"relrisk"(X, sigma = NULL, ..., varcov = NULL, at = "pixels", relative=FALSE, se=FALSE, casecontrol=TRUE, control=1, case)"ppp"
which has factor valued marks).
sigma may be a function which can be used
to select a different bandwidth for each type of point. See Details.
bw.relrisk to select the
bandwidth, or passed to density.ppp to control the
pixel resolution.
sigma.
at="pixels") or
only at the points of X (at="points").
FALSE (the default) the algorithm
computes the probabilities of each type of point.
If TRUE, it computes the
relative risk, the ratio of probabilities
of each type relative to the probability of a control.
control
in a bivariate point pattern.
Ignored if there are more than 2 types of points.
se=FALSE (the default), the format is described below.
If se=TRUE, the result is a list of two entries,
estimate and SE, each having the format described below.If X consists of only two types of points,
and if casecontrol=TRUE,
the result is a pixel image (if at="pixels")
or a vector (if at="points").
The pixel values or vector values
are the probabilities of a case if relative=FALSE,
or the relative risk of a case (probability of a case divided by the
probability of a control) if relative=TRUE.If X consists of more than two types of points,
or if casecontrol=FALSE, the result is:
at="pixels")
a list of pixel images, with one image for each possible type of point.
The result also belongs to the class "solist" so that it can
be printed and plotted.
at="points")
a matrix of probabilities, with rows corresponding to
data points $x[i]$, and columns corresponding
to types $j$.
relative=FALSE,
or the relative risk of each type (probability of each type divided by the
probability of a control) if relative=TRUE.If relative=FALSE, the resulting values always lie between 0
and 1. If relative=TRUE, the results are either non-negative
numbers, or the values Inf or NA.
relrisk is generic and can be used to
estimate relative risk in different ways.
This function relrisk.ppp is the method for point pattern
datasets. It computes nonparametric estimates of relative risk
by kernel smoothing. If X is a bivariate point pattern
(a multitype point pattern consisting of two types of points)
then by default,
the points of the first type (the first level of marks(X))
are treated as controls or non-events, and points of the second type
are treated as cases or events. Then by default this command computes
the spatially-varying probability of a case,
i.e. the probability $p(u)$
that a point at spatial location $u$
will be a case. If relative=TRUE, it computes the
spatially-varying relative risk of a case relative to a
control, $r(u) = p(u)/(1- p(u))$.
If X is a multitype point pattern with $m > 2$ types,
or if X is a bivariate point pattern
and casecontrol=FALSE,
then by default this command computes, for each type $j$,
a nonparametric estimate of
the spatially-varying probability of an event of type $j$.
This is the probability $p[j](u)$
that a point at spatial location $u$
will belong to type $j$.
If relative=TRUE, the command computes the
relative risk of an event of type $j$
relative to a control,
$r[j](u) = p[j](u)/p[k](u)$,
where events of type $k$ are treated as controls.
The argument control determines which type $k$
is treated as a control.
If at = "pixels" the calculation is performed for
every spatial location $u$ on a fine pixel grid, and the result
is a pixel image representing the function $p(u)$
or a list of pixel images representing the functions
$p[j](u)$ or $r[j](u)$
for $j = 1,...,m$.
An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as NA.
If at = "points" the calculation is performed
only at the data points $x[i]$. By default
the result is a vector of values
$p(x[i])$ giving the estimated probability of a case
at each data point, or a matrix of values
$p[j](x[i])$ giving the estimated probability of
each possible type $j$ at each data point.
If relative=TRUE then the relative risks
$r(x[i])$ or $r[j](x[i])$ are
returned.
An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as Inf.
Estimation is performed by a simple Nadaraja-Watson type kernel smoother (Diggle, 2003). The smoothing bandwidth can be specified in any of the following ways:
sigma is a single numeric value, giving the standard
deviation of the isotropic Gaussian kernel.
sigma is a numeric vector of length 2, giving the
standard deviations in the $x$ and $y$ directions of
a Gaussian kernel.
varcov is a 2 by 2 matrix giving the
variance-covariance matrix of the Gaussian kernel.
sigma is a function which selects
the bandwidth.
Bandwidth selection will be applied
separately to each type of point.
An example of such a function is bw.diggle.
sigma and varcov
are both missing or null. Then a common
smoothing bandwidth sigma
will be selected by cross-validation using bw.relrisk.
If se=TRUE then standard errors will also be computed,
based on asymptotic theory, assuming a Poisson process.
relrisk.ppm for point process
models which computes parametric
estimates of relative risk, using the fitted model. See also
bw.relrisk,
density.ppp,
Smooth.ppp,
eval.im
p.oak <- relrisk(urkiola, 20)
if(interactive()) {
plot(p.oak, main="proportion of oak")
plot(eval.im(p.oak > 0.3), main="More than 30 percent oak")
plot(split(lansing), main="Lansing Woods")
p.lan <- relrisk(lansing, 0.05, se=TRUE)
plot(p.lan$estimate, main="Lansing Woods species probability")
plot(p.lan$SE, main="Lansing Woods standard error")
wh <- im.apply(p.lan$estimate, which.max)
types <- levels(marks(lansing))
wh <- eval.im(types[wh])
plot(wh, main="Most common species")
}
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