Nonlinear curve fitting with confidence intervals

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Our goal is to fit this equation to data \(y = c1 exp(-x) + c2*x\) and compute the confidence intervals on the parameters.

This is actually could be a linear regression problem, but it is convenient to illustrate the use the nonlinear fitting routine because it makes it easy to get confidence intervals for comparison. The basic idea is to use the covariance matrix returned from the nonlinear fitting routine to estimate the student-t corrected confidence interval.

# Nonlinear curve fit with confidence interval
import numpy as np
from scipy.optimize import curve_fit
from scipy.stats.distributions import  t

x = np.array([ 0.1,  0.2,  0.3,  0.4,  0.5,  0.6,  0.7,  0.8,  0.9,  1. ])
y = np.array([ 4.70192769,  4.46826356,  4.57021389,  4.29240134,  3.88155125,
               3.78382253,  3.65454727,  3.86379487,  4.16428541,  4.06079909])

# this is the function we want to fit to our data
def func(x,c0, c1):
    return c0 * np.exp(-x) + c1*x

pars, pcov = curve_fit(func, x, y, p0=[4.96, 2.11])

alpha = 0.05 # 95% confidence interval

n = len(y)    # number of data points
p = len(pars) # number of parameters

dof = max(0, n-p) # number of degrees of freedom

tval = t.ppf(1.0 - alpha / 2.0, dof) # student-t value for the dof and confidence level

for i, p,var in zip(range(n), pars, np.diag(pcov)):
    sigma = var**0.5
    print 'c{0}: {1} [{2}  {3}]'.format(i, p,
                                  p - sigma*tval,
                                  p + sigma*tval)

import matplotlib.pyplot as plt
plt.plot(x,y,'bo ')
xfit = np.linspace(0,1)
yfit = func(xfit, pars[0], pars[1])
plt.plot(xfit,yfit,'b-')
plt.legend(['data','fit'],loc='best')
plt.savefig('images/nonlin-fit-ci.png')
c0: 4.96713966439 [4.62674476567  5.30753456311]
c1: 2.10995112628 [1.76711622427  2.45278602828]

Copyright (C) 2013 by John Kitchin. See the License for information about copying.

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