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

[Bruce Southey]

massimo sandal wrote:
> 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
> http://www.jpk.com/unfolding-of-individual-titin-i27-octamer-i91-8.media.f229c5303ada22eb0a4ebe759457750av2.gif 
>
>
> 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
>
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>   

What are the actual parameters that you think you are trying to estimate 
here?
What is y and x relative to the equation? In particular is y=F*P or just 
F or P?

Your parameter therm is a constant so I would first compute Y/therm 
before doing anything else or just ignore it.
Also, you probably need to rescale both x and y because these are either 
very small or very large. Perhaps even standardize x to mean 0 and 
variance 1. However, you do need to be very careful here. Getting 
nonlinear models to converge to 'correct' parameters is often an art 
than a science.

However, I still don't think you have the data to estimate this function 
as there are no clear 'high' and 'low' points. I also don't think this 
model will describe the patterns provided by the links (not even clear 
how your data relates to these images).

Bruce