[SciPy-User] How to estimate error in polynomial coefficients from scipy.polyfit?

[josef.pktd at gmail.com]

On Mon, Mar 29, 2010 at 1:31 PM, David Goldsmith
<d.l.goldsmith at gmail.com> wrote:
> On Mon, Mar 29, 2010 at 7:34 AM, Robert Kern <robert.kern at gmail.com> wrote:
>>
>> On Sun, Mar 28, 2010 at 22:26, David Goldsmith <d.l.goldsmith at gmail.com>
>> wrote:
>> > On Sun, Mar 28, 2010 at 11:36 AM, <alan at ajackson.org> wrote:
>> >>
>> >> Crack permeability goes like the third power of the opening (that is,
>> >> fluid flow through cracks - think gas or oil in a fractured rock).
>> >
>> > Power law or polynomial: from a regression stand point, there's quite a
>> > big
>> > difference.
>>
>> "Third power" == "x**3". He's not talking about a power law.
>
> Yes, I know that, but from a regression stand point, unless there's an
> offset (constant) term (in which case a two parameter polynomial fit is what
> you'll be doing) if your model is simply y = ax**3, aren't you better off
> doing the regression as if you were doing a power law, albeit w/ a fixed
> power (i.e., log transforming the data first, fixing the slope parameter at
> three, and then regressing to find the constant term, i.e., log(a))?
>
> In other words, I was soliciting examples of situations where a true
> polynomial (as opposed to a monomial) model was appropriate - I maintain
> that if your model is a monomial (integer power law model, w/ only one
> term), then, as far as regression is concerned, it is more appropriate to
> think of it as a power law model w/ a fixed parameter, not as a polynomial
> model.  From this perspective, the number of "naturally occurring"
> polynomial models is greatly reduced.

That looks to me like splitting hairs.

It depends on the statistical model for the regression error, e.g.

y = ax**3 + u   where u is normal, additive noise,
or
y = ax**3 *z   where z is log-normal, multiplicative noise
ln(y) = ln(a) + 3*ln(x) + u  with u = ln(z)

I would do it if I want to estimate or test if 3 is the correct power,
but not if 3 is known.

Josef

>
> DG
>
>>
>> --
>> Robert Kern
>>
>> "I have come to believe that the whole world is an enigma, a harmless
>> enigma that is made terrible by our own mad attempt to interpret it as
>> though it had an underlying truth."
>>  -- Umberto Eco
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