Version:	0.8.0
Title:	Kernel Local Polynomial Regression
Description:	Computes local polynomial estimators for the regression and also density. It comprises several different utilities to handle kernel estimators.
Depends:	R (≥ 2.5.0), graphics, stats
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation:	yes
Packaged:	2022-11-28 18:26:23 UTC; bquast
Author:	Jorge Luis Ojeda Cabrera [aut, cre], Bastiaan Quast [ctb]
Maintainer:	Jorge Luis Ojeda Cabrera <jojeda@unizar.es>
Repository:	CRAN
Date/Publication:	2022-11-29 07:35:21 UTC

Kernel characteristics

Description

For a given kernel these functions return some of the most commonly used numeric values related to them.

Usage

	RK(K)
	RdK(K)
	mu2K(K)
	mu0K(K)
	K4(K)
	dom(K)

Arguments

Details

Most of these functions are implemented as an attribute of every kernel. For the computations of the numeric value for these quantities, see references.

Value

A numeric value returning:

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

Kernels, Compute kernel values.

Examples

	##	Note that lower and upper params are set in the definition to
	##	use 'dom()' function.
	g <- function(kernels)
	{
		mu0 <- sapply(kernels,function(x) computeMu0(x,))
		mu0.ok <- sapply(kernels,mu0K)
		mu2 <- sapply(kernels,function(x) computeMu(2,x))
		mu2.ok <- sapply(kernels,mu2K)
		Rk.ok <- sapply(kernels,RK)
		RK <- sapply(kernels,function(x) computeRK(x))
		K4 <- sapply(kernels,function(x) computeK4(x))
		res <- data.frame(mu0,mu0.ok,mu2,mu2.ok,RK,Rk.ok,K4)
		res
	}
	g(kernels=c(EpaK,gaussK,TriweigK,TrianK))

Kernels.

Description

Definition of common kernels used in local polynomial estimation.

Usage

	CosK(x)
	EpaK(x)
	Epa2K(x)
	gaussK(x)

Arguments

Details

The implementation of these kernels is done by means functions that can operate on vectors.

Most common referred numeric values for these kernels are provided as attributes, see RK, mu0K, etc....

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

RK, mu0K.

Parzen–Rosenblatt denstiy estimator.

Description

Parzen–Rosenblat univariate density estimator.

Usage

PRDenEstC(x, xeval, bw, kernel, weig = rep(1, length(x)))

Arguments

Details

Simple Parzen–Rosenblat univariate density estimation, computed using definition.

Value

Returns an (x,den) data frame.

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

density, that uses FT to compute a kernel density estimator, bkde from package KernSmooth for a binned version, and bw.nrd0, dpik, denCVBwSelC for bandwidth selection.

Examples

	N <- 100
	x <-  runif(N)
	xeval <- 0:10/10
	b0.125 <- PRDenEstC(x, xeval, 0.125, EpaK)
	b0.05 <- PRDenEstC(x, xeval, 0.05, EpaK)
	cbind(x = xeval, fx = 1, b0.125 = b0.125$den, b0.05 = b0.05$den)

Bivariate Local estimation.

Description

Simple bivariate Local density and regression estimation with weights.

Usage

    bivDens(X,weig,K,H)
    bivReg(X,Y,weig,K,H)
    ## S3 method for class 'bivNpEst'
predict(object,newdata,...)
    ## S3 method for class 'bivNpEst'
plot(x,...)

Arguments

.

Details

The functions bivDens and bivReg provide a very basic interface that allows bivariate local estimation with weights. It implements basic kernel density estimator and Nadaraya–Watson estimator for bivariate data. Very simple interface methods allow the prediction and plotting of these estimators.

The only bivariate kernels provided are epaK2d and gauK2d. New ones can be added in the same way as functions with a vector of length 2.

The default bandwidth selector (see mayBeBwSel) that has been provided is not optimal or good in any sense. It has been added as a simple way to provide an easy, fast and simple way to be able to use the estimators.

The graphical parameters allowed for ... in plot(x,...) are those that appears in the function persp. The list plotBivNpEstOpts provide a default for some of these graphical parameters.

Value

A list containing:

Author(s)

Jorge Luis Ojeda Cabrera.

