Given a multitype point pattern, this function estimates the spatially-varying probability of each type of point, or the ratios of such probabilities, using kernel smoothing. The default smoothing bandwidth is selected by cross-validation.
# S3 method for ppp
relrisk(X, sigma = NULL, ...,
at = c("pixels", "points"),
weights = NULL, varcov = NULL,
relative=FALSE, normalise=FALSE,
adjust=1, edge=TRUE, diggle=FALSE,
se=FALSE, wtype=c("value", "multiplicity"),
casecontrol=TRUE, control=1, case, shrink=0, fudge=0)If 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:
(if 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.
(if at="points")
a matrix of probabilities, with rows corresponding to
data points \(x_i\), and columns corresponding
to types \(j\).
The pixel values or matrix entries
are the probabilities of each type of point if 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.
The command 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 (Bithell, 1990, 1991; Diggle, 2003; Baddeley,
Rubak and Turner, 2015).
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 \(j = 1,\ldots,m\).
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 (Bithell, 1990, 1991; Diggle, 2003; Baddeley, Rubak and Turner, 2015, section 14.4). 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.
An infinite smoothing bandwidth, sigma=Inf, is permitted
and yields a constant estimate of relative risk.
If se=TRUE then standard errors will also be computed,
based on asymptotic theory, assuming a Poisson process.
The optional argument weights may provide numerical weights
for the points of X. It should be a numeric vector of length
equal to npoints(X).
The argument weights can also be an expression.
It will be evaluated in the data frame as.data.frame(X)
to obtain a vector of weights. The expression may involve
the symbols x and y representing the Cartesian
coordinates, and the symbol marks representing the mark values.
The argument weights can also be a pixel image
(object of class "im"). numerical weights for the data points
will be extracted from this image (by looking up the pixel values
at the locations of the data points in X).
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Bithell, J.F. (1990) An application of density estimation to geographical epidemiology. Statistics in Medicine 9, 691--701.
Bithell, J.F. (1991) Estimation of relative risk functions. Statistics in Medicine 10, 1745--1751.
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
Diggle, P.J., Zheng, P. and Durr, P. (2005) Non-parametric estimation of spatial segregation in a multivariate point process: bovine tuberculosis in Cornwall, UK. Applied Statistics 54, 645--658.
There is another method 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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