Re: [R] Weird LM behaviour

From: Jason Barnhart <jasoncbarnhart_at_msn.com>
Date: Sat 20 May 2006 - 09:16:54 EST

I see what you mean. Thanks for the correction.

-jason

> 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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