How to fit a Gaussian best fit for the data - matplotlib

Question

I have a data set for which I am plotting a graph of time vs intensity at a particular frequency.
On the x axis is the time data set which is in a numpy array and on the y axis is the intensity array.
time = [ 0.3 1.3 2.3 3.3 4.3 5.3 6.3 7.3 8.3 9.3 10.3 11.3 12.3 13.3
14.3 15.3 16.3 17.3 18.3 19.3 20.3 21.3 22.3 23.3 24.3 25.3 26.3 27.3
28.3 29.3 30.3 31.3 32.3 33.3 34.3 35.3 36.3 37.3 38.3 39.3 40.3 41.3
42.3 43.3 44.3 45.3 46.3 47.3 48.3 49.3 50.3 51.3 52.3 53.3 54.3 55.3
56.3 57.3 58.3 59.3]
intensity = [1.03587, 1.03187, 1.03561, 1.02893, 1.04659, 1.03633, 1.0481 ,
1.04156, 1.02164, 1.02741, 1.02675, 1.03651, 1.03713, 1.0252 ,
1.02853, 1.0378 , 1.04374, 1.01427, 1.0387 , 1.03389, 1.03148,
1.04334, 1.042 , 1.04154, 1.0161 , 1.0469 , 1.03152, 1.22406,
5.4362 , 7.92132, 6.50259, 4.7227 , 3.32571, 2.46484, 1.74615,
1.51446, 1.2711 , 1.15098, 1.09623, 1.0697 , 1.06085, 1.05837,
1.04151, 1.0358 , 1.03574, 1.05095, 1.03382, 1.04629, 1.03636,
1.03219, 1.03555, 1.02886, 1.04652, 1.02617, 1.04363, 1.03591,
1.04199, 1.03726, 1.03246, 1.0408 ]
When I plot this using matplotlib, using;
plt.figure(figsize=(15,6))
plt.title('Single frequency graph at 636 kHz', fontsize=18)
plt.plot(time,intensity)
plt.xticks(time[::3], fontsize=12)
plt.yticks(fontsize=12)
plt.xlabel('Elapsed time (minutes:seconds)', fontsize=18)
plt.ylabel('Intensity at 1020 kHz', fontsize=18)
plt.savefig('WIND_Single_frequency_graph_1020_kHz')
plt.show()
The graph looks like -
I want to fit a gaussian curve for this data, and this is the code I used,
def Gauss(x, A, B):
y = A*np.exp(-1*B*x**2)
return y
parameters, covariance = curve_fit(Gauss, time, intensity_636)
fit_A = parameters[0]
fit_B = parameters[1]
fit_y = Gauss(time, fit_A, fit_B)
plt.figure(figsize=(15,6))
plt.plot(time, intensity, 'o', label = 'data')
plt.plot(time, fit_y, '-', label ='fit')
plt.legend()
And the best fit i obtain looks like this -
Where am I going wrong? How can I make the best fit curve fit the data better?

Answer 1

By inspection one obweve that the curve isn't symmetrical. This is even more visible in logarithmic y-scale zoomed in the range close to the peak.
This draw to think that the Gaussian model (which is symetrical) cannot be fitted correctly.
Also one observe that the part of the curve around the peak isn't far from to be linear. Thus a better model might be made of the combination of two exponential functions, for example :
I suppose that you can code this function in your nonlinear regression sofware. The above rough values of the parameters can be used as starting values for the iterative calculus.

Answer 2

You need to define a more flexible model (more parameters) and to define reasonable initial values for them:
def f(x, a, b, mu, sigma):
return a + b * np.exp(-(x - mu) ** 2 / (2 * sigma ** 2))
popt, pcov = curve_fit(f, time, intensity, p0=[1, 1, 30.3, 2])
plt.plot(time, intensity)
plt.plot(time, f(time, *popt))
plt.show()