Fitting a model to data in Python

curve_fit finds the parameters, and the covariance gives their error.

import numpy as np
from scipy import optimize


def learning(x, ceiling, rate):
    return ceiling * (1 - np.exp(-rate * x))


rng = np.random.default_rng(6)
days = np.arange(1, 31)
observed = learning(days, 120, 0.12) + rng.normal(0, 3, size=30)

params, covariance = optimize.curve_fit(learning, days, observed,
                                        p0=[100.0, 0.1])
errors = np.sqrt(np.diag(covariance))
print(params.round(3), errors.round(3))

residuals = observed - learning(days, *params)
print(round(np.std(residuals), 3))

How it works

The model's first argument is x, the rest are parameters.
p0 is the starting guess; a bad one can fail to converge.
The diagonal of the covariance gives one-sigma errors.

Keywords and builtins used here

as
def
learning
print
return
round

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Step 3 of 4 in Optimization, step 12 of 23 in Scientific computing with SciPy.

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