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

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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 refernce 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 n0 = length(x)*mean(bandwidth)/diff(range(x)) is a mean number of observations in a smoothing interval):

ridge = 5*sqrt(n0)*h^(2*d) / ((2*d+3)*(2*d+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

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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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