Examples

  n <- 100
  d <- data.frame(x=rexp(n,rate=1/2),y=rnorm(n))
  ## x is a length-biased version of an exp. dist. with rate 1.
  dDen <- bivDens(d,weig=1/d$x)
  plot(dDen,r=5)
  d <- data.frame(X1=runif(n),X2=runif(n))
  d$Y <-  exp(10*d$X1+d$X2^2) 
  dDen <- bivDens(d[,c("X1","X2")])
  plot(dDen,r=5)
  dReg <- bivReg(d[,c("X1","X2")],d$Y)
  plot(dReg,r=5)
  plot(dReg,r=5,phi=20,theta=40)

Compute kernel values.

Description

Some R code provided to compute kernel related values.

Usage

	computeRK(kernel, lower=dom(kernel)[[1]], upper=dom(kernel)[[2]],
			subdivisions = 25)
	computeK4(kernel, lower=dom(kernel)[[1]], upper=dom(kernel)[[2]],
			subdivisions = 25)
	computeMu(i, kernel, lower=dom(kernel)[[1]], upper=dom(kernel)[[2]],
			subdivisions = 25)
	computeMu0(kernel, lower=dom(kernel)[[1]], upper=dom(kernel)[[2]],
			subdivisions = 25)
	Kconvol(kernel,lower=dom(kernel)[[1]],upper=dom(kernel)[[2]],
			subdivisions = 25)

Arguments

Details

These functions uses function integrate.

Value

A numeric value returning:

These functions are implemented by means of integrate.

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

RK, Kernel characteristics, integrate.

Examples

	##	Note that lower and upper params are set in the definition to
	##	use 'dom()' function.
	g <- function(kernels)
	{
		mu0 <- sapply(kernels,function(x) computeMu0(x,))
		mu0.ok <- sapply(kernels,mu0K)
		mu2 <- sapply(kernels,function(x) computeMu(2,x))
		mu2.ok <- sapply(kernels,mu2K)
		Rk.ok <- sapply(kernels,RK)
		RK <- sapply(kernels,function(x) computeRK(x))
		K4 <- sapply(kernels,function(x) computeK4(x))
		res <- data.frame(mu0,mu0.ok,mu2,mu2.ok,RK,Rk.ok,K4)
		res
	}
	g(kernels=c(EpaK,gaussK,TriweigK,TrianK))

CV bandwidth selector for density

Description

Computes Cross Validation bandwidth selector for the Parzen–Rosenblatt density estimator...

Usage

denCVBwSelC(x, kernel = gaussK, weig = rep(1, length(x)),
            interval = .lokestOptInt)

Arguments

Details

The selector is implemented using its definition.

Value

A numeric value with the bandwidth.

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

bw.nrd0, dpik.

Examples

stdy <- function(size=100,rVar=rnorm,dVar=dnorm,kernel=gaussK,x=NULL)
{
	if( is.null(x) ) x <- rVar(size)
	Tc <- system.time( dbwc <- denCVBwSelC(x,kernel) )[3]
	ucvT <- system.time( ucvBw <- bw.ucv(x,lower=0.00001,upper=2.0) )[3]
	nrdT <- system.time( nrdBw <- bw.nrd(x) )[3]
	{
		xeval <- seq( min(x)+dbwc , max(x)-dbwc ,length=50)
		hist(x,probability=TRUE)
		lines(xeval,trueDen <- dVar(xeval),col="red")
		lines(density(x),col="cyan")
		xevalDenc <- PRDenEstC(x,xeval,dbwc,kernel)
		dencMSE <- mean( (trueDen-xevalDenc)^2 )
		xevalucvDen <- PRDenEstC(x,xeval,ucvBw,kernel)
		ucvMSE <- mean( (trueDen-xevalucvDen)^2 )
		xevalDenNrd <- PRDenEstC(x,xeval,nrdBw,kernel)
		nrdMSE <- mean( (trueDen-xevalDenNrd)^2 )
		lines(xevalDenc,col="green")
		lines(xevalucvDen,col="blue")
		lines(xevalDenNrd,col="grey")
	}
	return( cbind(  bwVal=c(evalC=dbwc,ucvBw=ucvBw,nrdBw=nrdBw),
				mse=c(dencMSE,ucvMSE,nrdMSE),
				time=c(Tc,ucvT,nrdT) ) )
}
stdy(100,kernel=gaussK)
stdy(100,rVar=rexp,dVar=dexp,kernel=gaussK)
##	check stdy with other kernel, distributions

Equivalent Kernel.

Description

Computes the Equivalent kernel for the local polynomial estimation.

