# On Standard Deviations¶

This essay is meant as a reflection on the implementation of Standard Deviations and/or measurement errors in symfit. Although reading this essay in it’s entirely will only be interesting to a select few, I urge anyone who uses symfit to read the following summarizing bullet points, as symfit is NOT backward-compatible with scipy.

• standard deviations are assumed to be measurement errors by default, not relative weights. This is the opposite of the scipy definition. Set absolute_sigma=False when calling Fit to get the scipy behavior.

## Analytical Example¶

The implementation of standard deviations should be in agreement with cases to which the analytical solution is known. symfit was build such that this is true. Let’s follow the example outlined by [taldcroft]. We’ll be sampling from a normal distribution with $$\mu = 0.0$$ and varying $$\sigma$$. It can be shown that given a sample from such a distribution:

$\mu = 0.0$
$\sigma_{\mu} = \frac{\sigma}{\sqrt{N}}$

where N is the size of the sample. We see that the error in the sample mean scales with the $$\sigma$$ of the distribution.

In order to reproduce this with symfit, we recognize that determining the avarage of a set of numbers is the same as fitting to a constant. Therefore we will fit to samples generated from distributions with $$\sigma = 1.0$$ and $$\sigma = 10.0$$ and check if this matches the analytical values. Let’s set $$N = 10000$$.

N = 10000
sigma = 10.0
np.random.seed(10)
yn = np.random.normal(size=N, scale=sigma)

a = Parameter('a')
y = Variable('y')
model = {y: a}

fit = Fit(model, y=yn, sigma_y=sigma)
fit_result = fit.execute()

fit_no_sigma = Fit(model, y=yn)
fit_result_no_sigma = fit_no_sigma.execute()


This gives the following results:

• a = 5.102056e-02 +- 1.000000e-01 when sigma_y is provided. This matches the analytical prediction.
• a = 5.102056e-02 +- 9.897135e-02 without sigma_y provided. This is incorrect.

If we run the above code example with sigma = 1.0, we get the following results:

• a = 5.102056e-03 +- 9.897135e-03 when sigma_y is provided. This matches the analytical prediction.
• a = 5.102056e-03 +- 9.897135e-03 without sigma_y provided. This is also correct, since providing no weights is the same as setting the weights to 1.

To conclude, if symfit is provided with the standard deviations, it will give the expected result by default. As shown in [taldcroft] and symfit‘s tests.py, scipy.optimize.curve_fit has to be provided with the absolute_sigma=True setting to do the same.

Important

We see that even if the weight provided to every data point is the same, the scale of the weight still effects the result. scipy was build such that the opposite is true: if all datapoints have the same weight, the error in the parameters does not depend on the scale of the weight.

This difference is due to the fact that symfit is build for area’s of science where one is dealing with measurement errors. And with measurement errors, the size of the errors obviously matters for the certainty of the fit parameters, even if the errors are the same for every measurement.

If you want the scipy behavior, initiate Fit with absolute_sigma=False.

# Comparison to Mathematica¶

In Mathematica, the default setting is also to use relative weights, which we just argued is not correct when dealing with measurement errors. In [mathematica] this problem is discussed very nicely, and it is shown how to solve this in Mathematica.

Since symfit is a fitting tool for the practical man, measurement errors are assumed by default.