[R] Prediction Intervals (reposting)
Uwe Ligges
ligges at statistik.uni-dortmund.de
Sat Aug 16 20:04:48 CEST 2003
Ronnen Levinson wrote:
> (I'm reposting this message because the original has not appeared after
> about 2 days. Sorry if it shows up twice.)
>
>
> Hello.
>
> First, thanks to those who responded to my recent inquiry about using
> contour() over arbitrary (x,y) by mentioning the interp() function in
> the akima package. That worked nicely. Now for a new question:
>
> I would like to use a pair of prediction intervals to graphically bound
> the noise in some y(x) measurements. Here's an artificial example
> showing a function y(x)=x + noise, where the noise diminishes as x
> increases from 0 to 1.
>
> x=seq(0,1,0.01)
> y=x+runif(length(x),-1,1)*((1-x)/5)
> fit=lm(y ~ 1 + x)
> pred=predict(fit, interval="prediction")
> matplot(x,pred,type="l",ylab="y")
> points(x,y)
>
> I would have expected the lower and upper prediction intervals to
> converge as x increases (and the noise decreases), but they seem to
> remain virtually equidistant. Can anyone explain (a) the behavior that I
> see, and (b) how to obtain curves that do bound the noise?
>
> Thanks,
>
> Ronnen.
>
Well, since you use an "ordinary" lm(), for your residuals Var(e)=const.
is assumed (among other assumptions), hence a global variance is also
considered and estimated for prediction, of course:
I(x_0) := x'_0 +- t_{n-h-1; 1-\alpha/2}
\sqrt{s^2 (x'_0 (X'X)^{-1} x_0 + 1)}
^^^
You have to choose a completely different model.
Uwe Ligges
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