Usage

equivKernel(kernel,nu,deg,lower=dom(kernel)[[1]],upper=dom(kernel)[[2]],
	subdivisions=25)

Arguments

Details

The definition of the Equivalent kernel for the local polynomial estimation can be found in page 64 in Fan and Gijbels(1996). The implementation uses computeMu to compute matrix S and then returns a function object

Value

Returns a vector whose components are the equivalent kernel used to compute the local polynomial estimator for the derivatives in nu.

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

See Also

cteNuK, adjNuK.

Examples

##	Some kernels and equiv. for higher order
##	compare with p=1
curve(EpaK(x),-3,3,ylim=c(-.5,1))
f <- equivKernel(EpaK,0,3)
curve(f(x),-3,3,add=TRUE,col="blue")
curve(gaussK(x),-3,3,add=TRUE)
f <- equivKernel(gaussK,0,3)
curve(f(x),-3,3,add=TRUE,col="blue")
##	Draw several Equivalent locl polynomial kernels
curve(EpaK(x),-3,3,ylim=c(-.5,1))
for(p in 1:5){
	curve(equivKernel(gaussK,0,p)(x),-3,3,add=TRUE)
    }

Kernel Constants used in Bandwidth Selection.

Description

These are values depending on the kernel and the local polynomial degrees that are used in bandwidth selection, as proposed in Fan and Gijbels(1996).

Usage

cteNuK(nu,p,kernel,lower=dom(kernel)[[1]],upper=dom(kernel)[[2]],
	subdivisions= 25)
adjNuK(nu,p,kernel,lower=dom(kernel)[[1]],upper=dom(kernel)[[2]],
	subdivisions= 25)

Arguments

Details

cteNuK is computed using Compute kernel values and link{equivKernel} jointly with the numerical integration utility integrate. adjNuK is implemented using quotients of previous functions. See Fan and Gijbels(1996) pages 67 and 119.

Value

Both functions returns numeric values.

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

regCVBwSelC, pluginBw, integrate.

Local Polynomial Weights

Description

Local Constant and local Linear estimator with weight.

Usage

  locCteWeightsC(x, xeval, bw, kernel, weig = rep(1, length(x)))
  locLinWeightsC(x, xeval, bw, kernel, weig = rep(1, length(x)))
  locPolWeights(x, xeval, deg, bw, kernel, weig = rep(1, length(x)))
  locWeightsEval(lpweig, y)
  locWeightsEvalC(lpweig, y)

Arguments

Details

locCteWeightsC and locLinWeightsC computes local constant and local linear weights, say any of the entries of the vector (X^TWX)^{-1}X^TW for p=0 and p=1 resp. locWeightsEvalC and locWeightsEval computes local the estimator for a given vector of responses y

Value

locCteWeightsC and locLinWeightsC returns a list with two components

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

Kernels, locpol.

Examples

	size <- 200
	sigma <- 0.25
	deg <- 1
	kernel <- EpaK
	bw <- .25
	xeval <- 0:100/100
	regFun <- function(x) x^3
	x <- runif(size)
	y <- regFun(x) + rnorm(x, sd = sigma)
	d <- data.frame(x, y)
	lcw <- locCteWeightsC(d$x, xeval, bw, kernel)$locWeig
	lce <- locWeightsEval(lcw, y)
	lceB <- locCteSmootherC(d$x, d$y, xeval, bw, kernel)$beta0
	mean((lce-lceB)^2)
    llw <- locLinWeightsC(d$x, xeval, bw, kernel)$locWeig
	lle <- locWeightsEval(llw, y)
	lleB <- locLinSmootherC(d$x, d$y, xeval, bw, kernel)$beta0
	mean((lle-lleB)^2)

Local Polynomial estimation.

Description

Formula interface for the local polynomial estimation.

Usage

    locpol(formula,data,weig=rep(1,nrow(data)),bw=NULL,kernel=EpaK,deg=1,
            xeval=NULL,xevalLen=100)
    confInterval(x)
    ## S3 method for class 'locpol'
residuals(object,...)
    ## S3 method for class 'locpol'
fitted(object,deg=0,...)
    ## S3 method for class 'locpol'
summary(object,...)
    ## S3 method for class 'locpol'
print(x,...)
    ## S3 method for class 'locpol'
plot(x,...)

Arguments

Details

This is an interface to the local polynomial estimation function that provides basic lm functionality. summary and print methods shows very basic information about the fit, fitted return the estimation of the derivatives if deg is larger than 0, and plot provides a plot of data, local polynomial estimation and the variance estimation.

Variance estimation is carried out by means of the local constant regression estimation of the squared residuals.

confInterval provides confidence intervals for all points in x$lpFit$[,x$X], say those in xeval.

