mppm

Fit Point Process Model to Several Point Patterns

Fits a Gibbs point process model to several point patterns simultaneously.

Usage

mppm(formula, data, interaction=Poisson(), ...,
            iformula=NULL,
        use.gam = FALSE)

Details

This function fits a common point process model to a dataset containing several different point patterns. It extends the capabilities of the function ppm to deal with data such as

• replicated observations of spatial point patterns
• two groups of spatial point patterns
• a designed experiment in which the response from each unit is a point pattern.

Schematically, the data are regarded as the results of a designed experiment involving $n$ experimental units. Each unit has a response, and optionally some explanatory variables (covariates) describing the experimental conditions for that unit. In this context, the response from each unit is a point pattern. The value of a particular covariate for each unit can be either a single value (numerical, logical or factor), or a spatial covariate. A spatial covariate is a quantity that depends on spatial location, for example, the soil acidity or altitude at each location. For the purposes of mppm, a spatial covariate must be stored as a pixel image (object of class "im") which gives the values of the covariate at a fine grid of locations.

The argument data is a hyperframe (a generalisation of a data frame, see hyperframe). This is like a data frame except that the entries can be objects of any class. The hyperframe has one row for each experimental unit, and one column for each variable (response or explanatory variable).

The formula should be an Rformula. The left hand side of formula determines the response variable. This should be a single name, which should correspond to a column in data.

The right hand side of formula determines the spatial trend of the model. It specifies the linear predictor, and effectively represents the logarithm of the spatial trend. Variables in the formula must be the names of columns of data, or one of the reserved names [object Object],[object Object],[object Object]

The column of responses in data must consist of point patterns (objects of class "ppp"). The individual point pattern responses can be defined in different spatial windows. If some of the point patterns are marked, then they must all be marked, and must have the same type of marks.

The scope of models that can be fitted to each pattern is the same as the scope of ppm, that is, Gibbs point processes with interaction terms that belong to a specified list, including for example the Poisson process, Strauss process, Geyer's saturation model, and piecewise constant pairwise interaction models. The stochastic part of the model is determined by the arguments interaction and (optionally) iformula.

• In the simplest case,interactionis an object of class"interact", determining the interpoint interaction structure of the point process model, for all experimental units.
• Alternatively,interactionmay be a hyperframe, whose entries are objects of class"interact". It should have the same number of rows asdata.
  • Ifinteractionconsists of only one column, then the entry in rowiis taken to be the interpoint interaction for theith experimental unit (corresponding to theith row ofdata).
  • Ifinteractionhas more than one column, then the argumentiformulais also required. Each row ofinteractiondetermines several interpoint interaction structures that might be applied to the corresponding row ofdata. The choice of interaction is determined byiformula; this should be anRformula, without a left hand side. For example ifinteractionhas two columns calledAandBtheniformula = ~Bindicates that the interpoint interactions are taken from the second column.

References

Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283--322. Baddeley, A., Bischof, L., Sintorn, I.-M., Haggarty, S., Bell, M. and Turner, R. Analysis of a designed experiment where the response is a spatial point pattern. In preparation.

See Also

ppm, print.mppm, summary.mppm, coef.mppm,

Aliases

• mppm

Examples

# Waterstriders data
 data(waterstriders)
 H <- hyperframe(Y = waterstriders)
 mppm(Y ~ 1,  data=H)
 mppm(Y ~ 1,  data=H, Strauss(7))
 mppm(Y ~ id, data=H)
 mppm(Y ~ x,  data=H)

# Synthetic data from known model
n <- 10
H <- hyperframe(V=1:n,
                U=runif(n, min=-1, max=1),
                M=factor(letters[1 + (1:n) %% 3]))
H$Z <- setcov(square(1))
H$U <- with(H, as.im(U, as.rectangle(Z)))
H$Y <- with(H, rpoispp(eval.im(exp(2+3*Z)), win=as.rectangle(Z)))

fit <- mppm(Y ~Z + U + V, data=H)