relrisk(X, sigma = NULL, ..., varcov = NULL, at = "pixels",
casecontrol=TRUE, case=2)
"ppp"
which has factor valued marks).sigma
may be a function which can be used
to select a different bandwidth for each type of pbw.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"
).X
consists of only two types of points,
the result is a pixel image (if at="pixels"
)
or a vector of probabilities (if at="points"
). If X
consists of more than two types of points,
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"listof"
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$j$.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 this command computes
the spatially-varying risk of an event,
i.e. the probability $p(u)$
that a point at spatial location $u$
will be a case. If X
is a multitype point pattern with $m > 2$ types,
or if X
is a bivariate point pattern
and casecontrol=FALSE
,
then this command computes, for each type $j$,
a nonparametric estimate of
the spatially-varying risk 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 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
$p_j(u)$ for $j = 1,\ldots,m$.
If at = "points"
the calculation is performed
only at the data points $x_i$. 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.
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 afunction
which selects
the bandwidth.
Bandwidth selection will be appliedseparately to each type of point.
An example of such a function isbw.diggle
.sigma
andvarcov
are both missing or null. Then acommonsmoothing bandwidthsigma
will be selected by cross-validation usingbw.relrisk
.bw.relrisk
,
density.ppp
,
Smooth.ppp
,
eval.im
data(urkiola)
p <- relrisk(urkiola, 20)
if(interactive()) {
plot(p, main="proportion of oak")
plot(eval.im(p > 0.3), main="More than 30 percent oak")
plot(split(lansing), main="Lansing Woods")
rr <- relrisk(lansing, 0.05)
plot(rr, main="Lansing Woods relative risk")
wh <- im.apply(rr, which.max)
types <- levels(marks(lansing))
wh <- eval.im(types[wh])
plot(wh, main="Most common species")
}
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