Fits a Gibbs point process model to several point patterns simultaneously.
mppm(formula, data, interaction=Poisson(), ...,
iformula=NULL,
random=NULL,
use.gam = FALSE,
reltol.pql=1e-3,
gcontrol=list())A formula describing the systematic part of the model.
Variables in the formula are names of columns in data.
A hyperframe (object of class "hyperframe",
see hyperframe) containing the
point pattern responses and the explanatory variables.
Interpoint interaction(s) appearing in the model.
Either an object of class "interact"
describing the point process interaction
structure, or a hyperframe (with the same number of
rows as data) whose entries are objects of class
"interact".
Arguments passed to ppm controlling
the fitting procedure.
Optional. A formula (with no left hand side)
describing the interaction to be applied to each case.
Each variable name in the formula should either be the name of a column
in the hyperframe interaction, or the name of a column
in the hyperframe data that is a vector or factor.
Optional. A formula (with no left hand side)
describing a random effect. Variable names in the formula
may be any of the column names of data and interaction.
The formula must be recognisable to lme.
Relative tolerance for successive steps in
the penalised quasi-likelihood algorithm,
used when the model includes random effects.
The algorithm terminates when the root mean square of the
relative change in coefficients is less than reltol.pql.
List of arguments to control the fitting algorithm.
Arguments are passed to glm.control
or gam.control
or lmeControl
depending on the kind of model being fitted.
If the model has random effects, the arguments are passed to
lmeControl. Otherwise,
if use.gam=TRUE the arguments are passed to
gam.control, and if
use.gam=FALSE (the default) they are passed to
glm.control.
An object of class "mppm" representing the
fitted model.
There are methods for
print, summary, coef,
AIC, anova, fitted, fixef, logLik,
plot, predict, ranef, residuals,
summary, terms and vcov for this class.
The default methods for update
and formula also work on this class.
It is also possible to include random effects in the
trend term. The argument random is a formula,
with no left-hand side, that specifies the structure of the
random effects. The formula should be recognisable to
lme (see the description of the argument random
for lme).
The names in the formula random may be any of the covariates
supplied by data.
Additionally the formula may involve the name
id, which is a factor representing the
serial number (\(1\) to \(n\)) of the point pattern in the
list X.
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.
The syntax of this function is similar to that of
standard R model-fitting functions like lm and
glm. The first argument formula is an R formula
describing the systematic part of the model. The second argument
data contains the responses and the explanatory variables.
Other arguments determine the stochastic structure of the model.
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 R formula.
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
Cartesian coordinates of location
Mark attached to point
which is a factor representing the serial number (\(1\) to \(n\)) of the point pattern, i.e. the row number in the data hyperframe.
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.
Additionally, it is possible to include random effects
as explained in the section on Random Effects below.
The stochastic part of the model is determined by
the arguments interaction and (optionally) iformula.
In the simplest case, interaction is
an object of class "interact",
determining the interpoint interaction structure of the point
process model, for all experimental units.
Alternatively, interaction may be a hyperframe,
whose entries are objects of class "interact".
It should have the same number of rows as data.
If interaction consists of only one column,
then the entry in row i is taken to be the
interpoint interaction for the ith experimental unit
(corresponding to the ith row of data).
If interaction has more than one column,
then the argument iformula is also required.
Each row of interaction determines
several interpoint interaction structures that might be applied
to the corresponding row of data.
The choice of interaction is determined by iformula;
this should be an R formula,
without a left hand side.
For example if interaction has two columns called
A and B then iformula = ~B indicates that the
interpoint interactions are taken from the second column.
Variables in iformula
typically refer to column names of interaction.
They can also be names of columns in
data, but only for columns of numeric, logical or factor
values. For example iformula = ~B * group (where group
is a column of data that contains a factor) causes the
model with interpoint interaction B to be fitted
with different interaction parameters for each level of group.
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.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
Bell, M. and Grunwald, G. (2004) Mixed models for the analysis of replicated spatial point patterns. Biostatistics 5, 633--648.
# NOT RUN {
# Waterstriders data
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))))
fit <- mppm(Y ~Z + U + V, data=H)
# }
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