How to fit a double Gaussian distribution in Python?
Question:
I am trying to obtain a double Gaussian distribution for data (link) using Python. The raw data is of the form:
For the given data, I would like to obtain two Gaussian profiles for the peaks seen in figure. I tried it with the following code (source):
from sklearn import mixture
import matplotlib.pyplot
import matplotlib.mlab
import numpy as np
from pylab import *
data = np.genfromtxt('gaussian_fit.dat', skiprows = 1)
x = data[:, 0]
y = data[:, 1]
clf = mixture.GMM(n_components=2, covariance_type='full')
clf.fit((y, x))
m1, m2 = clf.means_
w1, w2 = clf.weights_
c1, c2 = clf.covars_
fig = plt.figure(figsize = (5, 5))
plt.subplot(111)
plotgauss1 = lambda x: plot(x,w1*matplotlib.mlab.normpdf(x,m1,np.sqrt(c1))[0], linewidth=3)
plotgauss2 = lambda x: plot(x,w2*matplotlib.mlab.normpdf(x,m2,np.sqrt(c2))[0], linewidth=3)
fig.savefig('gaussian_fit.pdf')
But I am not able to get the desired output. So, how can a double Gaussian distribution be obtained in Python?
Update
I was able to fit a single Gaussian distribution with the following code:
import pylab as plb
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy import asarray as ar,exp
import numpy as np
data = np.genfromtxt('gaussian_fit.dat', skiprows = 1)
x = data[:, 0]
y = data[:, 1]
n = len(x)
mean = sum(x*y)/n
sigma = sum(y*(x-mean)**2)/n
def gaus(x,a,x0,sigma):
return a*exp(-(x-x0)**2/(2*sigma**2))
popt,pcov = curve_fit(gaus, x, y ,p0 = [1, mean, sigma])
fig = plt.figure(figsize = (5, 5))
plt.subplot(111)
plt.plot(x, y, label='Raw')
plt.plot(x, gaus(x, *popt), 'o', markersize = 4, label='Gaussian fit')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
fig.savefig('gaussian_fit.pdf')
Answers:
You can’t use scikit-learn for this, because the you are not dealing with a set of samples whose distribution you want to estimate. You could of course transform your curve to a PDF, sample it and then try to fit it using a Gaussian mixture model, but that seems to be a bit of an overkill to me.
Here’s a solution using simple least square curve fitting. To get it to work I had to remove the background, i.e. ignore all data points with y < 5, and also provide a good starting vector for leastsq, which can be estimated form a plot of the data.
Finding the Starting Vector
The parameter vector that that is found by the least squares method is the vector
params = [c1, mu1, sigma1, c2, mu2, sigma2]
Here, c1 and c2 are scaling factors for the two Gaussians, i.e. their height, mu1and mu2 are the means, i.e. the horizontal positions of the peaks and sigma1 and sigma2 the standard deviations that determine the width of the Gaussians. To find a starting vector I just looked at a plot of the data and estimated the height of the two peaks ( = c1, c2, respectively) and their horizontal position (= mu1, mu1, respectively). sigma1 and sigma2 were simply set to 1.0.
Code
from sklearn import mixture
import matplotlib.pyplot
import matplotlib.mlab
import numpy as np
from pylab import *
from scipy.optimize import leastsq
data = np.genfromtxt('gaussian_fit.dat', skiprows = 1)
x = data[:, 0]
y = data[:, 1]
def double_gaussian( x, params ):
(c1, mu1, sigma1, c2, mu2, sigma2) = params
res = c1 * np.exp( - (x - mu1)**2.0 / (2.0 * sigma1**2.0) )
+ c2 * np.exp( - (x - mu2)**2.0 / (2.0 * sigma2**2.0) )
return res
def double_gaussian_fit( params ):
fit = double_gaussian( x, params )
return (fit - y_proc)
# Remove background.
y_proc = np.copy(y)
y_proc[y_proc < 5] = 0.0
# Least squares fit. Starting values found by inspection.
fit = leastsq( double_gaussian_fit, [13.0,-13.0,1.0,60.0,3.0,1.0] )
plot( x, y, c='b' )
plot( x, double_gaussian( x, fit[0] ), c='r' )
enter image description hereTry the following code for multi-gaussian fitting:
# -*- coding: utf-8 -*-
"""
Created on Sat Jan 28 18:57:13 2023[enter image description here][1]
@author: Sagar Dam
"""
"""
This is the fitting function for Multi Gaussian data. The inputs are two
same length array of datatype float.
