lokerns {lokern} | R Documentation |
Nonparametric estimation of regression functions and their derivatives with kernel regression estimators and automatically adapted local plug-in bandwidth function.
lokerns(x , y, deriv = 0, n.out=300, x.out=NULL, korder= deriv + 2, hetero=FALSE, is.rand=TRUE, inputb= is.numeric(bandwidth) && bandwidth > 0, m1 = 400, xl=NULL, xu=NULL, s=NULL, sig=NULL, bandwidth=NULL)
x |
vector of design points, not necessarily ordered. |
y |
vector of observations of the same length as x. |
deriv |
order of derivative of the regression function to be estimated. Only deriv=0,1,2 are allowed for automatic smoothing, whereas deriv=0,1,2,3,4 is possible when smoothing with an input bandwidth array. The default value is deriv=0. |
n.out |
number of output design points where the function has to
be estimated; default is n.out=300 . |
x.out |
vector of output design points where the function has to be estimated. The default is an equidistant grid of n.out points from min(x) to max(x). |
korder |
nonnegative integer giving the kernel order; it defaults to
korder = deriv+2 or k = nu + 2 where k - nu
must be even. The maximal possible values are for automatic
smoothing, k <= 4, whereas for smoothing with input
bandwidth array, k <= 6.
|
hetero |
logical: if TRUE, heteroscedastic error variables are assumed for variance estimation, if FALSE the variance estimation is optimized for homoscedasticity. Default value is hetero=FALSE. |
is.rand |
logical: if TRUE (default), random x are assumed and the
s-array of the convolution estimator is computed as smoothed
quantile estimators in order to adapt this variability. If FALSE,
the s-array is choosen as mid-point sequences as the classical
Gasser-Mueller estimator, this will be better for equidistant and
fixed design.
|
inputb |
logical: if true, a local input bandwidth array is used; if
FALSE (default), a data-adaptive local plug-in bandwidths
array is calculated and used.
|
m1 |
integer, the number of grid points for integral approximation when estimating the plug-in bandwidth. The default, 400, may be increased if a very large number of observations are available. |
xl, xu |
numeric (scalars), the lower and upper bounds for integral
approximation and variance estimation when estimating the plug-in
bandwidth. By default (when xl and xu are not specified),
the 87% middle part of [xmin,xmax] is used.
|
s |
s-array of the convolution kernel estimator. If it is not given by input
it is calculated as midpoint-sequence of the ordered design points for
is.rand=FALSE or as quantiles estimators of the design density
for is.rand=TRUE .
|
sig |
variance of the error variables. If it is not given by
input or if hetero=TRUE (no default) it is calculated by a
nonparametric variance estimator. |
bandwidth |
local bandwidth array for kernel regression estimation. If it is
not given by input or if inputb=FALSE a data-adaptive local
plug-in bandwidth array is used instead.
|
This function calls an efficient and fast algorithm for automatically adaptive nonparametric regression estimation with a kernel method.
Roughly spoken, the method performs a local averaging of the observations when estimating the regression function. Analogously, one can estimate derivatives of small order of the regression function. Crucial for the kernel regression estimation used here is the choice the local bandwidth array. Too small bandwidths will lead to a wiggly curve, too large ones will smooth away important details. The function lokerns calculates an estimator of the regression function or derivatives of the regression function with an automatically chosen local plugin bandwidth function. It is also possible to use a local bandwidth array which are specified by the user.
Main ideas of the plugin method are to estimate the optimal bandwidths
by estimating the asymptotically optimal mean squared error optimal
bandwidths. Therefore, one has to estimate the variance for
homoscedastic error variables and a functional of a smooth variance
function for heteroscedastic error variables, respectively. Also, one
has to estimate an integral functional of the squared k-th derivative
of the regression function (k=korder
) for the global
bandwidth and the squared k-th derivative itself for the local
bandwidths.
Here, a further kernel estimator for this derivative is used with a bandwidth which is adapted iteratively to the regression function. A convolution form of the kernel estimator for the regression function and its derivatives is used. Thereby one can adapt the s-array for random design. Using this estimator leads to an asymptotically minimax efficient estimator for fixed and random design. Polynomial kernels and boundary kernels are used with a fast and stable updating algorithm for kernel regression estimation.
More details can be found in the references and on http://www.unizh.ch/biostat/Software/kernsplus.html.
a list including used parameters and estimator.
x |
vector of ordered design points. |
y |
vector of observations ordered with respect to x. |
bandwidth |
local bandwidth array which was used for kernel regression estimation. |
x.out |
vector of ordered output design points. |
est |
vector of estimated regression function or its derivative. |
sig |
variance estimation which was used for calculating the plug-in bandwidths if hetero=TRUE (default) and either inputb=FALSE (default) or is.rand=TRUE (default). |
deriv |
derivative of the regression function which was estimated. |
korder |
order of the kernel function which was used. |
xl |
lower bound for integral approximation and variance estimation. |
xu |
upper bound for integral approximation and variance estimation. |
s |
vector of midpoint values used for the convolution kernel regression estimator. |
All the references in glkerns
.
glkerns
for global bandwidth computation.
data(cars) lofit <- lokerns(cars$ speed, cars$ dist) (sb <- summary(lofit$bandwidth)) op <- par(fg = "gray90", tcl = -0.2, mgp = c(3,.5,0)) plot(lofit$band, ylim=c(0,3*sb["Max."]), type="h",#col="gray90", ann = FALSE, axes = FALSE) if(R.version$major > 1 || R.version$minor >= 3.0) boxplot(lofit$bandwidth, add = TRUE, at = 304, boxwex = 8, col = "gray90",border="gray", pars = list(axes = FALSE)) axis(4, at = c(0,pretty(sb)), col.axis = "gray") par(op) par(new=TRUE) plot(dist ~ speed, data = cars, main = "Local Plug-In Bandwidth Vector") lines(lofit$x.out, lofit$est, col=4) mtext(paste("bandwidth in [", paste(format(sb[c(1,6)], dig = 3),collapse=","), "]; Median b.w.=",formatC(sb["Median"])))