Uses cross-validation to select a smoothing bandwidth for the estimation of relative risk on a linear network.
# S3 method for lpp
bw.relrisk(X, ...,
method = c("likelihood", "leastsquares", "KelsallDiggle", "McSwiggan"),
distance=c("path", "euclidean"),
hmin = NULL, hmax = NULL, nh = NULL,
fast = TRUE, fastmethod = "onestep",
floored = TRUE, reference = c("thumb", "uniform", "sigma"),
allow.infinite = TRUE, epsilon = 1e-20, fudge = 0,
verbose = FALSE, warn = TRUE)A single numerical value giving the selected bandwidth.
The result also belongs to the class "bw.optim"
(see bw.optim.object)
which can be plotted to show the bandwidth selection criterion
as a function of sigma.
A multitype point pattern on a linear network (object of class
"lpp" which has factor-valued marks).
Arguments passed to density.lpp to control the
resolution of the algorithm.
Character string (partially matched) determining the cross-validation method. See Details.
Character string (partially matched)
specifying the type of smoothing kernel.
See density.lpp.
Optional. Numeric values.
Range of trial values of smoothing bandwith sigma
to consider. There is a sensible default.
Number of trial values of smoothing bandwidth sigma
to consider.
Logical value specifying whether the leave-one-out density estimates
should be computed using a fast approximation (fast=TRUE, the
default) or exactly (fast=FALSE).
Developer use only.
Character string (partially matched) specifying the
bandwidth for calculating the
reference intensities used in the McSwiggan method
(modified Kelsall-Diggle method).
reference="sigma" means the maximum bandwidth considered,
which is given by the argument sigma.
reference="thumb" means the bandwidths selected by
Scott's rule of thumb bw.scott.iso.
reference="uniform" means infinite bandwidth corresponding to
uniform intensity.
Logical value indicating whether an infinite bandwidth (corresponding to a constant relative risk) should be permitted as a possible choice of bandwidth.
A small constant value added to the reference density in some of the cross-validation calculations, to improve performance.
Fudge factor to prevent very small density estimates in the
leave-one-out calculation. If fudge > 0,
then the lowest permitted value
for a leave-one-out estimate of intensity is
fudge/L, where L is the total length of the
network.
Logical value indicating whether to print progress reports,
Logical. If TRUE, issue a warning if the minimum of
the cross-validation criterion occurs at one of the ends of the
search interval.
Greg McSwiggan and Adrian Baddeley Adrian.Baddeley@curtin.edu.au.
This function is a method for the generic bw.relrisk.
It computes an optimal value of smoothing bandwidth
for the nonparametric estimation of relative risk on a linear network
using relrisk.lpp.
The optimal value is found by minimising a cross-validation criterion.
The cross-validation criterion is selected by the argument method:
method="likelihood" | (negative) likelihood cross-validation |
method="leastsquares" | least squares cross-validation |
method="KelsallDiggle" | Kelsall and Diggle (1995) density ratio cross-validation |
method="McSwiggan" | McSwiggan et al (2019) modified density ratio cross-validation |
See McSwiggan et al (2019) for details.
The result is a numerical value giving the selected bandwidth sigma.
The result also belongs to the class "bw.optim"
allowing it to be printed and plotted. The plot shows the cross-validation
criterion as a function of bandwidth.
The ‘optimal’ bandwidth is the value of bandwidth
which minimises the cross-validation criterion.
The range of values for the smoothing bandwidth sigma
is set by the arguments hmin, hmax. There is a sensible default,
based on the linear network version of Scott's rule
bw.scott.iso.
If the optimal bandwidth is achieved at an endpoint of the
interval [hmin, hmax], the algorithm will issue a warning
(unless warn=FALSE). If this occurs, then it is probably advisable
to expand the interval by changing the arguments hmin, hmax.
The cross-validation procedure is based on kernel estimates
of intensity, which are computed by density.lpp.
Any arguments ... are passed to density.lpp
to control the kernel estimation procedure. This includes the
argument distance which specifies the type of kernel.
The default is distance="path";
the fastest option is distance="euclidean".
Kelsall, J.E. and Diggle, P.J. (1995) Kernel estimation of relative risk. Bernoulli 1, 3--16.
McSwiggan, G., Baddeley, A. and Nair, G. (2019) Estimation of relative risk for events on a linear network. Statistics and Computing 30 (2) 469--484.
relrisk.lpp, bw.relrisk,
bw.optim.object
set.seed(2020)
X <- superimpose(A=runiflpp(20, simplenet),
B=runifpointOnLines(20, as.psp(simplenet)[1]))
plot(bw.relrisk(X, hmin=0.13, hmax=0.22, method="McSwiggan"))
plot(bw.relrisk(X, hmin=0.1, hmax=0.2, nh=8, distance="euclidean"))
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