lokerns: Kernel Regression Smoothing with Local Plug-in Bandwidth
In lokern: Kernel Regression Smoothing with Local or Global Plug-in Bandwidth

View source: R/lokerns.R

lokerns

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, ...)

## Default S3 method:
lokerns(x, y=NULL, deriv = 0, n.out=300, x.out=NULL, x.inOut = TRUE,
        korder = deriv + 2, hetero=FALSE, is.rand=TRUE,
        inputb = is.numeric(bandwidth) && all(bandwidth > 0),
        m1 = 400, xl=NULL, xu=NULL,
        s=NULL, sig=NULL, bandwidth=NULL, trace.lev = 0, ...)

## S3 method for class 'formula'
lokerns(formula, data, subset, na.action, ...)

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).
`x.inOut`	logical or character string indicating if `x.out` should contain the input `x` values. Note that this argument did not exist, equivalently to being `FALSE`, up to lokern version `1.0-9`. In order for `residuals()` or `fitted()` methods to be applicable, it must be `TRUE` or a character string specifying one of the `methods`s of `seqXtend` (package sfsmisc). The default, `TRUE` corresponds to method `"aim"`.
`korder`	nonnegative integer giving the kernel order k; it defaults to `korder = deriv+2` or k = ν + 2 where k - ν 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` (by default when `bandwidth` is not specified), 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` 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.
`trace.lev`	integer indicating how much the internal (Fortran level) computations should be “traced”, i.e., be reported. The default, `0`, does not print anything.

`formula`	a `formula` of the form `y ~ pred`, specifying the response variable `y` and predictor variable `pred` which must be in `data`.
`data`	an optional matrix or data frame (or similar: see `model.frame`) containing the variables in the formula `formula`. By default the variables are taken from `environment(formula)`.
`subset`	an optional vector specifying a subset of observations to be used.
`na.action`	a function which indicates what should happen when the data contain `NA`s. Defaults to `getOption("na.action")`.
`...`	for the `formula` method: Optional arguments all passed to `lokerns.default()`.

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=\code{korder}) for the global bandwidth and the squared k-th derivative itself for the local bandwidths.

Some more details are in glkerns.

Value

an object of class(es) c("lokerns", "KernS"), which is a list including used parameters and estimator, containing among others

`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 (at `x.out`).
`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.

Examples

data(cars)
lofit <- lokerns(dist ~ speed, data=cars)
lofit # print() method

if(require("sfsmisc")) {
  TA.plot(lofit)
} else { plot(residuals(lofit) ~ fitted(lofit)); abline(h = 0, lty=2) }
qqnorm(residuals(lofit), ylab = "residuals(lofit)")

## nice simple plot of data + smooth
plot(lofit)

(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)

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, col=4, lwd=2)
mtext(paste("bandwidth in [",
            paste(format(sb[c(1,6)], dig = 3),collapse=","),
            "];  Median b.w.=",formatC(sb["Median"])))

## using user-specified bandwidth array
myBW <- round(2*lofit$bandwidth, 2)
(lofB <- lokerns(dist ~ speed, data=cars, bandwidth = myBW)) # failed (for a while)
## can use deriv=3 (and 4) here:
lofB3 <- lokerns(dist ~ speed, data=cars, bandwidth = myBW, deriv=3)
plot(lofB)
lines(lofB3, col=3)
stopifnot(inherits(lofB3, "KernS"), identical(lofB3$korder, 5L))

lokern documentation built on March 7, 2023, 7:26 p.m.