Package 'lpridge'

Local polynomial regression fitting with Epanechnikov weights

Description

Fast and stable algorithm for nonparametric estimation of regression functions and their derivatives via local polynomials with Epanechnikov weight function.

Usage

lpepa(x, y, bandwidth, deriv = 0, n.out = 200, x.out = NULL,
      order = deriv+1, mnew = 100, var = FALSE)

Arguments

x	vector of design points, not necessarily ordered.
y	vector of observations of the same length as x.
bandwidth	bandwidth(s) for nonparametric estimation. Either a number or a vector of the same length as x.out.
deriv	order of derivative of the regression function to be estimated; defaults to deriv = 0.
n.out	number of output design points where the function has to be estimated. The default is n.out=200.
x.out	vector of output design points where the function has to be estimated. The default value is an equidistant grid of n.out points from min(x) to max(x).
order	integer, order of the polynomial used for local polynomials. Must be $\le 10$ and defaults to order = deriv+1.
mnew	integer forcing to restart the algorithm after mnew updating steps. The default is mnew = 100. For mnew = 1 you get a numerically “super-stable” algorithm (see reference SBE&G below).
var	logical flag: if TRUE, the variance of the estimator proportional to the residual variance is computed (see details).

Details

More details are described in the first reference SBE&G (1994) below. In S&G, a bad finite sample behaviour of local polynomials for random designs was found. For practical use, we therefore propose local polynomial regression fitting with ridging, as implemented in the function lpridge. In lpepa, several parameters described in SBE&G are fixed either in the fortran routine or in the R-function. There, you find comments how to change them.

For var=TRUE, the variance of the estimator proportional to the residual variance is computed, i.e., the exact finite sample variance of the regression estimator is var(est) = est.var * sigma^2.

Value

a list including used parameters and estimator.

x	vector of ordered design points.
y	vector of observations ordered according to x.
bandwidth	vector of bandwidths actually used for nonparametric estimation.
deriv	order of derivative of the regression function estimated.
x.out	vector of ordered output design points.
order	order of the polynomial used for local polynomials.
mnew	force to restart the algorithm after mnew updating steps.
var	logical flag: whether the variance of the estimator was computed.
est	estimator of the derivative of order deriv of the regression function.
est.var	estimator of the variance of est (proportional to residual variance).

References

Originally available from Biostats, University of Zurich under ‘Manuscripts’, but no longer.

- Numerical stability and computational speed:
B. Seifert, M. Brockmann, J. Engel and T. Gasser (1994) Fast algorithms for nonparametric curve estimation. J. Computational and Graphical Statistics 3, 192–213.

- Statistical properties:
Seifert, B. and Gasser, T. (1996) Finite sample variance of local polynomials: Analysis and solutions. J. American Statistical Association 91(433), 267–275.

Seifert, B. and Gasser, T. (2000) Data adaptive ridging in local polynomial regression. J. Computational and Graphical Statistics 9, 338–360.

Seifert, B. and Gasser, T. (1998) Ridging Methods in Local Polynomial Regression. in: S. Weisberg (ed), Dimension Reduction, Computational Complexity, and Information, Vol.30 of Computing Science & Statistics, Interface Foundation of North America, 467–476.

Seifert, B. and Gasser, T. (1998) Local polynomial smoothing. in: Encyclopedia of Statistical Sciences, Update Vol.2, Wiley, 367–372.

Seifert, B., and Gasser, T. (1996) Variance properties of local polynomials and ensuing modifications. in: Statistical Theory and Computational Aspects of Smoothing, W. Härdle, M. G. Schimek (eds), Physica, 50–127.

See Also

lpridge, and also lowess and loess which do local linear and quadratic regression quite a bit differently.

Examples

data(cars)
attach(cars)

epa.sd <- lpepa(speed,dist, bandw=5)		# local polynomials

plot(speed, dist, main = "data(cars) & lp epanechnikov regression")
lines(epa.sd$x.out, epa.sd$est,  col="red")
lines(lowess(speed,dist, f= .5), col="orange")
detach()

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Local polynomial regression fitting with ridging

Description

Fast and stable algorithm for nonparametric estimation of regression functions and their derivatives via local polynomials and local polynomial ridge regression with polynomial weight functions.

