How Does Fit Work?

In this section we describe in more detail how Fit gets from a (named) model and a bunch of data to a fit. We will use NumericalLeastSquares as example.

Consider the following example:

from symfit import parameters, variables, Fit

a, b = parameters('a, b')
x, y = variables('x, y')
model = {y: a * x + b}

fit = Fit(model, x=x_data, y=y_data, sigma_y=sigma_data)
fit_result = fit.execute()

The first thing symfit does is build \(\chi^2\) for your model:

chi_squared = sum((y - f)**2/sigmas[y]**2 for y, f in model.items())

In this line sigmas is a dict which contains all vars that where given a value, or returns 1 otherwise.

This \(\chi^2\) is then transformed into a python function which can then be used to do the numerical calculations:

vars, params = seperate_symbols(chi_squared)
py_chi_squared = lambdify(vars + params, chi_squared)

We are now almost there. Just two steps left. The first is to wrap all the data into the py_chi_squared function using partial() into the function to be optimized:

from functools import partial

error = partial(py_chi_squared, **data_per_var)

where data_per_var is a dict containing variable names: value pairs.

Now all that is left is to minimize error as a function of the parameters. NumericalLeastSquares does this by calling leastsqbound() and have it find the best fit parameters:

best_fit_parameters, covariance_matrix = leastsqbound(
    error,
    self.guesses,
    self.eval_jacobian,
    self.bounds,
)

That’s it! Finally there are some steps to generate a FitResults object, but these are not important for our current discussion.