From: Jason Barnhart <jasoncbarnhart_at_msn.com>

Date: Sat 20 May 2006 - 09:16:54 EST

R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Sat May 20 09:30:46 2006

Date: Sat 20 May 2006 - 09:16:54 EST

I see what you mean. Thanks for the correction.

-jason

- Original Message ----- From: "Thomas Lumley" <tlumley@u.washington.edu> To: "Jason Barnhart" <jasoncbarnhart@msn.com> Cc: <R-help@stat.math.ethz.ch>; "Rense Nieuwenhuis" <r.nieuwenhuis@student.ru.nl> Sent: Friday, May 19, 2006 2:39 PM Subject: Re: [R] Weird LM behaviour

> On Fri, 19 May 2006, Jason Barnhart wrote:

*>
**>> No, not weird.
**>>
**>> Think of it this way. As you move point (0,2) to (1,2) the slope which
**>> was
**>> 0 is moving towards infinity. Eventually the 3 points are perfectly
**>> vertical and so must have infinite slope.
**>>
**>> Your delta-x is not sufficiently granular to show the slope change for
**>> x-values very close to 1 but not yet 1, like 0.999999999. Note lm
**>> returns
**>> NA when x=1.
**>
**> This turns out not to be the case. Worked to infinite precision the mean
**> of y is 2 at x and at 1, so the infinite-precision slope is exactly zero
**> for all x!=1 and undefined for x=1.
**>
**> Now, we are working to finite precision and the slope is obtained by
**> solving a linear system that gets increasingly poorly conditioned as x
**> approaches 1. This means that for x not close to 1 the answer should be 0
**> to withing a small multiple of machine epsilon (and it is) and that for x
**> close to 1 the answer should be zero to within an increasingly large
**> multiple of machine epsilon (and it is).
**>
**> Without a detailed error analysis of the actual algorithm being used, you
**> can't really predict whether the answer will follow a more-or-less
**> consistent trend or oscillate violently. You can estimate a bound for the
**> error: it should be a small multiple of the condition number of the design
**> matrix times machine epsilon.
**>
**> As an example of how hard it is to predict exactly what answer you get, if
**> R used the textbook formula for linear regression the bound would be a lot
**> worse, but in this example the answer is slightly closer to zero done that
**> way.
**>
**> Unless you really need to know, trying to understand why the fourteenth
**> decimal place of a result has the value it does is not worth the effort.
**>
**>
**> -thomas
**>
*

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