Usage

lpridge(x, y, bandwidth, deriv=0, n.out=200, x.out=NULL,
	order = NULL, ridge = NULL, weight = "epa", mnew = 100,
	var = FALSE)

Arguments

x	vector of design points, not necessarily ordered.
y	vector of observations of the same length as x.
bandwidth	bandwidth for nonparametric estimation. Either a number or a vector of the same length as x.out.
deriv	order of derivative of the regression function to be estimated; default is 0.
n.out	number of output design points at which to evaluate the estimator; defaults to 200.
x.out	vector of output design points at which to evaluate the estimator; By default, an equidistant grid of n.out points from min(x) to max(x).
order	order of the polynomial used for local polynomials. The default value is deriv + 1.
ridge	ridging parameter. The default value performs a slight ridging (see "Details"). ridge = 0 leads to the local polynomial estimator without ridging.
weight	kernel weight function. The default value is weight = "epa" for Epanechnikov weights. Other weights are "bi" for biweights (square of "epa") and "tri" for triweights (cube of "epa"). If weight is a vector, it is interpreted as vector of coefficients of the polynomial weight function. Thus, weight = "epa" is equivalent to weight = c(1,0,-1).
mnew	force to restart the algorithm after mnew updating steps. The default value is mnew = 100. For mnew = 1 you get a numerically "super-stable" algorithm (see reference SBE&G below).
var	logical flag: if TRUE, the variance of the estimator proportional to the residual variance is computed (see "Details" below).

Details

described in the reference SBE&G below. Several parameters described there are fixed either in the fortran routine or in the R-function. There, you find comments how to change them.

In S&G, a bad finite sample behavior of local polynomials for random design was found, and ridging of the estimator was proposed. In lpridge(), we use a ridging matrix corresponding to the smoothness assumption “The squared difference of the derivative of order deriv of the regression function at the point of estimation and the weighted mean of design points is bounded by the residual variance divided by the ridge parameter.”

Thus, without any smoothness assumption, ridge = 0 would be appropriate, and for a nearly constant derivative of order deriv, a ridge parameter going to infinity behaves well. For equidistant design, ridging influences the estimator only at the boundary. Asymptotically, the influence of any non-increasing ridge parameter vanishes.

So far, our experience with the choice of a ridging parameter is limited. Therefore we have chosen a default value which performs a slight modification of the local polynomial estimator (with denotations $h =$ bandwidth, $d =$ deriv, and where $n_0$ = length(x)*mean(bandwidth)/diff(range(x)) is a mean number of observations in a smoothing interval):

$ridge = 5\sqrt{n_0} h^{2d} / ((2d+3)(2d+5))$

For var=TRUE, the variance of the estimator proportional to the residual variance is computed, i.e., the exact finite sample variance of the regression estimator is var(est) = est.var * sigma^2.

Value

a list including used parameters and estimator.

x	vector of ordered design points.
y	vector of observations ordered according to x.
bandwidth	vector of bandwidths actually used for nonparametric estimation.
deriv	order of derivative of the regression function estimated.
x.out	vector of ordered output design points.
order	order of the polynomial used for local polynomials.
ridge	ridging parameter used.
weight	vector of coefficients of the kernel weight function.
mnew	force to restart the algorithm after mnew updating steps.
var	logical flag: whether the variance of the estimator was computed.
est	estimator of the derivative of order deriv of the regression function.
est.var	estimator of the variance of est (proportional to residual variance).

References

The same as for lpepa.

Examples

data(cars)
attach(cars)
plot(speed, dist, main = "data(cars) & lpRIDGE Regression")


myfit <- lpridge(speed,dist,bandw = 5, ridge=0)		# local polynomials
lines(myfit$x.out,myfit$est,col=2)

myridge <- lpridge(speed,dist,bandw = 5)		# local pol. ridge
lines(myridge$x.out,myridge$est,col=3)
mtext("bandw = 5")
legend(5,120, c("ridge = 0", "default ridging"), col = 2:3, lty = 1)
detach()

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