Given a fitted point process model obtained by ppm
,
evaluate the spatial trend or the conditional intensity of the model
at new locations.
1 2 3 4 5 6 7 8 9 
object 
A fitted point process model, typically obtained from
the modelfitting algorithm 
window 
Optional. A window (object of class 
ngrid 
Optional. Dimensions of a rectangular grid of locations
inside 
locations 
Optional. Data giving the exact
x,y coordinates (and marks, if required)
of locations at which predictions should be computed.
Either a point pattern, or a data frame with columns named 
covariates 
Values of external covariates required by the model. Either a data frame or a list of images. See Details. 
type 
Character string.
Indicates which property of the fitted model should be predicted.
Options are 
se 
Logical value indicating whether to calculate standard errors as well. 
interval 
String (partially matched) indicating whether to produce
estimates ( 
level 
Coverage probability for the confidence or prediction interval. 
X 
Optional. A point pattern (object of class 
correction 
Name of the edge correction to be used
in calculating the conditional intensity.
Options include 
... 
Ignored. 
new.coef 
Numeric vector of parameter values to replace the
fitted model parameters 
check 
Logical value indicating whether to check the internal format
of 
repair 
Logical value indicating whether to repair the internal format
of 
This function computes properties of a fitted spatial point process
model (object of class "ppm"
). For a Poisson point process
it can compute the fitted intensity function, or the expected number of
points in a region. For a Gibbs point process it can compute the
spatial trend (first order potential), conditional intensity,
and approximate intensity of the process.
Point estimates, standard errors,
confidence intervals and prediction intervals are available.
Given a point pattern dataset, we may fit
a point process model to the data using the
modelfitting algorithm ppm
. This
returns an object of class "ppm"
representing
the fitted point process model (see ppm.object
).
The parameter estimates in this fitted model can be read off
simply by printing the ppm
object.
The spatial trend, conditional intensity and intensity of the
fitted model are evaluated using this function predict.ppm
.
The default action is to create a rectangular grid of points in the observation window of the data point pattern, and evaluate the spatial trend at these locations.
The argument type
specifies the values that are desired:
type="trend"
:the “spatial trend” of the fitted model is evaluated at each required spatial location u. See below.
type="cif"
:the conditional intensity lambda(u,X) of the fitted model is evaluated at each required spatial location u, with respect to the data point pattern X.
type="intensity"
:the intensity lambda(u) of the fitted model is evaluated at each required spatial location u.
type="count"
:the expected total number of points (or the expected number
of points falling in window
) is evaluated.
If window
is a tessellation,
the expected number of points in each tile of the tessellation
is evaluated.
The spatial trend, conditional intensity, and intensity are all equivalent if the fitted model is a Poisson point process. However, if the model is not a Poisson process, then they are all different. The “spatial trend” is the (exponentiated) first order potential, and not the intensity of the process. [For example if we fit the stationary Strauss process with parameters beta and gamma, then the spatial trend is constant and equal to beta, while the intensity is a smaller value.]
The default is to compute an estimate of the desired quantity.
If interval="confidence"
or interval="prediction"
,
the estimate is replaced by a confidence interval or prediction interval.
If se=TRUE
, then a standard error is also calculated,
and is returned together with the (point or interval) estimate.
The spatial locations where predictions are required,
are determined by the (incompatible)
arguments ngrid
and locations
.
If the argument ngrid
is present, then
predictions are performed at a rectangular
grid of locations in the window window
.
The result of prediction will be a pixel image or images.
If locations
is present, then predictions
will be performed at the spatial locations given by
this dataset. These may be an arbitrary list of spatial locations,
or they may be a rectangular grid.
The result of prediction will be either a numeric vector
or a pixel image or images.
If neither ngrid
nor locations
is given, then
ngrid
is assumed. The value of ngrid
defaults to
spatstat.options("npixel")
, which is initialised to 128
when spatstat is loaded.
The argument locations
may be a point pattern,
a data frame or a list specifying arbitrary locations;
or it may be a binary image mask (an object of class "owin"
with type "mask"
) or a pixel image (object of class
"im"
) specifying (a subset of) a rectangular
grid of locations.
If locations
is a point pattern (object of class "ppp"
),
then prediction will be performed at the points of the point pattern.
The result of prediction will be a vector of predicted values,
one value for each point.