Value

A list containing among other components:

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

Crist'obal, J. A. and Alcal\'a, J. T. (2000). Nonparametric regression estimators for length biased data\/. J. Statist. Plann. Inference, 89, pp. 145-168.

Ahmad, Ibrahim A. (1995) On multivariate kernel estimation for samples from weighted distributions\/. Statistics & Probability Letters, 22, num. 2, pp. 121-129

See Also

locpoly from package KernSmooth, ksmooth and loess in stats (but from earlier package modreg).

Examples

    N <- 250
    xeval <- 0:100/100
    ##  ex1
    d <- data.frame(x = runif(N))
    d$y <- d$x^2 - d$x + 1 + rnorm(N, sd = 0.1)
    r <- locpol(y~x,d)
    plot(r)
    ##  ex2
    d <- data.frame(x = runif(N))
    d$y <- d$x^2 - d$x + 1 + (1+d$x)*rnorm(N, sd = 0.1)
    r <- locpol(y~x,d)
    plot(r)
    ## notice:
    rr <- locpol(y~x,d,xeval=runif(50,-1,1))
    ## notice x has null dens. outside (0,1)
    ## plot(rr) raises an error, no conf. bands outside (0,1).
    ## length biased data !!
    d <- data.frame(x = runif(10*N))
    d$y <- d$x^2 - d$x + 1 + (rexp(10*N,rate=4)-.25)
    posy <- d$y[ whichYPos <- which(d$y>0) ];
    d <- d[sample(whichYPos, N,prob=posy,replace=FALSE),]
    rBiased <- locpol(y~x,d)
    r <- locpol(y~x,d,weig=1/d$y)
    plot(d)
    points(r$lpFit[,r$X],r$lpFit[,r$Y],type="l",col="blue")
    points(rBiased$lpFit[,rBiased$X],rBiased$lpFit[,rBiased$Y],type="l")
    curve(x^2 - x + 1,add=TRUE,col="red")

Local Polynomial estimation.

Description

Computes the local polynomial estimation of the regression function.

Usage

locCteSmootherC(x, y, xeval, bw, kernel, weig = rep(1, length(y)))
locLinSmootherC(x, y, xeval, bw, kernel, weig = rep(1, length(y)))
locCuadSmootherC(x, y, xeval, bw, kernel, weig = rep(1, length(y)))
locPolSmootherC(x, y, xeval, bw, deg, kernel, DET = FALSE,
	weig = rep(1, length(y)))
looLocPolSmootherC(x, y, bw, deg, kernel, weig = rep(1, length(y)),
        DET = FALSE)

Arguments

Details

All these function perform the estimation of the regression function for different degrees. While locCteSmootherC, locLinSmootherC, and locCuadSmootherC uses direct computations for the degrees 0,1 and 2 respectively, locPolSmootherC implements a general method for any degree. Particularly useful can be looLocPolSmootherC(Leave one out) which computes the local polynomial estimator for any degree as locPolSmootherC does, but estimating m(x_i) without using i–th observation on the computation.

Value

A data frame whose components gives the evaluation points, the estimator for the regression function m(x) and its derivatives at each point, and the estimation of the marginal density for x to the p+1 power. These components are given by:

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

locpoly from package KernSmooth, ksmooth and loess in stats (but from earlier package modreg).

Examples

N <- 100
xeval <- 0:10/10
d <- data.frame(x = runif(N))
bw <- 0.125
fx <- xeval^2 - xeval + 1
##	Non random
d$y <- d$x^2 - d$x + 1
cuest <- locCuadSmootherC(d$x, d$y ,xeval, bw, EpaK)
lpest2 <- locPolSmootherC(d$x, d$y , xeval, bw, 2, EpaK)
print(cbind(x = xeval, fx, cuad0 = cuest$beta0,
lp0 = lpest2$beta0, cuad1 = cuest$beta1, lp1 = lpest2$beta1))
##	Random
d$y <- d$x^2 - d$x + 1 + rnorm(d$x, sd = 0.1)
cuest <- locCuadSmootherC(d$x,d$y , xeval, bw, EpaK)
lpest2 <- locPolSmootherC(d$x,d$y , xeval, bw, 2, EpaK)
lpest3 <- locPolSmootherC(d$x,d$y , xeval, bw, 3, EpaK)
cbind(x = xeval, fx, cuad0 = cuest$beta0, lp20 = lpest2$beta0,
lp30 = lpest3$beta0, cuad1 = cuest$beta1, lp21 = lpest2$beta1,
lp31 = lpest3$beta1)

Plugin Bandwidth selector.

