R help - Aug 2003 - Prediction Intervals (reposting)

(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.

-- 
Ronnen Levinson, Ph.D.            \/      RML27 at cornell.edu
scientist                         ||      http://ronnen.com
Lawrence Berkeley National Lab    /\      fax 425.955.1992
 
=====================================I took a speed reading course and read
'War and Peace' in twenty minutes. It involves Russia.
-- Woody Allen

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

The usual assumption in regression is that noise is iid, that is has the same
distribution except for a location shift at all values of x.  Prediction
intervals
adopting this premise are based on an average of the noise over the whole sample
and thus produce the result you see in your figure.

Your model is what I would call a linear location scale model, that is both
location and scale of the noise are linear in x...one way to generate prediction
bands
for this sort of model is to fit conditional quantile functions.  Here is an
example:


     x=seq(0,1,0.01)
     y=x+runif(length(x),-1,1)*((1-x)/5)
     require(quantreg)
     abline(rq(y~x,tau=.9)$coef[,1])
     abline(rq(y~x,tau=.1)$coef[,1])

the resulting band contains 80 percent of the sample observations.  You can
get some further details from ?rq and the references their.


url:	www.econ.uiuc.edu/~roger/my.html	Roger Koenker
email	rkoenker at uiuc.edu			Department of Economics
vox: 	217-333-4558				University of Illinois
fax:   	217-244-6678				Champaign, IL 61820

On Sat, 16 Aug 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.
>
> --
> Ronnen Levinson, Ph.D.            \/      RML27 at cornell.edu
> scientist                         ||      http://ronnen.com
> Lawrence Berkeley National Lab    /\      fax 425.955.1992
>
> =====================================> I took a speed reading course and
read 'War and Peace' in twenty minutes. It involves Russia.
> -- Woody Allen
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
>