.formula

Examples

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

 ppm(nztrees ~ 1)

 Q <- quadscheme(nztrees) 
 ppm(Q ~ 1) 
 # 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

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.


# Compare Maximum Pseudolikelihood and Huang-Ogata fits:
ppm(swedishpines ~ 1, Strauss(9))

ppm(swedishpines ~ 1, Strauss(9), method="ho")
<testonly>ppm(swedishpines ~ 1, Strauss(9), method="ho", nd=16, nsim=8)</testonly>

 # COVARIATES
 #
 X <- rpoispp(42)
 weirdfunction <- function(x,y){ 10 * x^2 + 5 * sin(10 * y) }
 #
 # (a) covariate values as function
 ppm(X ~ y + 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, data=data.frame(Z=Zvalues))
 # Note Q not X

 # COVARIATE FUNCTION WITH EXTRA ARGUMENTS
 #
f <- function(x,y,a){ y - a }
ppm(X ~ x + f, covfunargs=list(a=1/2))

 # COVARIATE: inside/outside window
 b <- owin(c(0.1, 0.6), c(0.1, 0.9))
 ppm(X ~ b)

 ## 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>