Description

Implements a plugin bandwidth selector for the regression function.

Usage

pluginBw(x, y, deg, kernel, weig = rep(1,length(y)))

Arguments

Details

Computes the plug-in bandwidth selector as shown in Fan and Gijbels(1996) book using pilots estimates as given in page 110-112 (Rule of thumb for bandwidth selection). Currently, only even values of p are can be used.

Value

A numeric value.

Note

Currently, only even values of p are can be used.

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

thumbBw, regCVBwSelC.

Examples

	size <- 200
	sigma <- 0.25
	deg <- 1
	kernel <- EpaK
	xeval <- 0:100/100
	regFun <- function(x) x^3
	x <- runif(size)
	y <- regFun(x) + rnorm(x, sd = sigma)
	d <- data.frame(x, y)
	cvBwSel <- regCVBwSelC(d$x,d$y, deg, kernel, interval = c(0, 0.25))
	thBwSel <- thumbBw(d$x, d$y, deg, kernel)
	piBwSel <- pluginBw(d$x, d$y, deg, kernel)
	est <- function(bw, dat, x) return(locPolSmootherC(dat$x,dat$y, x, bw, deg,
					kernel)$beta0)
	ise <- function(val, est) return(sum((val - est)^2 * xeval[[2]]))
	plot(d$x, d$y)
	trueVal <- regFun(xeval)
	lines(xeval, trueVal, col = "red")
	xevalRes <- est(cvBwSel, d, xeval)
	cvIse <- ise(trueVal, xevalRes)
	lines(xeval, xevalRes, col = "blue")
	xevalRes <- est(thBwSel, d, xeval)
	thIse <- ise(trueVal, xevalRes)
	xevalRes <- est(piBwSel, d, xeval)
	piIse <- ise(trueVal, xevalRes)
	lines(xeval, xevalRes, col = "blue", lty = "dashed")
	res <- rbind(	bw = c(cvBwSel, thBwSel, piBwSel),
					ise = c(cvIse, thIse, piIse) )
	colnames(res) <- c("CV", "th", "PI")
	res

Cross Validation Bandwidth selector.

Description

Implements Cross validation bandwidth selector for the regression function.

Usage

regCVBwSelC(x, y, deg, kernel=gaussK,weig=rep(1,length(y)),
interval=.lokestOptInt)

Arguments

Details

Computes the weighted ASE for every bandwidth returning the minimum. The function is implemented by means of a C function that computes for a single bandwidth the ASE, and a call to optimise on a given interval.

Value

A numeric value.

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

H\"ardle W.(1990) Smoothing techniques. Springer Series in Statistics, New York (1991).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

thumbBw, pluginBw.

Examples

	size <- 200
	sigma <- 0.25
	deg <- 1
	kernel <- EpaK
	xeval <- 0:100/100
	regFun <- function(x) x^3
	x <- runif(size)
	y <- regFun(x) + rnorm(x, sd = sigma)
	d <- data.frame(x, y)
	cvBwSel <- regCVBwSelC(d$x,d$y, deg, kernel, interval = c(0, 0.25))
	thBwSel <- thumbBw(d$x, d$y, deg, kernel)
	piBwSel <- pluginBw(d$x, d$y, deg, kernel)
	est <- function(bw, dat, x) return(locPolSmootherC(dat$x,dat$y, x, bw, deg,
					kernel)$beta0)
	ise <- function(val, est) return(sum((val - est)^2 * xeval[[2]]))
	plot(d$x, d$y)
	trueVal <- regFun(xeval)
	lines(xeval, trueVal, col = "red")
	xevalRes <- est(cvBwSel, d, xeval)
	cvIse <- ise(trueVal, xevalRes)
	lines(xeval, xevalRes, col = "blue")
	xevalRes <- est(thBwSel, d, xeval)
	thIse <- ise(trueVal, xevalRes)
	xevalRes <- est(piBwSel, d, xeval)
	piIse <- ise(trueVal, xevalRes)
	lines(xeval, xevalRes, col = "blue", lty = "dashed")
	res <- rbind(	bw = c(cvBwSel, thBwSel, piBwSel),
					ise = c(cvIse, thIse, piIse) )
	colnames(res) <- c("CV", "th", "PI")
	res

Kernel selection.

Description

Uses kernel attributes to selects kernels. This function is mainly used for internal purposes.

Usage

selKernel(kernel)

Arguments

Details

Uses RK(K) to identify a kernel. The integer is used in the C code part to perform computations with given kernel. It allows for a kernel selection in C routines. It is used only for internal purposes.

Value

An integer that is unique for each kernel.

Warning

Used only for internal purposes.

Author(s)

Jorge Luis Ojeda Cabrera.

Simple smoother

Description

Computes simple kernel smoothing

Usage

  simpleSmootherC(x, y, xeval, bw, kernel, weig = rep(1, length(y)))
  simpleSqSmootherC(x, y, xeval, bw, kernel)

Arguments

Details

Computes simple smoothing, that is to say: it averages y values times kernel evaluated on x values. simpleSqSmootherC does the average with the square of such values.

Value

Both functions returns a data.frame with

...

Author(s)

Jorge Luis Ojeda Cabrera.

See Also

PRDenEstC, Kernel characteristics

Examples

	size <- 1000
	x <- runif(100)
	bw <- 0.125
	kernel <- EpaK
	xeval <- 1:9/10
	y <- rep(1,100)	
	##	x kern. aver. should give density f(x)
	prDen <- PRDenEstC(x,xeval,bw,kernel)$den
	ssDen <- simpleSmootherC(x,y,xeval,bw,kernel)$reg
	all(abs(prDen-ssDen)<1e-15)
	##	x kern. aver. should be f(x)*R2(K) aprox.
	s2Den <- simpleSqSmootherC(x,y,xeval,bw,kernel)$reg
	summary( abs(prDen*RK(kernel)-s2Den) )
	summary( abs(1*RK(kernel)-s2Den) )
	##	x kern. aver. should be f(x)*R2(K) aprox.
	for(n in c(1000,1e4,1e5))
	{
		s2D <- simpleSqSmootherC(runif(n),rep(1,n),xeval,bw,kernel)$reg
		cat("\n",n,"\n")
		print( summary( abs(1*RK(kernel)-s2D) ) )
	}

Rule of thumb for bandwidth selection.

Description

Implements Fan and Gijbels(1996)'s Rule of thumb for bandwidth selection

Usage

thumbBw(x, y, deg, kernel, weig = rep(1, length(y)))
compDerEst(x, y, p, weig = rep(1, length(y)))

Arguments

Details

See Fan and Gijbels(1996) book, Section 4.2. This implementation is also considering weights. compDerEst computes the p+1 derivative of the regression function in a simple manner, assuming it is a polynomial in x. thumbBw gives a bandwidth selector by means of pilot estimator given by compDerEst and the mean of residuals.

Value

thumbBw returns a single numeric value, while compDerEst returns a data frame whose components are:

Author(s)

Jorge Luis Ojeda Cabrera.

References

Fan, J. and Gijbels, I. Local polynomial modelling and its applications\/. Chapman & Hall, London (1996).

Wand, M.~P. and Jones, M.~C. Kernel smoothing\/. Chapman and Hall Ltd., London (1995).

See Also

regCVBwSelC, pluginBw.

Examples

	size <- 200
	sigma <- 0.25
	deg <- 1
	kernel <- EpaK
	xeval <- 0:100/100
	regFun <- function(x) x^3
	x <- runif(size)
	y <- regFun(x) + rnorm(x, sd = sigma)
	d <- data.frame(x, y)
	cvBwSel <- regCVBwSelC(d$x,d$y, deg, kernel, interval = c(0, 0.25))
	thBwSel <- thumbBw(d$x, d$y, deg, kernel)
	piBwSel <- pluginBw(d$x, d$y, deg, kernel)
	est <- function(bw, dat, x) return(locPolSmootherC(dat$x,dat$y, x, bw, deg,
					kernel)$beta0)
	ise <- function(val, est) return(sum((val - est)^2 * xeval[[2]]))
	plot(d$x, d$y)
	trueVal <- regFun(xeval)
	lines(xeval, trueVal, col = "red")
	xevalRes <- est(cvBwSel, d, xeval)
	cvIse <- ise(trueVal, xevalRes)
	lines(xeval, xevalRes, col = "blue")
	xevalRes <- est(thBwSel, d, xeval)
	thIse <- ise(trueVal, xevalRes)
	xevalRes <- est(piBwSel, d, xeval)
	piIse <- ise(trueVal, xevalRes)
	lines(xeval, xevalRes, col = "blue", lty = "dashed")
	res <- rbind(	bw = c(cvBwSel, thBwSel, piBwSel),
					ise = c(cvIse, thIse, piIse) )
	colnames(res) <- c("CV", "th", "PI")
	res