Confidence interval for exponential curve fit

Question

I'm trying to obtain a confidence interval on an exponential fit to some x,y data (available here). Here's the MWE I have to find the best exponential fit to the data:

from pylab import *
from scipy.optimize import curve_fit

# Read data.
x, y = np.loadtxt('exponential_data.dat', unpack=True)

def func(x, a, b, c):
    '''Exponential 3-param function.'''
    return a * np.exp(b * x) + c

# Find best fit.
popt, pcov = curve_fit(func, x, y)
print popt

# Plot data and best fit curve.
scatter(x, y)
x = linspace(11, 23, 100)
plot(x, func(x, *popt), c='r')
show()

which produces:

How can I obtain the 95% (or some other value) confidence interval on this fit preferably using either pure python, numpy or scipy (which are the packages I already have installed)?

Answer 1

You can use the uncertainties module to do the uncertainty calculations. uncertainties keeps track of uncertainties and correlation. You can create correlated uncertainties.ufloat directly from the output of curve_fit.

To be able to do those calculation on non-builtin operations such as exp you need to use the functions from uncertainties.unumpy.

You should also avoid your from pylab import * import. This even overwrites python built-ins such as sum.

A complete example:

import numpy as np
from scipy.optimize import curve_fit
import uncertainties as unc
import matplotlib.pyplot as plt
import uncertainties.unumpy as unp


def func(x, a, b, c):
    '''Exponential 3-param function.'''
    return a * np.exp(b * x) + c

x, y = np.genfromtxt('data.txt', unpack=True)

popt, pcov = curve_fit(func, x, y)

a, b, c = unc.correlated_values(popt, pcov)

# Plot data and best fit curve.
plt.scatter(x, y, s=3, linewidth=0, alpha=0.3)

px = np.linspace(11, 23, 100)
# use unumpy.exp
py = a * unp.exp(b * px) + c

nom = unp.nominal_values(py)
std = unp.std_devs(py)

# plot the nominal value
plt.plot(px, nom, c='r')

# And the 2sigma uncertaintie lines
plt.plot(px, nom - 2 * std, c='c')
plt.plot(px, nom + 2 * std, c='c')
plt.savefig('fit.png', dpi=300)

And the result: