fittingtools 0.1.5

Tools to get prediction intervals when using scipy's least_squares minimizer

fittingtools

Some tools built upon scipy's least squares fitting.

Installation

pip install fittingtools

Basic usage: exponential model

import fittingtools as fls

Define the exponential model:

import numpy as np
from numpy.random import default_rng
from scipy.optimize import least_squares
rng = default_rng()

def model(t, b):
    return b[0] + b[1] * np.exp(b[2] * t)

Generate noisy data

def gen_data(t, a, b, c, noise=0., n_outliers=0, seed=None):
    rng = default_rng(seed)

    y = a + b * np.exp(t * c)

    error = noise * rng.standard_normal(t.size)
    outliers = rng.integers(0, t.size, n_outliers)
    error[outliers] *= 10

    return y + error

a = 0.5
b = 2.0
c = -1
t_min = 0
t_max = 10
n_points = 15

t_train = np.linspace(t_min, t_max, n_points)
y_train = gen_data(t_train, a, b, c, noise=0.1, n_outliers=3)

Define the residuals

def res(b, t, y):
    return model(t, b) - y

Optional:

Define the jacobian for the model

def jac(b, t, y):
    n, p = len(t), len(b)
    j = np.empty((n, p), dtype=np.float64)
    j[:, 0] = np.ones(n, dtype=np.float64)
    e = np.exp(b[2] * t)
    j[:, 1] = e
    j[:, 2] = b[1] * t * e
    return j

Fit the data using scipy.optimize.least_squares:

x0 = np.array([1.0, 1.0, 0.0])
res_lsq = least_squares(res, x0, args=(t_train, y_train), loss='cauchy', f_scale=0.1, jac=jac)

Get the confidence intervals for the optimzied parameters:

popt = res_lsq.x
parameters_ci = fls.confidence_interval(ls_res=res_lsq, level=0.95)
for i, p, lci, uci in zip(range(len(popt)), popt, parameters_ci[:,0], parameters_ci[:,1]):
  print(f'beta[{i}]: {p:>7.3f}, 95% CI: [{lci:>7.3f}, {uci:>7.3f}]')

beta[0]:   0.492, 95% CI: [  0.344,   0.640]
beta[1]:   2.096, 95% CI: [  1.829,   2.364]
beta[2]:  -1.039, 95% CI: [ -1.391,  -0.686]

Generate the prediction intervals for the fit (95% confidence level)

t_test = np.linspace(t_min, t_max, n_points * 10)
y_true = gen_data(t_test, a, b, c)

y_pred, delta = fls.prediction_intervals(
    model=model, x_pred=t_test, ls_res=res_lsq, jac=jac
)

Plot the data

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1, constrained_layout=True)
ax.plot(t_train, y_train, 'o')
ax.fill_between(t_test, y_pred - delta, y_pred + delta, color='C0', alpha=0.5, label='95% intervals')
ax.plot(t_test, y_true, 'k', linewidth=2, label='true')
ax.plot(t_test, y_pred, linewidth=2, label='cauchy')

ax.set_xlabel('x')
ax.set_ylabel('y')
ax.legend(loc='best', frameon=True)
plt.show()