If the model is a marked point process, then
locations
should be a marked point pattern, with marks of the
same kind as the model; prediction will be performed at these
marked points.
The result of prediction will be a vector of predicted values,
one value for each (marked) point.
If locations
is a data frame or list, then it must contain
vectors locations$x
and locations$y
specifying the
x,y coordinates of the prediction locations. Additionally, if
the model is a marked point process, then locations
must also contain
a factor locations$marks
specifying the marks of the
prediction locations. These vectors must have equal length.
The result of prediction will be a vector of predicted values,
of the same length.
If locations
is a binary image mask, then prediction will be
performed at each pixel in this binary image where the pixel value
is TRUE
(in other words, at each pixel that is inside the
window). If the fitted model is an unmarked point process, then the
result of prediction will be an image. If the fitted model is a
marked point process, then prediction will
be performed for each possible value of the mark at each such
location, and the result of prediction will be a
list of images, one for each mark value.
If locations
is a pixel image (object of class "im"
),
then prediction will be performed at each pixel in this image where
the pixel value is defined (i.e.\ where the pixel value is not
NA
).
The argument covariates
gives the values of any spatial covariates
at the prediction locations.
If the trend formula in the fitted model
involves spatial covariates (other than
the Cartesian coordinates x
, y
)
then covariates
is required.
The format and use of covariates
are analogous to those of the
argument of the same name in ppm
.
It is either a data frame or a list of images.
If covariates
is a list of images, then
the names of the entries should correspond 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.
If covariates
is a data frame, then the
i
th row of covariates
is assumed to contain covariate data for the i
th location.
When locations
is a data frame,
this just means that each row of covariates
contains the
covariate data for the location specified in the corresponding row of
locations
. When locations
is a binary image
mask, the row covariates[i,]
must correspond to the location
x[i],y[i]
where x = as.vector(raster.x(locations))
and y = as.vector(raster.y(locations))
.
Note that if you only want to use prediction in order to
generate a plot of the predicted values,
it may be easier to use plot.ppm
which calls
this function and plots the results.
If total
is given:
a numeric vector or matrix.
If locations
is given and is a data frame:
a vector of predicted values for the spatial locations
(and marks, if required) given in locations
.
If ngrid
is given, or if locations
is given
and is a binary image mask or a pixel image:
If object
is an unmarked point process,
the result is a pixel image object (of class "im"
, see
im.object
) containing the predictions.
If object
is a multitype point process,
the result is a list of pixel images, containing the predictions
for each type at the same grid of locations.
The “predicted values” are either values of the spatial trend
(if type="trend"
), values of the conditional intensity
(if type="cif"
or type="lambda"
),
values of the intensity (if type="intensity"
)
or numbers of points (if type="count"
).
If se=TRUE
, then the result is a list with two entries,
the first being the predicted values in the format described above,
and the second being the standard errors in the same format.
The current implementation invokes predict.glm
so that prediction is wrong if the trend formula in
object
involves terms in ns()
,
bs()
or poly()
.
This is a weakness of predict.glm
itself!
Error messages may be very opaque,
as they tend to come from deep in the workings of
predict.glm
.
If you are passing the covariates
argument
and the function crashes,
it is advisable to start by checking that all the conditions
listed above are satisfied.
and \rolf
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.
ppm
,
ppm.object
,
plot.ppm
,
print.ppm
,
fitted.ppm
,
spatstat.options
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 
m < ppm(cells ~ polynom(x,y,2), Strauss(0.05))
trend < predict(m, type="trend")
## Not run:
image(trend)
points(cells)
## End(Not run)
cif < predict(m, type="cif")
## Not run:
persp(cif)
## End(Not run)
data(japanesepines)
mj < ppm(japanesepines ~ harmonic(x,y,2))
se < predict(mj, se=TRUE)
# prediction interval for total number of points
predict(mj, type="count", interval="p")
# prediction at arbitrary locations
predict(mj, locations=data.frame(x=0.3, y=0.4))
X < runifpoint(5, Window(japanesepines))
predict(mj, locations=X, se=TRUE)
# multitype
rr < matrix(0.06, 2, 2)
ma < ppm(amacrine ~ marks, MultiStrauss(rr))
Z < predict(ma)
Z < predict(ma, type="cif")
predict(ma, locations=data.frame(x=0.8, y=0.5,marks="on"), type="cif")

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