Given a fitted point process model obtained by `ppm`

,
evaluate the spatial trend or the conditional intensity of the model
at new locations.

```
# S3 method for ppm
predict(object, window=NULL, ngrid=NULL, locations=NULL,
covariates=NULL,
type=c("trend", "cif", "intensity", "count"),
se=FALSE,
interval=c("none", "confidence", "prediction"),
level = 0.95,
X=data.ppm(object), correction,
…, new.coef=NULL, check=TRUE, repair=TRUE)
```

object

A fitted point process model, typically obtained from
the model-fitting algorithm `ppm`

. An object of
class `"ppm"`

(see `ppm.object`

).

window

Optional. A window (object of class `"owin"`

)
*delimiting* the locations where predictions
should be computed. Defaults to the window of the
original data used to fit the model `object`

.

ngrid

Optional. Dimensions of a rectangular grid of locations
inside `window`

where the predictions should be computed.
An integer, or an integer vector of length 2,
specifying the number of grid points in the \(y\) and \(x\)
directions. (Incompatible with `locations`

)

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 `x`

and
`y`

, or a binary image mask, or a pixel image.
(Incompatible with `ngrid`

)

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 `"trend"`

for the spatial trend,
`"cif"`

or `"lambda"`

for the conditional intensity,
`"intensity"`

for the intensity, and
`"count"`

for the total number of points in `window`

.

se

Logical value indicating whether to calculate standard errors as well.

interval

String (partially matched) indicating whether to produce
estimates (`interval="none"`

, the default)
or a confidence interval (`interval="confidence"`

)
or a prediction interval (`interval="prediction"`

).

level

Coverage probability for the confidence or prediction interval.

X

Optional. A point pattern (object of class `"ppp"`

)
to be taken as the data point pattern when calculating the
conditional intensity. The default is to use the original data
to which the model was fitted.

correction

Name of the edge correction to be used
in calculating the conditional intensity.
Options include `"border"`

and `"none"`

.
Other options may include `"periodic"`

,
`"isotropic"`

and `"translate"`

depending on the model.
The default correction is the one that was used to fit `object`

.

…

Ignored.

new.coef

Numeric vector of parameter values to replace the
fitted model parameters `coef(object)`

.

check

Logical value indicating whether to check the internal format
of `object`

. If there is any possibility that this object
has been restored from a dump file, or has otherwise lost track of
the environment where it was originally computed, set
`check=TRUE`

.

repair

Logical value indicating whether to repair the internal format
of `object`

, if it is found to be damaged.

*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.

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
model-fitting 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:

- If
`type="trend"`

: the ``spatial trend'' of the fitted model is evaluated at each required spatial location \(u\). See below.

- If
`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 \(X\).

- If
`type="intensity"`

: the intensity \(\lambda(u)\) of the fitted model is evaluated at each required spatial location \(u\).

- If
`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.

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`

```
# NOT RUN {
# }
# NOT RUN {
m <- ppm(cells ~ polynom(x,y,2), Strauss(0.05))
trend <- predict(m, type="trend")
# }
# NOT RUN {
image(trend)
points(cells)
# }
# NOT RUN {
cif <- predict(m, type="cif")
# }
# NOT RUN {
persp(cif)
# }
# 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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