Package 'lokern' reference manual

Title:	Kernel Regression Smoothing with Local or Global Plug-in Bandwidth
Description:	Kernel regression smoothing with adaptive local or global plug-in bandwidth selection.
Authors:	Eva Herrmann [aut] (F77 & S original), Martin Maechler [cre, aut] (Packaged for R and enhanced quite a bit, <https://orcid.org/0000-0002-8685-9910>)
Maintainer:	Martin Maechler <[email protected]>
License:	GPL (>= 2)
Version:	1.1-12
Built:	2024-10-18 05:16:14 UTC
Source:	https://github.com/r-forge/curves-etc

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, ...)

## Default S3 method:
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 class 'formula'
glkerns(formula, data, subset, na.action, ...)
glkerns(x, ...)

## Default S3 method:
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 class 'formula'
glkerns(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 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 `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 $k = \nu + 2$ 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 `NA`s. Defaults to `getOption("na.action")`.
`...`	for the `formula` method: Optional arguments all passed to `glkerns.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 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.

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.

Author(s)

- 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.

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. 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. 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

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))
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))

Methods for ("KernS" classed) Results of lokerns() and glkerns()

Description

Methods for results of glkerns() and lokerns() which are of (S3) class "KernS".

Usage

## S3 method for class 'KernS'
fitted(object, ...)
## S3 method for class 'KernS'
plot(x, type = "l", lwd = 2.5, col = 3, ...)
## S3 method for class 'KernS'
predict(object, x, deriv = object[["deriv"]],
        korder = deriv+2, trace.lev = 0, ...)
## S3 method for class 'KernS'
print(x, digits = getOption("digits"), ...)
## S3 method for class 'KernS'
residuals(object, ...)
## S3 method for class 'KernS'
fitted(object, ...)
## S3 method for class 'KernS'
plot(x, type = "l", lwd = 2.5, col = 3, ...)
## S3 method for class 'KernS'
predict(object, x, deriv = object[["deriv"]],
        korder = deriv+2, trace.lev = 0, ...)
## S3 method for class 'KernS'
print(x, digits = getOption("digits"), ...)
## S3 method for class 'KernS'
residuals(object, ...)

Arguments

`x`, `object`	an R object, of S3 class `"KernS"`, typically result either from `glkerns()` or `lokerns()`.
`type`, `lwd`, `col`	arguments for `plot()` only for the case when `x$deriv` is not 0.
`deriv`	integer, $\ge 0$ , specifiying order of derivative that should be predicted.
`korder`	nonnegative integer giving the kernel order; see `lokerns` or `glkerns`.
`digits`	number of significant digits, see `print`.
`trace.lev`	integer; level of tracing of Fortran level computations; see `lokerns`.
`...`	potentially further arguments passed to and from methods. For the `plot(*, deriv=0)` method, these are passed to `plotDS` from package sfsmisc.

Details

Note that fitted() and residuals() rely on x.inOut having been true or x.out having contained the data x, in the lokerns or glkerns call.

The plot() method calls plotDS from package sfsmisc.

predict(object, x, deriv) when either some x are not in x.out or deriv is not 0, basically recalls the original lokerns or glkerns function (keeping the bandwidths for lokerns).

Value

(differing, depending on the generic function)

Examples

## "interesting" artificial data:
set.seed(47)
x <- sort(round(10*runif(250),2))
fn <- function(x) 5 - x/2 + 3*exp(-(x-5)^2)
y <- fn(x) + rnorm(x)/4
plot(x,y)
## Tracing the phases in the Fortran code: trace=1 gives some, trace=3 gives *much*
lof <- lokerns(x,y, trace=2)
plot(lof)
plot(lof, cex = 1/4)# maybe preferable
plot(fn, 0, 10, add=TRUE, col=adjustcolor("gray40",1/2), lwd=2, lty=2)
## Simpler, using the lines() method:
plot(x,y); lines(lof, lwd=2, col=2)

