Fitting function with curve_fit, but the fitted curve is wrong

Question

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

times, amplitudes = openFile("../datafiles/number_of_counts.txt")

mean = sum(np.array(times)*np.array(amplitudes))/sum(amplitudes)            
sigma = np.sqrt(sum(np.array(amplitudes)*(np.array(times)-mean)**2)/sum(amplitudes))        

params,pcov = curve_fit(gaus,times, amplitudes,p0=[max(amplitudes),mean,sigma])


plt.plot(times, amplitudes)
plt.plot(times ,gaus(np.array(times),params[0],params[1],params[2]),'r',  label="fitted curve")

plt.ylabel("Coincidents")
plt.title("Coincident plot")
plt.legend()
plt.show()

My gaussian fit doesn't work properly, but looks like a soft curve, instead of for fitting to the sharp peak, I assume I have some super silly error in my script, but not sure what. Someone who can see it?

Answer 1

Your data has a constant offset of about 3750, but your gaus model function cannot account for that, so you are fitting a normal distribution with offset 0.

It needs one more parameter:

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

Then:

offset_guess = 3750  # maybe calculate it from the data as well
params, pcov = curve_fit(
    gaus, times, amplitudes,
    p0=[max(amplitudes), mean, sigma, offset_guess])

plt.plot(times, gaus(np.array(times), params[0], params[1], params[2], params[3]), ...)

Result:

>>> print(params)
[1545.00193331  -20.45639132  -43.28484495 3792.41050636]

Answer 2

I extracted data from the plot for analysis, and found that with the extracted data a Weibull peak plus offset gave me a better fit than a Gaussian peak with offset. Here is a graphical Python fitter with the extracted data and a Weibull peak plus offset equation, you should be able to substitute in the actual data and run it directly. Note the simple determination of initial parameter estimates.

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

xData = numpy.array([4.914e+03, 5.600e+03, 7.886e+03, 8.571e+03, 1.063e+04, 1.154e+04, 1.269e+04, 1.566e+04, 1.634e+04, 1.817e+04, 1.886e+04, 2.114e+04, 2.389e+04, 2.526e+04, 2.754e+04, 3.051e+04, 3.257e+04, 3.417e+04, 3.554e+04, 3.669e+04, 4.011e+04, 4.240e+04, 4.491e+04, 4.583e+04, 4.697e+04, 4.880e+04, 4.994e+04, 5.154e+04, 5.246e+04, 5.474e+04, 5.634e+04, 5.886e+04, 6.091e+04, 6.366e+04, 6.731e+04, 7.051e+04, 7.257e+04, 7.394e+04, 7.691e+04, 7.851e+04, 7.966e+04, 8.103e+04, 8.240e+04, 8.514e+04, 8.720e+04, 8.834e+04, 8.949e+04, 9.109e+04, 9.223e+04, 9.360e+04, 9.566e+04, 9.726e+04, 9.909e+04, 1.005e+05, 1.014e+05, 1.030e+05, 1.059e+05, 1.073e+05, 1.089e+05, 1.101e+05, 1.119e+05, 1.130e+05, 1.139e+05, 1.162e+05, 1.178e+05, 1.190e+05, 1.203e+05, 1.222e+05, 1.233e+05, 1.247e+05, 1.281e+05, 1.299e+05, 1.309e+05, 1.329e+05, 1.341e+05, 1.357e+05, 1.370e+05, 1.382e+05, 1.395e+05, 1.407e+05, 1.430e+05, 1.439e+05])
yData = numpy.array([3.300e+03, 8.100e+03, 6.100e+03, 1.010e+04, 9.700e+03, 7.300e+03, 7.500e+03, 6.900e+03, 8.100e+03, 3.900e+03, 5.700e+03, 4.900e+03, 4.500e+03, 8.300e+03, 4.100e+03, 8.100e+03, 5.300e+03, 8.100e+03, 6.700e+03, 1.130e+04, 9.300e+03, 6.300e+03, 9.500e+03, 8.900e+03, 1.490e+04, 6.300e+03, 1.190e+04, 6.300e+03, 7.700e+03, 1.310e+04, 9.500e+03, 1.590e+04, 1.050e+04, 1.930e+04, 4.890e+04, 7.350e+04, 5.230e+04, 5.130e+04, 2.350e+04, 1.950e+04, 1.010e+04, 1.510e+04, 9.500e+03, 9.500e+03, 6.900e+03, 6.900e+03, 1.210e+04, 6.300e+03, 7.700e+03, 5.700e+03, 1.410e+04, 8.700e+03, 1.390e+04, 4.900e+03, 7.500e+03, 4.900e+03, 9.500e+03, 5.300e+03, 9.300e+03, 6.300e+03, 1.250e+04, 4.300e+03, 7.700e+03, 6.900e+03, 9.700e+03, 8.500e+03, 1.130e+04, 5.300e+03, 5.100e+03, 1.700e+03, 8.700e+03, 7.300e+03, 6.300e+03, 2.100e+03, 3.100e+03, 7.100e+03, 4.900e+03, 6.100e+03, 3.700e+03, 9.300e+03, 5.500e+03, 5.700e+03])


def func(x, a, b, c, offset): # Weibull peak with offset from zunzun.com
    return a * numpy.exp(-0.5 * numpy.power(numpy.log(x/b) / c, 2.0)) + offset

a_est = max(yData)
b_est = max(yData)
c_est = 1.0
offset_est = min(yData)
initialParameters = numpy.array([a_est, b_est, c_est, offset_est])

# curve fit the test data
fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)

modelPredictions = func(xData, *fittedParameters) 

absError = modelPredictions - yData

SE = numpy.square(absError) # squared errors
MSE = numpy.mean(SE) # mean squared errors
RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))

print('Parameters:', fittedParameters)
print('RMSE:', RMSE)
print('R-squared:', Rsquared)

print()


##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
    axes = f.add_subplot(111)

    # first the raw data as a scatter plot
    axes.plot(xData, yData,  'D')

    # create data for the fitted equation plot
    xModel = numpy.linspace(min(xData), max(xData), 500)
    yModel = func(xModel, *fittedParameters)

    # now the model as a line plot
    axes.plot(xModel, yModel)

    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label

    plt.show()
    plt.close('all') # clean up after using pyplot

graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)