Binned fit

Imagine we have the following data distribution over \(x\):

import numpy as np

sample_size = 1_000
rng = np.random.default_rng(seed=0)
x_distribution = rng.power(3, size=sample_size)

Alternatively, we could have sampled the distribution with numpy.histogram().

We sample the distribution into a histogram with matplotlib.pyplot.hist(). The bin edges will serve as \(x\)-values and the bin heights will be the observed \(y\)-values.

%config InlineBackend.figure_formats = ['svg']

import matplotlib.pyplot as plt

fig, ax = plt.subplots(figsize=(8, 5))
ax.set_xlabel("$x$")
n_bins = 50
bin_values, bin_edges, _ = ax.hist(x_distribution, bins=n_bins)
plt.show()

x_values = (bin_edges[1:] + bin_edges[:-1]) / 2
y_values = bin_values

The data distribution has been generated by numpy.random.power() and can therefore be described by the following expression:

import sympy as sp

n, a, x = sp.symbols("n a x")
expression = n * x ** (a - 1)
expression

\[\displaystyle n x^{a - 1}\]

We would like to find which values for \(a, n\) describe this distribution best. For fast computations, we first formulate this expression as a ParametrizedFunction with numpy as computational backend (numpy is fast enough for the small data samples with which we are working here). We also provide a first guess for the values of \(a\) and \(n\).

from tensorwaves.function.sympy import create_parametrized_function

function = create_parametrized_function(
    expression,
    parameters={a: 2, n: 15},
    backend="numpy",
)
initial_parameters = function.parameters

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function.update_parameters(initial_parameters)
fig, ax = plt.subplots(figsize=(8, 5))
ax.set_xlabel("$x$")
ax.hist(x_distribution, bins=n_bins, label="Data distribution")
ax.plot(x_values, function({"x": x_values}), label="Initial fit model", c="red")
ax.legend()
plt.show()

This binned distribution lends itself well to be optimized with a ChiSquared estimator.

from tensorwaves.estimator import ChiSquared
from tensorwaves.optimizer import Minuit2

estimator = ChiSquared(
    function,
    domain={"x": x_values},
    observed_values=y_values,
    backend="numpy",
)
optimizer = Minuit2()
fit_result = optimizer.optimize(estimator, initial_parameters)
fit_result

FitResult(
 minimum_valid=True,
 execution_time=0.0008292198181152344,
 function_calls=62,
 estimator_value=799.3094411131799,
 parameter_values={ 'a': 3.158993714718434, 'n': 58.1379343010243,
 },
 parameter_errors={ 'a': 0.020006328109296395, 'n': 0.311408749372934,
 },
)

The optimized value of \(a\) indeed lies close to the value of \(a=3\) with which we generated the distribution.

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function.update_parameters(fit_result.parameter_values)
fig, ax = plt.subplots(figsize=(8, 5))
ax.set_xlabel("$x$")
ax.hist(x_distribution, bins=n_bins, label="Data distribution")
ax.plot(x_values, function({"x": x_values}), label="Optimized model", c="red")
ax.legend()
plt.show()

Also note that the integral over the expression with these optimized parameter values is close to the surface of the bins:

bin_width = bin_edges[1] - bin_edges[0]
surface = np.sum(bin_values) * bin_width
surface

np.float64(18.424306915235626)

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from IPython.display import Math

substitutions = {
    a: fit_result.parameter_values["a"],
    n: fit_result.parameter_values["n"],
}
integral = sp.Integral(expression, (x, 0, 1))
evaluated_integral = integral.subs(substitutions).doit()
latex = sp.multiline_latex(integral, evaluated_integral, environment="eqnarray")
Math(latex)

\[\displaystyle \begin{eqnarray} \int\limits_{0}^{1} n x^{a - 1}\, dx & = & 18.4039411126863 \end{eqnarray}\]