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IWETHEY Home / IWETHEY Board / Science Forum / Let's see if I got this right
Post #144,417
by tablizer
3/4/04 1:36:40 PM
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New Let's see if I got this right
So what you are saying is that IF the variance follows a normal distribution, then 2 is theoretically the best matching exponent. However, if the actual variance is not a normal curve, then it may not be.

However, since most "natural" things follow a normal variance, least-squred is used because for the normal assumption, it is the simplest known way to calculate the answer.

One would have to know more about the nature of the actual error distribution and know that it is not a normal distribution if they want to possibly do better than least-squares.

I can live with that.
________________
oop.ismad.com
Post #144,423
by ben_tilly
3/4/04 1:46:05 PM
Reply
New Yes
You can strengthen your "it may not be" though. The 1809 result from Gauss proved that if you have a family of possible error distributions which are the same up to translation (ie move the center) and scale (multiply by a constant factor), then least squares can only be best if that family is the normal distribution.

That is, if the real error curve differs from the normal at all, then least squares cannot be the best possible technique. The best one is very likely, however, to be far more complicated than "use a different exponent". There would be little point in going through this effort unless you had good reason to believe that the distribution was not normal, and you had a pretty good idea what the real distribution was.

Cheers,
Ben
"good ideas and bad code build communities, the other three combinations do not"
- [link|http://archives.real-time.com/pipermail/cocoon-devel/2000-October/003023.html|Stefano Mazzocchi]
     Spectrum matching? - (tablizer) - (12) - March 3, 2004, 05:15:22 PM EST
         It's a standard fitting technique. - (Another Scott) - (11) - March 3, 2004, 05:53:15 PM EST
             We had a big argument about least squares once - (tablizer) - (10) - March 3, 2004, 07:49:55 PM EST
                 It's not arbitrary... - (Another Scott) - (4) - March 3, 2004, 08:23:39 PM EST
                     re: "Proof" - (tablizer) - (3) - March 3, 2004, 08:35:32 PM EST
                         I'll give a brief rundown... - (Another Scott) - (1) - March 3, 2004, 11:52:26 PM EST
                             Ouch. I'll have to reread it several times. My math is rusty - (tablizer) - March 4, 2004, 12:19:39 AM EST
                         There is - (deSitter) - March 4, 2004, 09:18:22 AM EST
                 The exponent 2 is NOT arbitrary - (ben_tilly) - (4) - March 4, 2004, 11:32:22 AM EST
                     Physical events - (ChrisR) - (1) - March 4, 2004, 11:50:10 AM EST
                         Not in an obvious way to me - (ben_tilly) - March 4, 2004, 01:38:47 PM EST
                     Let's see if I got this right - (tablizer) - (1) - March 4, 2004, 01:36:40 PM EST
                         Yes - (ben_tilly) - March 4, 2004, 01:46:05 PM EST

That's not actually how law works.
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