# [SciPy-user] scipy.optimize.leastsq and covariance matrix meaning

massimo sandal massimo.sandal@unibo...
Mon Nov 10 09:29:02 CST 2008

```Bruce Southey wrote:
> massimo sandal wrote:
>> massimo sandal wrote:
>>
>>> I'll try to sketch up a script reproducing the core of the problem
>>> with actual data.
>> Here it is. Can anyone give it a look to help me understand if and how
>> to make sense of the covariance matrix?
>>
>> m.
> There is some problem with your model with respect to your data. Looking
> at the plot of x and y, the relationship is linear with a correlation of
> 0.86. There is no hint of a non-linear relationship although a spline or
> similar local polynomial method could give a nicer fit. I do not know
> what you would expect to see from your function but you should also plot
> the expected model using typical values of your parameters.
>
> I would suggest you explore fitting polynomial models first (could only
> get a linear term for x in what you provided) and splines before doing
> nonlinear models.

The kind of things I am fitting is a single molecule force spectroscopy
force curve: see for example

and I am fitting peaks using the worm-like chain equation (actually, in
the script I use the inverse valus of the parameters) that you can find
here:
http://en.wikipedia.org/wiki/Worm-like_chain

with results looking like that:
http://www.jpk.com/titin-force-extension-profile.media.37bc90d1dbd105742098ed8317385a48v1.gif
http://biology.plosjournals.org/perlserv/?request=slideshow&type=figure&doi=10.1371/journal.pbio.0060006&id=93367

The model is non-linear because the physics underlying the data is
non-linear. I am not "choosing" the equation*, I am applying that
equation to find parameters from the curve.

What I have pasted is just the section of a much larger data plot. The
section can seem almost linear, but the non-linear fit on that section
fits perfectly also the remaining sections -as expected.
Fitting the whole peak or only the last portion of it does not change
significantly the fit or the output parameters.

The whole software I am working on is Hooke, available at
http://code.google.com/p/hooke , in case anyone is interested.

m.

*strictly speaking the are subtly different models to choose from indeed
(WLC, FJC, etc.), but WLC is the simpler and more widespread and is
enough for what I mean to do

--
Massimo Sandal , Ph.D.
University of Bologna
Department of Biochemistry "G.Moruzzi"

snail mail:
Via Irnerio 48, 40126 Bologna, Italy

email:
massimo.sandal@unibo.it

web:
http://www.biocfarm.unibo.it/samori/people/sandal.html

tel: +39-051-2094388
fax: +39-051-2094387
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