Gaussian fit for Python

Question

I\'m trying to fit a Gaussian for my data (which is already a rough gaussian). I\'ve already taken the advice of those here and tried curve_fit and leasts

Answer 1

Explanation

You need good starting values such that the curve_fit function converges at "good" values. I can not really say why your fit did not converge (even though the definition of your mean is strange - check below) but I will give you a strategy that works for non-normalized Gaussian-functions like your one.

Example

The estimated parameters should be close to the final values (use the weighted arithmetic mean - divide by the sum of all values):

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

x = np.arange(10)
y = np.array([0, 1, 2, 3, 4, 5, 4, 3, 2, 1])

# weighted arithmetic mean (corrected - check the section below)
mean = sum(x * y) / sum(y)
sigma = np.sqrt(sum(y * (x - mean)**2) / sum(y))

def Gauss(x, a, x0, sigma):
    return a * np.exp(-(x - x0)**2 / (2 * sigma**2))

popt,pcov = curve_fit(Gauss, x, y, p0=[max(y), mean, sigma])

plt.plot(x, y, 'b+:', label='data')
plt.plot(x, Gauss(x, *popt), 'r-', label='fit')
plt.legend()
plt.title('Fig. 3 - Fit for Time Constant')
plt.xlabel('Time (s)')
plt.ylabel('Voltage (V)')
plt.show()

I personally prefer using numpy.

Comment on the definition of the mean (including Developer's answer)

Since the reviewers did not like my edit on #Developer's code, I will explain for what case I would suggest an improved code. The mean of developer does not correspond to one of the normal definitions of the mean.

Your definition returns:

>>> sum(x * y)
125

Developer's definition returns:

>>> sum(x * y) / len(x)
12.5 #for Python 3.x

The weighted arithmetic mean:

>>> sum(x * y) / sum(y)
5.0

Similarly you can compare the definitions of standard deviation (sigma). Compare with the figure of the resulting fit:

Comment for Python 2.x users

In Python 2.x you should additionally use the new division to not run into weird results or convert the the numbers before the division explicitly:

from __future__ import division

or e.g.

sum(x * y) * 1. / sum(y)