# [SciPy-user] negative values in diagonal of covariance matrix

Bruce Southey bsouthey@gmail....
Thu Dec 11 08:53:21 CST 2008

```Pavlo Shchelokovskyy wrote:
> Hi all,
>
> I'm a moderately new user of scipy, trying to make some curve-fitting
> with it. I wanted to use cov matrix output of leastsq to estimate
> errors of fitted parameters, but stumbled upon strange discrepancy
> (for one particular dataset):
>
> On Linux (Fedora 8), using Python 2.5.1, numpy 1.2.0 and scipy 0.6.0
>
>
>>>> from scipy import *
>>>> from scipy import optimize
>>>> y = asarray([217, 182, 162, 170, 255])
>>>> x = linspace(0, y.size - 1, y.size)
>>>> gauss = lambda p, x: p[0] + p[1] * exp(-(x-p[2])**2/(2*p[3]**2))
>>>> errfunc = lambda p, x, y: y - gauss(p, x)
>>>> pinit = (max(y), -y.ptp(), argmin(y), y.size/4)
>>>> fit,cov,info,mesg,success = optimize.leastsq(errfunc, pinit, args=(x,y), full_output = 1)
>>>> fit
>>>>
> array([  3.01602865e+05,  -3.01444487e+05,   1.83283239e+00, 8.87273494e+01])
>
>>>> cov
>>>>
> array([[ -2.27903048e+16,   2.27903047e+16,  -4.72378743e+05, -3.35454486e+12],
>        [  2.27903047e+16,  -2.27903046e+16,   4.72378733e+05, 3.35454485e+12],
>        [ -4.72378950e+05,   4.72378947e+05,   6.38886491e-05, -6.95317302e+01],
>        [ -3.35454486e+12,   3.35454485e+12,  -6.95316881e+01, -4.93761332e+08]])
>
> I know that the fit is very bad (I can reject it afterwards in my
> procedure), but what bothers me the most are the negative numbers on
> the diagonal of the covariance matrix - as far as I understand, there
> shouldn't be any... especially when the same script run on Windows XP,
> using Python 2.5.2, numpy 1.2.0 and scipy 0.6.0 gives no such values
>
>>>> fit
>>>>
>
> array([  3.58426106e+05,  -3.58267727e+05,   1.83283417e+00, 9.67300286e+01])
>
>
>>>> cov
>>>>
>
> array([[  6.40320190e+14,  -6.40320188e+14,   4.24641077e+04, 8.64500286e+10],
>
>        [ -6.40320188e+14,   6.40320186e+14,  -4.24641083e+04, -8.64500284e+10],
>
>        [  4.24641250e+04,  -4.24641094e+04,   7.64931901e-05, 5.73152924e+00],
>
>        [  8.64500286e+10,  -8.64500284e+10,   5.73152989e+00, 1.16716728e+07]])
>
> Is it my error or misunderstanding somewhere, or is it really a bug in
> Linux implementation?
>
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>
Actually you do not have sufficient data points to estimate this model
because you have four parameters and only five observations resulting in
no degrees of freedom for the error (if you allow correction for the
mean). I do not know scipy.optimize but I am very doubtful that the
model even converges correctly (a solution does not mean convergence).
If the model has not converged then everything else is usually invalid
(especially when those parameters depend on the solution).

Bruce
```