[SciPy-User] leastsq - When to scale covariance matrix by reduced chi square for confidence interval estimation

[josef.pktd at gmail.com]

On Fri, Jun 1, 2012 at 8:21 AM, Gregor Thalhammer
<gregor.thalhammer at gmail.com> wrote:
>
> Am 1.6.2012 um 11:21 schrieb Markus Baden:
>
> Hi Gregor,
>
> Thanks for the fast reply.
>
>> If you have knowledge about the statistical errors of your data, then
>> skipping step 2 and 3 is the recommended, and you can use the chi square to
>> assess the validity of the fit and your assumptions about the errors. On the
>> other hand, if you have insufficient knowledge about the errors, you can use
>> the reduced chi square as an estimate for the variance of your data (at
>> least under the assumption that the error is the same for all data points).

when you use the weighting and the estimated sigma, you still leave
the relative weighting unchanged. So we don't really assume that the
error is the same for all data points.

>> This is the idea behind steps 2 and 3.
>
>
> I just want to get that straight: So basically in the case where I either
> don't have errors, or I don't trust them, multiplying the covariance by the
> reduced chi square amounts to "normalizing" the covariance such that the fit
> would have a chi square of one (?). Maybe your point could go into the docs
> for curve_fit... or there could be a comment about standard procedure a bit
> like in origin
> (http://www.originlab.com/www/support/resultstech.aspx?ID=452&language=English&Version=All)
>
>
> Yes, I think you correctly got the idea.

My interpretation is that you have to trust your error estimate more
than the error estimate from the reduced chi square. In the two
extreme scaled_x_errors cases, your belief about the errors is far
from the model fit.

So either the residual sum of squares is a bad measurement of the
error (outliers, ....) or what you think the measurement errors should
be doesn't agree with the data.

In the example the measurement error is translated from x_errors to
y_errors by a linear approximation (at least according to the
comments). If this approximation is not good, then it introduces
another type of error, that would create a discrepancy between your
measurement errors and the reduced chisquare.


>
>
>>
>>
>> > Now in the particular problem I am working at, we have a couple of fits
>> > like [5] and some of them have a slightly worse reduced chi square of say
>> > about 1.4 or 0.7. At this point the two methods start to deviate and I am
>> > wondering which would be the correct way of quoting the errors estimated
>> > from the fit. Even a basic reference to some text book that explains the
>> > method used in scipy would be very helpful.
>>
>> I didn't look at your data, but I guess that these values of the reduced
>> chi square are still in range such that they are not a significant deviation
>> from the expected value of one. The chi-squared distribution is rather
>> broad. So I would omit steps 2 and 3. Only if you have good reasons not to
>> trust your assumptions about the errors of the data, then apply steps 2 and
>> 3.
>
>
> We looked at which part of the CDF these values are and they are still ok.
> And our errors are all inferred from measurements, so we trust them quite a
> bit. We use the fitting described to obtain a particular property of an ion
> via spectroscopy... that's also why we want to get our errors on that
> property correct :)
>
>
> As e.g. described nicely in http://mljohnson.pharm.virginia.edu/pdfs/174.pdf
> you have to be careful about the parameter error estimates obtained by this
> procedure. In general the obtained results are too optimistic.


Thanks, helpful example and discussion. I've never seen an estimated
curve that fits so well.

extra: With the measurement error in x, the least squares estimator is
usually biased. But I only know part of the theory for the linear case
and don't have much idea about how this will affect the estimates in
the non-linear case.

Josef

>
> Gregor
>
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