lpepa: Local polynomial regression fitting with Epanechnikov weights

In lpridge: Local Polynomial (Ridge) Regression

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()

lpridge documentation built on May 29, 2024, 5:12 a.m.