scipy.optimize.curve_fit uncertainty calculation

[willflanagan]

Our team is using scipy.optimize.curve_fit to return least-squares fitting and I am unable to return the "textbook uncertainties" that I expect. In fact, the uncertainty on the slope seems to be incorrect by exactly a factor of 1.5.

My toy example is three data points with only uncertainty on the y values:
(0,0+-1)
(0,1+-1)
(1,1+-2)

The best fit returned is 0+0.2x, which is correct based on my by-hand and excel calculations (attached). The uncertainty on the y-intercept is 1.0, which is also what I expect. The uncertainty on the slope is 0.8944271909999176 (python) rather than 1.341640786 (excel/hand calculations) which is a factor of exactly 1.5 to 10 digits.

I'm referencing the formulas in problem 8.19 in John Taylor's 'An Introduction to Error Analysis'.

This is a toy problem to make sure we trust the package. We will be using more points and higher-order polynomials.

Any suggestions? Thanks in advance for your time and help.

[willflanagan]

Following up on my own thread - we have found that setting the method to Trust Region Reflective algorithm ('trf') or dogleg algorithm ('dogbox') fixes this issue.

curve_fit(linear_function, x_data, y_data, sigma=y_error, absolute_sigma=True, method='trf')

The documentation does warn against using the Levenberg-Marquardt algorithm ('lm') with a sparse number of observations:

"The method ‘lm’ won’t work when the number of observations is less than the number of variables, use ‘trf’ or ‘dogbox’ in this case."

In this toy example we are fitting two parameters with three points, so I don't believe we are violating this rule above. Regardless, it hints that these other two algorithms are better for sparse data.

https://docs.scipy.org/doc/scipy/referen...e_fit.html
https://docs.scipy.org/doc/scipy/referen...st_squares