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

[Robert Kern]

On Mon, Nov 10, 2008 at 05:13, massimo sandal <massimo.sandal at unibo.it> 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?

The covariance matrix does need some scaling before it can be
interpreted statistically. Basically, if you are doing nonlinear least
squares as a statistical procedure, rather than a purely numerical
one, the residuals need to be scaled so that they are in units of
standard deviations of the measurement error for each individual
measurement. If you don't know what that is, then you can estimate it
from the fitted residuals. The parameter estimate is unchanged, but
you will need to rescale the covariance matrix of the estimate by
multiplying it by the residual variance.

scipy.odr does most of this for you. Attached is a version of your
code using scipy.odr. Here is the text output:

Fitted parameters: [  4.90666526e+06   4.78090340e+09]
Covariance: [[  1.72438988e+31  -1.64258997e+35]
 [ -1.64258997e+35   1.57791262e+39]]
Residual variance: 2.83606592894e-22
Scaled error bars: [  6.99319913e+04   6.68959208e+08]
Scaled covariance: [[  4.89048340e+09  -4.65849344e+13]
 [ -4.65849344e+13   4.47506422e+17]]

-- 
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
  -- Umberto Eco
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