qqnorm(residuals(lof)) # hmm... overfitting?
stopifnot(all.equal(y, fitted(lof) + residuals(lof), tolerance = 1e-15),
          predict(lof)$y == fitted(lof))
lof$iter # negative ?
tt <- seq(0, 10, by=1/32)
## again with 'tracing' [not for the average user]
p0 <- predict(lof, x=tt,          trace=1)
p1 <- predict(lof, x=tt, deriv=1, trace=1)
p2 <- predict(lof, x=tt, deriv=2)
plot(p2, type="l"); abline(h=0, lty=3) # not satisfactory, but lokerns(*,deriv=2) is
lof2 <- lokerns(x,y, deriv=2)
plot(lof2, ylim = c(-12,4), main=
   "lokerns(*, deriv=2) -- much more smooth than predict(*,deriv=2)")
mtext("as lokerns(*, deriv=2) chooses larger bandwidths[] !")
lines(predict(lof2, x=tt), col=adjustcolor("tomato", 1/3), lwd=5)
lines(p2, col="gray50"); abline(h=0, lty=3)
## add 2nd derivative of underlying fn():
f2 <- fn; body(f2) <- D(D(body(fn), "x"),"x")
lines(tt, f2(tt), col="blue")
## "interesting" artificial data:
set.seed(47)
x <- sort(round(10*runif(250),2))
fn <- function(x) 5 - x/2 + 3*exp(-(x-5)^2)
y <- fn(x) + rnorm(x)/4
plot(x,y)
## Tracing the phases in the Fortran code: trace=1 gives some, trace=3 gives *much*
lof <- lokerns(x,y, trace=2)
plot(lof)
plot(lof, cex = 1/4)# maybe preferable
plot(fn, 0, 10, add=TRUE, col=adjustcolor("gray40",1/2), lwd=2, lty=2)
## Simpler, using the lines() method:
plot(x,y); lines(lof, lwd=2, col=2)

qqnorm(residuals(lof)) # hmm... overfitting?
stopifnot(all.equal(y, fitted(lof) + residuals(lof), tolerance = 1e-15),
          predict(lof)$y == fitted(lof))
lof$iter # negative ?
tt <- seq(0, 10, by=1/32)
## again with 'tracing' [not for the average user]
p0 <- predict(lof, x=tt,          trace=1)
p1 <- predict(lof, x=tt, deriv=1, trace=1)
p2 <- predict(lof, x=tt, deriv=2)
plot(p2, type="l"); abline(h=0, lty=3) # not satisfactory, but lokerns(*,deriv=2) is
lof2 <- lokerns(x,y, deriv=2)
plot(lof2, ylim = c(-12,4), main=
   "lokerns(*, deriv=2) -- much more smooth than predict(*,deriv=2)")
mtext("as lokerns(*, deriv=2) chooses larger bandwidths[] !")
lines(predict(lof2, x=tt), col=adjustcolor("tomato", 1/3), lwd=5)
lines(p2, col="gray50"); abline(h=0, lty=3)
## add 2nd derivative of underlying fn():
f2 <- fn; body(f2) <- D(D(body(fn), "x"),"x")
lines(tt, f2(tt), col="blue")

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, ...)
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. 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 `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 $k = \nu + 2$ where $k - \nu$ must be even. The maximal possible values are for automatic smoothing, $k \le 4$ , whereas for smoothing with input bandwidth array, $k \le 6$ .
`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`	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))
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))

Nonparametric Variance Estimator

Description

Estimates the error variance $\sigma^2$ nonparametrically in the model

$Y_i = m(x_i) + E_i,$

where $E_i \sim (0,\sigma^2)$ , i.i.d.

Computes leave-one-out residuals (local linear approximation followed by reweighting) and their variance.

Usage

varNPreg(x, y)
varNPreg(x, y)

Arguments

`x`	abscissae values, ordered increasingly.
`y`	observations at `y[i]` at `x[i]`.

