glkerns: Kernel Regression Smoothing with Adaptive Plug-in Bandwidth

Description

Nonparametric estimation of regression functions and their derivatives with kernel regression estimators and automatically adapted (global) plug-in bandwidth.

Usage

glkerns(x, ...)
# S3 method for default
glkerns(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 formula
glkerns(formula, data, subset, na.action, ...)

Value

an object of class(es) c("glkerns", "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: bandwidth 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 bandwidth
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.

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 a global input bandwidth. By default, 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 methodss 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 \(k = \nu + 2\) where \(k - \nu\) must be even. The maximal possible values are for automatic smoothing, \(k \le 4\), whereas for smoothing with input bandwidth, \(k \le 6\).

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

global bandwidth for kernel regression estimation. If it is not given by input or if inputb=FALSE a data-adaptive global plug-in bandwidth 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 NAs. Defaults to getOption("na.action").
...: for the formula method: Optional arguments all passed to glkerns.default().

Author

- Eva Herrmann, TU Darmstadt(1995-1997): principal code (origianl Fortran and S+), see the references.
- Martin Maechler, 2001 ff: translated to R, created the package, refactored src/, added class, methods (predict, plot ..), arguments, docu, tweaks, help, examples, etc.
- The formula method was added in 2014 after proposals by Andri Signorell.

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 of a global bandwidth. Too small bandwidths will lead to a wiggly curve, too large ones will smooth away important details. The function glkerns calculates an estimator of the regression function or derivatives of the regression function with an automatically chosen global plugin bandwidth. It is also possible to use global bandwidths which are specified by the user.

Main ideas of the plugin method are to estimate the optimal bandwidths by estimating the asymptotically optimal mean integrated 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.

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 previously at Biostats, University of Zurich under software/kernel.html, but no longer.

References

- global plug-in bandwidth estimator:
Theo Gasser, Alois Kneip & Walter Koehler (1991) A flexible and fast method for automatic smoothing. Journal of the American Statistical Association 86, 643--652. tools:::Rd_expr_doi("10.2307/2290393")

Muller, H.-G. (1984) Smooth optimum kernel estimators of densities, regression curves and modes. The Annals of Statistics 12, 766--774.

- variance estimation:
Theo Gasser, Lothar Sroka & Christine Jennen-Steinmetz (1986) Residual Variance and Residual Pattern in Nonlinear Regression. Biometrika 73, 625--633. tools:::Rd_expr_doi("10.2307/2336527")

- adapting heteroscedasticity:
E. Herrmann (1997) Local bandwidth choice in kernel regression estimation. Journal of Graphical and Computational Statistics 6, 35--54.

- fast algorithm for kernel regression estimator:
T. Gasser & A. Kneip (1989) discussion of Buja, A., Hastie, T. and Tibshirani, R.: Linear smoothers and additive models, The Annals of Statistics 17, 532--535.

B. Seifert, M. Brockmann, J. Engel & T. Gasser (1994) Fast algorithms for nonparametric curve estimation. J. Computational and Graphical Statistics 3, 192--213.

- on the special kernel estimator for random design point:
E. Herrmann (1996) On the convolution type kernel regression estimator; Preprint 1833, FB Mathematik, Technische Universitaet Darmstadt; currently available from https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.6383

Examples

Run this code

data(xSim)## linear plus an exponential peak, see help(xSim)
n <- length(xSim)
tt <- ((1:n) - 1/2)/n # equidistant x ==> is.rand = FALSE
gk <- glkerns(tt, xSim, is.rand = FALSE)
gk # print method
plot(gk) # nice plot() method
if(require("sfsmisc")) {
  TA.plot(gk)
} else { plot(residuals(gk) ~ fitted(gk)); abline(h = 0, lty=2) }
qqnorm(residuals(gk), ylab = "residuals(gk)")

cat("glkerns() bandwidth:",format(gk$bandwidth, dig=10),"\n")
## local bandwidth: fit is very similar :
(lk <- lokerns(tt, xSim, is.rand = FALSE))
nobs(lk)

cols <- c(gl="PaleGreen", lo="Firebrick")
plot(lk$x.out, lk$bandwidth, axes = FALSE, xlab="", ylab="",
     ylim=c(0,max(lk$bandwidth)), type="h", col = "gray90")
axis(4); mtext("bandwidth(s)", side=4)
lines(lk$x.out, lk$bandwidth, col = cols["lo"], lty = 3)
abline(     h = gk$bandwidth, col = cols["gl"], lty = 4)
par(new=TRUE)
plot(tt, xSim, main = "global and local bandwidth kernel regression")
lines(gk$x.out, gk$est, col = cols["gl"], lwd = 1.5)
lines(lk$x.out, lk$est, col = cols["lo"])
# the red curve (local bw) is very slightly better
legend(0.7,4.4, c("global bw","local bw"), col = cols, lwd=1)

## This should look
op <- par(mfrow = c(3,1), mar = .1 + c(4,4,2,1), oma = c(0,0,3,0),
          mgp = c(1.5, 0.6,0))
plot(gk, main = expression(paste("Data & ", hat(f))))
## calling extra plot() method
gk1 <- glkerns(tt, xSim, deriv = 1, is.rand = FALSE)
plot(gk1$x.out, gk1$est, col = "green", lwd = 1.5, type = "l",
     main = expression(widehat(paste(f,"'"))))
abline(h=0, col="gray", lty = 3)
gk2 <- glkerns(tt, xSim, deriv = 2, is.rand = FALSE)
plot(gk2$x.out, gk2$est, col = "orange", lwd = 1.5, type = "l",
     main = expression(widehat(paste(f,"''"))))
abline(h=0, col="gray", lty = 3)
mtext("Example from www.unizh.ch/biostat/..../kernf77.html",side=3,
      outer = TRUE, cex = 1, font = par("font.main"))

par(op)
data(cars)
plot(dist ~ speed, data = cars,
     main = "Global Plug-In Bandwidth")
## these two are equivalent
m1glk <- glkerns(dist ~ speed, data = cars)
m.glk <- glkerns(cars$ speed, cars$ dist)
lines(m.glk, col=2) # using the lines() method
mtext(paste("bandwidth = ", format(m.glk$bandwidth, dig = 4)))
ii <- names(m1glk) != "call"
stopifnot(all.equal(m1glk[ii], m.glk[ii], tol = 1e-15))

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