Polyfit with scaling

[Ben Abbott]

On Feb 2, 2008, at 3:00 PM, Dmitri A. Sergatskov wrote:

> Though I was partially responsible for all this discussions,
> I am now rethinking this approach.
> polyfit should be as stable numerically as we can make it,
> but I do not think it should try to make a "better"
> fit than the one it was asked for.
>
> Polyfit does fit to a*x^n+b*x^(n-1)... polynomial.
> If people want it to fit to a*(x-x0)^n+b*(x-x0)^(n-1)
> they should pre-condition data accordingly prior to
> passing it to polyfit.
>
> So to make long story short -- I think the original
> Thomas' suggestion (just to normalize the data to max(x))
> was the good one.
>
> On related note -- it appears that wpolifit does a
> better job than polyfit on un-conditioned data as well.
> It appears that the difference is in polyfit
> uses straight "\",  while the wpolyfit does QR decomposition.
> Perhaps it is worth to port this to polyfit as well.
> Frankly I do not see why would not we copy wpolyfit to
> octave as polyfit (with an additional change to scale x
> unconditionally).

Regarding "scale x unconditionally", do you refer to the scaling used  
by wpolyfit;

	(x - mean (x)) / std (x)

or to Thomas' suggestion to just scale the magnitude?

	x / max (x)

If you refer to Thomas' suggestion, the maximum value will result in  
as much trouble/benefit as the minimum value.

Perhaps a better solution would be (a) the geometric mean of the  
magnitudes, (b) the median of the magnitudes, (c) the mean of the  
magnitudes, (d) consider several normalization options and select the  
most numerically stable one.

In any event, what should be done about s.R and s.X? Are they to  
represent the scaled dependent variable?

Ben