varNPreg: 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)

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, $\hat{\sigma^2}$.

Arguments

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

Author

Martin Maechler

Examples

Run this code

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))