[SciPy-User] Least Square fit and goodness of fit

[josef.pktd at gmail.com]

On Mon, May 17, 2010 at 12:18 AM, Benedikt Riedel <briedel at wisc.edu> wrote:
>
>
> On Sun, May 16, 2010 at 22:33, <josef.pktd at gmail.com> wrote:
>>
>> On Sun, May 16, 2010 at 9:05 PM, Benedikt Riedel <briedel at wisc.edu> wrote:
>> > What I still do not understand is the fact that curve_fit gives me a
>> > different output then leastsq, even though curve_fit calls leastsq.
>> >
>> > I tried to get the chi-squared because we want to plot contours of
>> > chi-square from the minimum to the maximum. I used following code:
>> >
>> > fitfunc = lambda p,x: p[0]+ p[1]*exp(-x)
>> > errfunc = lambda p, x, y: (y-fitfunc(p,x))
>> > pinit = [20,20.]
>> >
>> > def func(x, a, b):
>> >      return a*exp(-x) + b
>> >
>> > pfinal, covar = curve_fit(func,tau, R4ctsdataselect, p0=pinit,
>> > sigma=R4errctsdataselect)
>>
>> this uses weighted least squares
>> sigma : None or N-length sequence
>>    If not None, it represents the standard-deviation of ydata. This
>> vector, if given, will be used as weights in the least-squares problem
>>
>> In your initial example with leastsq you don't have any weighting,
>> it's just ordinary least squares
>>
>> maybe that's the difference.
>>
>>
>
> Yeah I guess that will be it.
>
>>
>> > print pfinal
>> > print covar
>> > dof=size(tau)-size(pinit)
>> > print dof
>> > chi2=(sum(pow(R4ctsdataselect-fitfunc(pinit, tau), 2)/fitfunc(pinit,
>> > tau)))/dof
>> > print chi2
>> >
>> > I am not 100% sure I am doing the degrees of freedom calculation right.
>> > I
>> > got the chi-square formula from the Pearson chi-squared test.
>>
>> I don't recognize your formula for chi2, and I don't see the
>> connection to Pearson chi-squared test .
>>
>> Do you have a reference?
>>
>
> I based my use of the Pearson test from what I read in an Econometrics book,
> but wiki has the a pretty good description. I basically based it off the
> example there. Where the expected would be what comes out of the fit and
> what you is the "R4ctsdataselect" for those specific values.
>
> http://en.wikipedia.org/wiki/Pearson%27s_chi-square_test

I looked at that, but it's a completely different case, the values in
the formulas are frequencies

    Oi = an observed frequency;
    Ei = an expected (theoretical) frequency, asserted by the null hypothesis;

