# profilepl

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##### Fit Models by Profile Maximum Pseudolikelihood or AIC

Fits point process models by maximising the profile likelihood, profile pseudolikelihood, profile composite likelihood or AIC.

Keywords
models, spatial
##### Usage
profilepl(s, f, …, aic=FALSE, rbord=NULL, verbose = TRUE, fast=TRUE)
##### Arguments
s

Data frame containing values of the irregular parameters over which the criterion will be computed.

f

Function (such as Strauss) that generates an interpoint interaction object, given values of the irregular parameters.

Data passed to ppm to fit the model.

aic

Logical value indicating whether to find the parameter values which minimise the AIC (aic=TRUE) or maximise the profile likelihood (aic=FALSE, the default).

rbord

Radius for border correction (same for all models). If omitted, this will be computed from the interactions.

verbose

Logical value indicating whether to print progress reports.

fast

Logical value indicating whether to use a faster, less accurate model-fitting technique when computing the profile pseudolikelihood. See Section on Speed and Accuracy.

##### Details

The model-fitting function ppm fits point process models to point pattern data. However, only the ‘regular’ parameters of the model can be fitted by ppm. The model may also depend on ‘irregular’ parameters that must be fixed in any call to ppm.

This function profilepl is a wrapper which finds the values of the irregular parameters that give the best fit. If aic=FALSE (the default), the best fit is the model which maximises the likelihood (if the models are Poisson processes) or maximises the pseudolikelihood or logistic likelihood. If aic=TRUE then the best fit is the model which minimises the Akaike Information Criterion AIC.ppm.

The argument s must be a data frame whose columns contain values of the irregular parameters over which the maximisation is to be performed.

An irregular parameter may affect either the interpoint interaction or the spatial trend.

interaction parameters:

in a call to ppm, the argument interaction determines the interaction between points. It is usually a call to a function such as Strauss. The arguments of this call are irregular parameters. For example, the interaction radius parameter $$r$$ of the Strauss process, determined by the argument r to the function Strauss, is an irregular parameter.

trend parameters:

in a call to ppm, the spatial trend may depend on covariates, which are supplied by the argument covariates. These covariates may be functions written by the user, of the form function(x,y,...), and the extra arguments … are irregular parameters.

The argument f determines the interaction for each model to be fitted. It would typically be one of the functions Poisson, AreaInter, BadGey, DiggleGatesStibbard, DiggleGratton, Fiksel, Geyer, Hardcore, LennardJones, OrdThresh, Softcore, Strauss or StraussHard. Alternatively it could be a function written by the user.

Columns of s which match the names of arguments of f will be interpreted as interaction parameters. Other columns will be interpreted as trend parameters.

The data frame s must provide values for each argument of f, except for the optional arguments, which are those arguments of f that have the default value NA.

To find the best fit, each row of s will be taken in turn. Interaction parameters in this row will be passed to f, resulting in an interaction object. Then ppm will be applied to the data ... using this interaction. Any trend parameters will be passed to ppm through the argument covfunargs. This results in a fitted point process model. The value of the log pseudolikelihood or AIC from this model is stored. After all rows of s have been processed in this way, the row giving the maximum value of log pseudolikelihood will be found.

The object returned by profilepl contains the profile pseudolikelihood (or profile AIC) function, the best fitting model, and other data. It can be plotted (yielding a plot of the log pseudolikelihood or AIC values against the irregular parameters) or printed (yielding information about the best fitting values of the irregular parameters).

In general, f may be any function that will return an interaction object (object of class "interact") that can be used in a call to ppm. Each argument of f must be a single value.

##### Value

An object of class "profilepl". There are methods for plot, print, summary, simulate, as.ppm, fitin and parameters for objects of this class.

The components of the object include

fit

Best-fitting model

param

The data frame s

iopt

Row index of the best-fitting parameters in s

To extract the best fitting model you can also use as.ppm.

##### Speed and Accuracy

Computation of the profile pseudolikelihood can be time-consuming. We recommend starting with a small experiment in which s contains only a few rows of values. This will indicate roughly the optimal values of the parameters. Then a full calculation using more finely spaced values can identify the exact optimal values.

It is normal that the procedure appears to slow down at the end. During the computation of the profile pseudolikelihood, the model-fitting procedure is accelerated by omitting some calculations that are not needed for computing the pseudolikelihood. When the optimal parameter values have been identified, they are used to fit the final model in its entirety. Fitting the final model can take longer than computing the profile pseudolikelihood.

If fast=TRUE (the default), then additional shortcuts are taken in order to accelerate the computation of the profile log pseudolikelihood. These shortcuts mean that the values of the profile log pseudolikelihood in the result (\$prof) may not be equal to the values that would be obtained if the model was fitted normally. Currently this happens only for the area interaction AreaInter. It may be wise to do a small experiment with fast=TRUE and then a definitive calculation with fast=FALSE.

##### References

Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283--322.

plot.profilepl

• profilepl
##### Examples
# NOT RUN {
# one irregular parameter
rr <- data.frame(r=seq(0.05,0.15, by=0.01))

# }
# NOT RUN {
ps <- profilepl(rr, Strauss, cells)
ps
plot(ps)

# two irregular parameters
rs <- expand.grid(r=seq(0.05,0.15, by=0.01),sat=1:3)

# }
# NOT RUN {
pg <- profilepl(rs, Geyer, cells)
pg
as.ppm(pg)

# multitype pattern with a common interaction radius

# }
# NOT RUN {
RR <- data.frame(R=seq(0.03,0.05,by=0.01))
MS <- function(R) { MultiStrauss(radii=diag(c(R,R))) }
pm <- profilepl(RR, MS, amacrine ~marks)

# }
# NOT RUN {