lokerns {lokern}R Documentation

Kernel Regression Smoothing with Local Plug-in Bandwidth

Description

Nonparametric estimation of regression functions and their derivatives with kernel regression estimators and automatically adapted local plug-in bandwidth function.

Usage

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)

Arguments

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.

Details

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.

Value

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.

References

All the references in glkerns.

See Also

glkerns for global bandwidth computation.

Examples

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"])))

[Package lokern version 1.0-4 Index]