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

Fri Jun 1 04:21:53 CDT 2012

```Hi Gregor,

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). 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
)

>
> > 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 :)

In any case, thanks again,

Markus
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