There are 3 outputs:
1. The array after fitting the data. That can be plotted.
2. The used parameters set as the name parameters.
3. The string to describe the parameters set and the fit function.
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit as fit
from decimal import Decimal
import pandas as pd
import matplotlib
matplotlib.rcParams['figure.dpi']=300 # highres display
def Gauss1(x,b,x0):
y=np.exp(-(x-x0)**2/(2*b**2))
return y
def Gaussfit(w,I):
xdata=w #Taking the x axis data
ydata=I #Taking the y axis data
'''
here the code fits only the normalized Gaussian
So, we first normalize the array and later multiply with the
amplitude factor to get the main array
'''
y_maxval=max(ydata) #taking the maximum value of the y array
ymax_index=(list(ydata)).index(y_maxval)
xmax_val=xdata[ymax_index] #Shifting the array as a non-shifted Gausian
xdata=xdata-xmax_val #Shifting the array as a non-shifted Gausian
ydata=ydata/y_maxval
parameters, covariance = fit(Gauss1, xdata, ydata,maxfev=100000)
fit_y = Gauss1(xdata, *parameters)
xdata=xdata+xmax_val
#parameters[1]+=xmax_val
fit_y=np.asarray(fit_y)
fit_y=fit_y*y_maxval # again multiplying the data to get the actual value
string1=r"Fit: $f(x)=Ae^{-frac{(x-x_0)^2}{2b^2}}$;"
string2=rf"with A={Decimal(str(y_maxval)).quantize(Decimal('1.00'))}, b={Decimal(str(parameters[0])).quantize(Decimal('1.00'))}, $x_0$={Decimal(str(parameters[1])).quantize(Decimal('1.00'))}"
string=string1+string2
return fit_y,parameters,string
def Multi_Gaussfit(x,y):
fit_y1,parameters1,string1=Gaussfit(x,y)
y2=y-fit_y1
fit_y2,parameters2,string2=Gaussfit(x,y2)
y3=y-fit_y1-fit_y2
fit_y3,parameters3,string3=Gaussfit(x,y3)
y4=y-fit_y1-fit_y2-fit_y3
fit_y4,parameters4,string4=Gaussfit(x,y4)
y5=y-fit_y1-fit_y2-fit_y3-fit_y4
fit_y5,parameters5,string5=Gaussfit(x,y5)
fit_y=fit_y1+fit_y2+fit_y3+fit_y4+fit_y5
parameters_data=[parameters1[0],parameters1[1],parameters2[0],parameters2[1],parameters3[0],parameters3[1],parameters4[0],parameters4[1],parameters5[0],parameters5[1]]
parameters_name=[r"$sigma_1$",r"$x_{01}$",r"$sigma_2$",r"$x_{02}$",r"$sigma_3$",r"$x_{03}$",r"$sigma_4$",r"$x_{04}$",r"$sigma_5$",r"$x_{05}$"]
parameters=pd.Series(parameters_data,index=parameters_name)
return(fit_y,parameters)
x=np.linspace(-30,30,601) #data along x axis
y=50*np.exp(-(x+20)**2/5)+30*np.exp(-(x+10)**2/7)+10*np.exp(-(x-20)**2/50)+40*np.exp(-(x-10)**2/0.5)+30*np.exp(-(x-0)**2/5) #data along y axis
random_noise=np.random.uniform(low=-2,high=2,size=(len(y)))
y=y+random_noise
fit_y,parameters=Multi_Gaussfit(x, y)
print(parameters)
plt.figure()
plt.plot(x,y,color='k')
#plt.title(string)
plt.plot(x,fit_y,'b-')
plt.figtext(0.9,0.9,str(parameters))
#plt.plot(x,y2,'r-')
plt.show()
#print(*parameters)
#print('FWHM= ', 2.355*parameters[0])
I am trying to obtain a double Gaussian distribution for data (link) using Python. The raw data is of the form:
For the given data, I would like to obtain two Gaussian profiles for the peaks seen in figure. I tried it with the following code (source):
from sklearn import mixture
import matplotlib.pyplot
import matplotlib.mlab
import numpy as np
from pylab import *
data = np.genfromtxt('gaussian_fit.dat', skiprows = 1)
x = data[:, 0]
y = data[:, 1]
clf = mixture.GMM(n_components=2, covariance_type='full')
clf.fit((y, x))
m1, m2 = clf.means_
w1, w2 = clf.weights_
c1, c2 = clf.covars_
fig = plt.figure(figsize = (5, 5))
plt.subplot(111)
plotgauss1 = lambda x: plot(x,w1*matplotlib.mlab.normpdf(x,m1,np.sqrt(c1))[0], linewidth=3)
plotgauss2 = lambda x: plot(x,w2*matplotlib.mlab.normpdf(x,m2,np.sqrt(c2))[0], linewidth=3)
fig.savefig('gaussian_fit.pdf')
But I am not able to get the desired output. So, how can a double Gaussian distribution be obtained in Python?