not points on a regression curve

Josef

>
>
>>
>> Josef
>>
>
> Thanks again
>
> Ben
>
>
>>
>> >
>> > Thank you very much for the help so far.
>> >
>> > Cheers,
>> >
>> > Ben
>> >
>> > On Sun, May 16, 2010 at 05:50, <josef.pktd at gmail.com> wrote:
>> >>
>> >> On Sun, May 16, 2010 at 12:12 AM, Benedikt Riedel <briedel at wisc.edu>
>> >> wrote:
>> >> >
>> >> >
>> >> > On Fri, May 14, 2010 at 14:51, <josef.pktd at gmail.com> wrote:
>> >> >>
>> >> >> On Fri, May 14, 2010 at 3:01 PM, Benedikt Riedel <briedel at wisc.edu>
>> >> >> wrote:
>> >> >> > Hey,
>> >> >> >
>> >> >> > I am fairly new Scipy and am trying to do a least square fit to a
>> >> >> > set
>> >> >> > of
>> >> >> > data. Currently, I am using following code:
>> >> >> >
>> >> >> > fitfunc = lambda p,x: p[0]+ p[1]*exp(-x)
>> >> >> > errfunc = lambda p, x, y: (y-fitfunc(p,x))
>> >> >> > pinit = [20,20.]
>> >> >> > out = leastsq(errfunc, pinit, args=(tau,R4ctsdataselect),
>> >> >> > full_output=1)
>> >> >> >
>> >> >> > I am now trying to get the goodness of fit out of this data. I am
>> >> >> > sort
>> >> >> > of
>> >> >> > running into a brick wall because I found a lot of conflicting
>> >> >> > ways
>> >> >> > of
>> >> >> > how
>> >> >> > to calculate it.
>> >> >>
>> >> >> For regression the usual is
>> >> >> http://en.wikipedia.org/wiki/Coefficient_of_determination
>> >> >> coefficient of determination is
>> >> >>
>> >> >>    R^2 = 1 - {SS_{err} / SS_{tot}}
>> >> >>
>> >> >> Note your fitfunc is linear in parameters and can be better
>> >> >> estimated
>> >> >> by linear least squares, OLS.
>> >> >> linear regression is handled in statsmodels and you can get lot's of
>> >> >> statistics without worrying about the formulas.
>> >> >> If you only have one slope parameter, then scipy.stats.linregress
>> >> >> also
>> >> >> works
>> >> >>
>> >> >
>> >> > Thanks for the information. I am still note quite sure if this is
>> >> > what
>> >> > my
>> >> > boss wants because there should not be an average y value.
>> >>
>> >> The definition of Rsquared is pretty uncontroversial with the y.mean()
>> >> correction, if there is a constant in the regression (although I know
>> >> mainly the linear case for this).
>> >>
>> >> If there is no constant in the regression, the definition or Rsquared
>> >> is not clear/unambiguous, but usually used without mean correction of
>> >> y.
>> >>
>> >> Josef
>> >>
>> >> >
>> >> >>
>> >> >> scipy.optimize.curve_fit (scipy 0.8) can also give you the
>> >> >> covariance
>> >> >> of the parameter estimates.
>> >> >> http://docs.scipy.org/scipy/docs/scipy.optimize.minpack.curve_fit
>> >> >
>> >> > I have been trying this out, but the fit just looks horrid compared
>> >> > to
>> >> > using
>> >> > leastsq method even though they call the same function according to
>> >> > the
>> >> > documentation.
>> >> >
>> >> >>
>> >> >> > I am aware of the chisquare function in stats function, but the
>> >> >> > documentation seems a little confusing to me. Any help would be
>> >> >> > greatly
>> >> >> > appreciates.
>> >> >>
>> >> >> chisquare and others like kolmogorov-smirnov are more for testing
>> >> >> the
>> >> >> goodness-of-fit of entire distributions, not for how well a curve or
>> >> >> line fits the data.
>> >> >>
>> >> >
>> >> > That is what I thought, which brought up my confusion when I asked
>> >> > other
>> >> > people and they told me to use that
>> >> >
>> >> >>
>> >> >> Josef
>> >> >>
>> >> >> >
>> >> >> > Thanks very much in advance.
>> >> >> >
>> >> >> > Cheers,
>> >> >> >
>> >> >> > Ben
>> >> >> >
>> >> >> >
>> >> >> >
>> >> >> > _______________________________________________
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>> >> >> >
>> >> >> >
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>> >> >
>> >> >
>> >> >
>> >> > --
>> >> > Benedikt Riedel
>> >> > Graduate Student University of Wisconsin-Madison
>> >> > Department of Physics
>> >> > Office: 2304 Chamberlin Hall
>> >> > Lab: 6247 Chamberlin Hall
>> >> > Tel:  (608) 301-5736
>> >> > Cell: (213) 519-1771
>> >> > Lab: (608) 262-5916
>> >> >
>> >> > _______________________________________________
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>> >> >
>> >> >
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>> >
>> >
>> > --
>> > Benedikt Riedel
>> > Graduate Student University of Wisconsin-Madison
>> > Department of Physics
>> > Office: 2304 Chamberlin Hall
>> > Lab: 6247 Chamberlin Hall
>> > Tel:  (608) 301-5736
>> > Cell: (213) 519-1771
>> > Lab: (608) 262-5916
>> >
>> > _______________________________________________
>> > SciPy-User mailing list
>> > SciPy-User at scipy.org
>> > http://mail.scipy.org/mailman/listinfo/scipy-user
>> >
>> >
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>
>
>
> --
> Benedikt Riedel
> Graduate Student University of Wisconsin-Madison
> Department of Physics
> Office: 2304 Chamberlin Hall
> Lab: 6247 Chamberlin Hall
> Tel:  (608) 301-5736
> Cell: (213) 519-1771
> Lab: (608) 262-5916
>
> _______________________________________________
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