[R] Prediction Intervals (reposting)

[Uwe Ligges]

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