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I have some data which I have fitted a normal distribution to using the scipy.stats.normal objects fit function like so: 

import numpy as np                                                                                                                                                                                                                       
import matplotlib.pyplot as plt                                                                                                                                                                                                          
from scipy.stats import norm                                                                                                                                                                                                             
import matplotlib.mlab as mlab                                                                                                                                                                                                           

x = np.random.normal(size=50000)                                                                                                                                                                                                         

fig, ax = plt.subplots()                                                                                                                                                                                                                 

nbins = 75                                                                                                                                                                                                                               
mu, sigma = norm.fit(x)                                                                                                                                                                                                                  
n, bins, patches = ax.hist(x,nbins,normed=1,facecolor = 'grey', alpha = 0.5, label='before');                                                                                                                                            
y0 = mlab.normpdf(bins, mu, sigma) # Line of best fit                                                                                                                                                                                    
ax.plot(bins,y0,'k--',linewidth = 2, label='fit before')                                                                                                                                                                                 
ax.set_title('$\mu$={}, $\sigma$={}'.format(mu, sigma))                                                                                                                                                                                  

plt.show()

I would now like to extract the uncertainty/error in the fitted mu and sigma values. How can I go about this?

SomeRandomPhysicist
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2 Answers2

9

You can use scipy.optimize.curve_fit: This method does not only return the estimated optimal values of the parameters, but also the corresponding covariance matrix:

popt : array

Optimal values for the parameters so that the sum of the squared residuals of f(xdata, *popt) - ydata is minimized

pcov : 2d array

The estimated covariance of popt. The diagonals provide the variance of the parameter estimate. To compute one standard deviation errors on the parameters use perr = np.sqrt(np.diag(pcov)).

How the sigma parameter affects the estimated covariance depends on absolute_sigma argument, as described above.

If the Jacobian matrix at the solution doesn’t have a full rank, then ‘lm’ method returns a matrix filled with np.inf, on the other hand ‘trf’ and ‘dogbox’ methods use Moore-Penrose pseudoinverse to compute the covariance matrix.

You can calculate the standard deviation errors of the parameters from the square roots of the diagonal elements of the covariance matrix as follows:

import numpy as np 
import matplotlib.pyplot as plt
from scipy.stats import norm 
from scipy.optimize import curve_fit

x = np.random.normal(size=50000)
fig, ax = plt.subplots() 
nbins = 75
n, bins, patches = ax.hist(x,nbins, density=True, facecolor = 'grey', alpha = 0.5, label='before'); 

centers = (0.5*(bins[1:]+bins[:-1]))
pars, cov = curve_fit(lambda x, mu, sig : norm.pdf(x, loc=mu, scale=sig), centers, n, p0=[0,1])

ax.plot(centers, norm.pdf(centers,*pars), 'k--',linewidth = 2, label='fit before') 
ax.set_title('$\mu={:.4f}\pm{:.4f}$, $\sigma={:.4f}\pm{:.4f}$'.format(pars[0],np.sqrt(cov[0,0]), pars[1], np.sqrt(cov[1,1 ])))

plt.show()

This results in the following plot:

enter image description here

Tonechas
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jdamp
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    please note that the uncertainties reported here are due entirely to the sampling of the data into 75 bins. There is no source of noise or non-normal distribution besides the otherwise arbitrary binning. – M Newville Mar 07 '18 at 01:30
  • @MNewville So, does the `norm.fit` not suffer from these uncertainties? Other than reporting on these uncertainties, how are `curve_fit` and `norm.fit` different? – Always Learning Forever Oct 04 '18 at 06:26
  • @AlwaysLearningForever . I think my earlier comment is not correct -- there is a natural distribution, and with a large enough number of bins, the uncertainty on centroid and width will stabilize to non-zero values. For what `norm.fit` does: I'm not 100% certain, but I believe that `scipy.stats.norm.fit()` uses Nelder-Mead to do the fit, while `curve_fit` uses Levenberg-Marquardt. I do not know if `scipy.stats.norm.fit()` even tries to estimates uncertainties, but I suspect not. – M Newville Oct 05 '18 at 04:22
3

See also lmfit (https://github.com/lmfit/lmfit-py) which gives an easier interface and reports uncertainties in fitted variables. To fit data to a normal distribution, see http://lmfit.github.io/lmfit-py/builtin_models.html#example-1-fit-peak-data-to-gaussian-lorentzian-and-voigt-profiles

and use something like

from lmfit.models import GaussianModel

model = GaussianModel()

# create parameters with initial guesses:
params = model.make_params(center=9, amplitude=40, sigma=1)  

result = model.fit(ydata, params, x=xdata)
print(result.fit_report())

The report will include the 1-sigma errors like

[[Variables]]
    sigma:       1.23218358 +/- 0.007374 (0.60%) (init= 1.0)
    center:      9.24277047 +/- 0.007374 (0.08%) (init= 9.0)
    amplitude:   30.3135620 +/- 0.157126 (0.52%) (init= 40.0)
    fwhm:        2.90157055 +/- 0.017366 (0.60%)  == '2.3548200*sigma'
    height:      9.81457817 +/- 0.050872 (0.52%)  == '0.3989423*amplitude/max(1.e-15, sigma)'
M Newville
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  • What method did you use to determine the initial parameter values in the example code that you provided? – James Phillips Mar 06 '18 at 16:52
  • @JamesPhillips : I looked at the data (not even posted here, but in the lmfit example) and guessed. Lmfit's GaussianModel actually has a `guess` method to help guess center, amplitude, and sigma -- the linked example uses that method. Peak-finding utilities in scipy or other libraries can also be useful at identifying peak centers. But also: for isolated Gaussian peaks, you don't need to be that close in the initial guess for the fit to converge. – M Newville Mar 06 '18 at 18:52
  • It does look like your guessing worked well in this case. – James Phillips Mar 06 '18 at 20:58