Value

A list with components

`res`	numeric; residuals at `x[]` of length `n`.
`snr`	explained variance of the true curve, i.e., an $R^2$ , defined as $1 - \hat{\sigma^2}/ \hat{\sigma_0^2}$ , where $\hat{\sigma^2} =$ `sigma2`, and $\hat{\sigma_0^2} := var(Y) = E[Y^2] - (E[Y])^2$ , see the example below.
`sigma2`	estimation of residual variance, $\hat{\sigma^2}$ .

Note

This is an R interface to the resest Fortran subroutine, used in lokerns and glkerns, see the latter's help page for references and context.

Earlier version of the lokern package accidentally contained varest() which has been an identical copy of varNPreg().

Author(s)

Martin Maechler

Examples

n <- 100
x <- sort(runif(n))
y <- sin(pi*x) + rnorm(n)/10
str(ve <- varNPreg(x,y))

plot(x, y)
## "fitted" = y - residuals:
lines(x, y - ve$res, col=adjustcolor(2, 1/2), lwd=3)
segments(x,y,x,y-ve$res, col=3:4, lty=2:3, lwd=1:2)

## sigma2 :=  1/n  sum_i res_i^2  :
          with(ve,         c(sigma2, sum(res^2)/n))
stopifnot(with(ve, all.equal(sigma2, sum(res^2)/n)))

## show how 'snr' is computed, given 'sigma2'  { in ../src/auxkerns.f }
dx2 <- diff(x, 2) # (x[i+1] - x[i-1]) i= 2..{n-1}
dx.n <- c(x[2]-x[1], dx2, x[n]-x[n-1])
SY  <- sum(dx.n * y)
SY2 <- sum(dx.n * y^2)
rx <- 2*(x[n]-x[1]) # 'dn'
(sigm2.0 <- SY2/rx - (SY/rx)^2)
(R2 <- 1 - ve$sigma2 / sigm2.0)
stopifnot(all.equal(ve$snr, R2))
n <- 100
x <- sort(runif(n))
y <- sin(pi*x) + rnorm(n)/10
str(ve <- varNPreg(x,y))

plot(x, y)
## "fitted" = y - residuals:
lines(x, y - ve$res, col=adjustcolor(2, 1/2), lwd=3)
segments(x,y,x,y-ve$res, col=3:4, lty=2:3, lwd=1:2)

## sigma2 :=  1/n  sum_i res_i^2  :
          with(ve,         c(sigma2, sum(res^2)/n))
stopifnot(with(ve, all.equal(sigma2, sum(res^2)/n)))

## show how 'snr' is computed, given 'sigma2'  { in ../src/auxkerns.f }
dx2 <- diff(x, 2) # (x[i+1] - x[i-1]) i= 2..{n-1}
dx.n <- c(x[2]-x[1], dx2, x[n]-x[n-1])
SY  <- sum(dx.n * y)
SY2 <- sum(dx.n * y^2)
rx <- 2*(x[n]-x[1]) # 'dn'
(sigm2.0 <- SY2/rx - (SY/rx)^2)
(R2 <- 1 - ve$sigma2 / sigm2.0)
stopifnot(all.equal(ve$snr, R2))

Simulated Linear plus Exponential Peak

Description

This is simulated data, a linear plus an exponential peak. In similar form, data like this appears in the smoothing literature since at least the eighties.

Usage

data(xSim)data(xSim)

Format

A vector of 75 numbers between -3.1323 and 4.4505, all rounded to 4 digits after the decimal.

Source

https://www.biostat.uzh.ch/en/research/software/kernel.html

Examples

data(xSim)
plot(xSim, main = "`xSim' - N=75 simulated linear + peak")
data(xSim)
plot(xSim, main = "`xSim' - N=75 simulated linear + peak")

Package 'lokern'

Help Index

Kernel Regression Smoothing with Adaptive Plug-in Bandwidth

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Methods for ("KernS" classed) Results of lokerns() and glkerns()

Description

Usage

Arguments

Details

Value

See Also

Examples

Kernel Regression Smoothing with Local Plug-in Bandwidth

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Nonparametric Variance Estimator

Description

Usage

Arguments

Value

Note

Author(s)

See Also

Examples

Simulated Linear plus Exponential Peak

Description

Usage

Format

Source

See Also

Examples