LSCV.risk  R Documentation 
Methods to find a jointly optimal, common casecontrol isotropic bandwidth for use in estimation of the fixed or adaptive kernelsmoothed relative risk function.
LSCV.risk(
f,
g = NULL,
hlim = NULL,
hseq = NULL,
type = c("fixed", "adaptive"),
method = c("kelsalldiggle", "hazelton", "davies"),
resolution = 64,
edge = TRUE,
hp = NULL,
pilot.symmetry = c("none", "f", "g", "pooled"),
auto.optim = TRUE,
seqres = 30,
parallelise = NA,
verbose = TRUE,
...
)
f 
Either a precalculated object of class 
g 
As for 
hlim 
An optional vector of length 2 giving the limits of the optimisation routine with respect to the bandwidth. If unspecified, the function attempts to choose this automatically. 
hseq 
An optional increasing sequence of bandwidth values at which to
manually evaluate the optimisation criterion. Used only in the case

type 
A character string; 
method 
A character string controlling the selector to use. There are
three types, based on either the mean integrated squared error (MISE)
(Kelsall and Diggle, 1995; default – 
resolution 
Spatial grid size; the optimisation will be based on a
[ 
edge 
Logical value indicating whether to edgecorrect the density estimates used. 
hp 
A single numeric value or a vector of length 2 giving the pilot
bandwidth(s) to be used for estimation of the pilot
densities for adaptive risk surfaces. Ignored if 
pilot.symmetry 
A character string used to control the type of
symmetry, if any, to use for the bandwidth factors when computing an
adaptive relative risk surface. See ‘Details’. Ignored if 
auto.optim 
Logical value indicating whether to automate the numerical
optimisation using 
seqres 
Optional resolution of an increasing sequence of bandwidth
values. Only used if 
parallelise 
Numeric argument to invoke parallel processing, giving
the number of CPU cores to use when 
verbose 
Logical value indicating whether to provide function progress commentary. 
... 
Additional arguments such as 
Given the established preference of using a common bandwidth for both case
and control density estimates when constructing a relative
risk surface, This function calculates a ‘jointly optimal’, common isotropic
LSCV bandwidth for the (Gaussian) kernelsmoothed relative risk function
(casecontrol densityratio). It can be shown that choosing a bandwidth that
is equal for both case and control density estimates is preferable to
computing ‘separately optimal’ bandwidths (Kelsall and Diggle, 1995). The user
can choose to either calculate a common smoothing parameter for a fixedbandwidth
relative risk surface (type = "fixed"
; default), or a common global bandwidth for
an adaptive risk surface (type = "adaptive"
). See further comments below.
method = "kelsalldiggle"
: the function computes the
common bandwidth which minimises the approximate mean integrated squared
error (MISE) of the logtransformed risk surface (Kelsall and Diggle, 1995).
method = "hazelton"
: the function minimises a
weightedbycontrol MISE of the (raw) relative risk function
(Hazelton, 2008).
method = "davies"
: the optimal bandwidth is
one that minimises a crude plugin approximation to the asymptotic
MISE (Davies, 2013). Only possible for type = "fixed"
.
For jointly optimal, common global bandwidth selection when type = "adaptive"
, the
optimisation routine utilises multiscale.density
. Like LSCV.density
,
the leaveoneout procedure does not affect the pilot density, for which additional
control is offered via the hp
and pilot.symmetry
arguments. The user has the option of
obtaining a socalled symmetric estimate (Davies et al. 2016) via
pilot.symmetry
. This amounts to choosing the same pilot density for
both case and control densities. By choosing "none"
(default), the
result uses the case and control data separately for the fixedbandwidth
pilots, providing the original asymmetric densityratio of Davies and
Hazelton (2010). By selecting either of "f"
, "g"
, or
"pooled"
, the pilot density is calculated based on the case, control,
or pooled case/control data respectively (using hp[1]
as the fixed
bandwidth). Davies et al. (2016) noted some beneficial practical behaviour
of the symmetric adaptive surface over the asymmetric. (The pilot bandwidth(s), if not supplied in hp
, are calculated
internally via default use of LSCV.density
, using the requested symmetricbased data set, or separately with respect to the case and control datasets f
and g
if
pilot.symmetry = "none"
.)
A single numeric value of the estimated bandwidth (if
auto.optim = TRUE
). Otherwise, a list of two numeric vectors of equal
length giving the bandwidth sequence (as hs
) and corresponding CV
function value (as CV
).
The jointly optimal bandwidth selector can be
computationally expensive for large data sets and fine evaluation grid
resolutions. The user may need to experiment with adjusting hlim
to
find a suitable minimum.
T. M. Davies
Davies, T. M. (2013), Jointly optimal bandwidth selection for the planar kernelsmoothed densityratio, Spatial and Spatiotemporal Epidemiology, 5, 5165.
Davies, T.M. and Baddeley A. (2018), Fast computation of spatially adaptive kernel estimates, Statistics and Computing, 28(4), 937956.
Davies, T.M. and Hazelton, M.L. (2010), Adaptive kernel estimation of spatial relative risk, Statistics in Medicine, 29(23) 24232437.
Davies, T.M., Jones, K. and Hazelton, M.L. (2016), Symmetric adaptive smoothing regimens for estimation of the spatial relative risk function, Computational Statistics & Data Analysis, 101, 1228.
Hazelton, M. L. (2008), Letter to the editor: Kernel estimation of risk surfaces without the need for edge correction, Statistics in Medicine, 27, 22692272.
Kelsall, J.E. and Diggle, P.J. (1995), Kernel estimation of relative risk, Bernoulli, 1, 316.
Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis, Chapman & Hall, New York.
Wand, M.P. and Jones, C.M., 1995. Kernel Smoothing, Chapman & Hall, London.
bivariate.density
data(pbc)
pbccas < split(pbc)$case
pbccon < split(pbc)$control
# FIXED (for common h)
LSCV.risk(pbccas,pbccon)
LSCV.risk(pbccas,pbccon,method="hazelton")
hcv < LSCV.risk(pbccas,pbccon,method="davies",auto.optim=FALSE)
plot(hcv[,1],log(hcv[,2]));abline(v=hcv[which.min(hcv[,2]),1],col=2,lty=2)
# ADAPTIVE (for common h0)
LSCV.risk(pbccas,pbccon,type="adaptive")
# change pilot bandwidths used
LSCV.risk(pbccas,pbccon,type="adaptive",hp=c(OS(pbccas)/2,OS(pbccon)/2))
# specify pooleddata symmetric relative risk estimator
LSCV.risk(pbccas,pbccon,type="adaptive",hp=OS(pbc),pilot.symmetry="pooled")
# as above, for Hazelton selector
LSCV.risk(pbccas,pbccon,type="adaptive",method="hazelton")
LSCV.risk(pbccas,pbccon,type="adaptive",method="hazelton",hp=c(OS(pbccas)/2,OS(pbccon)/2))
LSCV.risk(pbccas,pbccon,type="adaptive",method="hazelton",hp=OS(pbc),pilot.symmetry="pooled")
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