Update
I was able to fit a single Gaussian distribution with the following code:
import pylab as plb
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy import asarray as ar,exp
import numpy as np
data = np.genfromtxt('gaussian_fit.dat', skiprows = 1)
x = data[:, 0]
y = data[:, 1]
n = len(x)
mean = sum(x*y)/n
sigma = sum(y*(x-mean)**2)/n
def gaus(x,a,x0,sigma):
return a*exp(-(x-x0)**2/(2*sigma**2))
popt,pcov = curve_fit(gaus, x, y ,p0 = [1, mean, sigma])
fig = plt.figure(figsize = (5, 5))
plt.subplot(111)
plt.plot(x, y, label='Raw')
plt.plot(x, gaus(x, *popt), 'o', markersize = 4, label='Gaussian fit')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
fig.savefig('gaussian_fit.pdf')
You can’t use scikit-learn for this, because the you are not dealing with a set of samples whose distribution you want to estimate. You could of course transform your curve to a PDF, sample it and then try to fit it using a Gaussian mixture model, but that seems to be a bit of an overkill to me.
Here’s a solution using simple least square curve fitting. To get it to work I had to remove the background, i.e. ignore all data points with y < 5, and also provide a good starting vector for leastsq, which can be estimated form a plot of the data.
Finding the Starting Vector
The parameter vector that that is found by the least squares method is the vector
params = [c1, mu1, sigma1, c2, mu2, sigma2]
Here, c1 and c2 are scaling factors for the two Gaussians, i.e. their height, mu1and mu2 are the means, i.e. the horizontal positions of the peaks and sigma1 and sigma2 the standard deviations that determine the width of the Gaussians. To find a starting vector I just looked at a plot of the data and estimated the height of the two peaks ( = c1, c2, respectively) and their horizontal position (= mu1, mu1, respectively). sigma1 and sigma2 were simply set to 1.0.
Code
from sklearn import mixture
import matplotlib.pyplot
import matplotlib.mlab
import numpy as np
from pylab import *
from scipy.optimize import leastsq
data = np.genfromtxt('gaussian_fit.dat', skiprows = 1)
x = data[:, 0]
y = data[:, 1]
def double_gaussian( x, params ):
(c1, mu1, sigma1, c2, mu2, sigma2) = params
res = c1 * np.exp( - (x - mu1)**2.0 / (2.0 * sigma1**2.0) )
+ c2 * np.exp( - (x - mu2)**2.0 / (2.0 * sigma2**2.0) )
return res
def double_gaussian_fit( params ):
fit = double_gaussian( x, params )
return (fit - y_proc)
# Remove background.
y_proc = np.copy(y)
y_proc[y_proc < 5] = 0.0
# Least squares fit. Starting values found by inspection.
fit = leastsq( double_gaussian_fit, [13.0,-13.0,1.0,60.0,3.0,1.0] )
plot( x, y, c='b' )
plot( x, double_gaussian( x, fit[0] ), c='r' )
enter image description hereTry the following code for multi-gaussian fitting:
# -*- coding: utf-8 -*-
"""
Created on Sat Jan 28 18:57:13 2023[enter image description here][1]
@author: Sagar Dam
"""
"""
This is the fitting function for Multi Gaussian data. The inputs are two
same length array of datatype float.
There are 3 outputs:
1. The array after fitting the data. That can be plotted.
2. The used parameters set as the name parameters.
3. The string to describe the parameters set and the fit function.
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit as fit
from decimal import Decimal
import pandas as pd
import matplotlib
matplotlib.rcParams['figure.dpi']=300 # highres display
def Gauss1(x,b,x0):
y=np.exp(-(x-x0)**2/(2*b**2))
return y
def Gaussfit(w,I):
xdata=w #Taking the x axis data
ydata=I #Taking the y axis data
'''
here the code fits only the normalized Gaussian
So, we first normalize the array and later multiply with the
amplitude factor to get the main array
'''
y_maxval=max(ydata) #taking the maximum value of the y array
ymax_index=(list(ydata)).index(y_maxval)
xmax_val=xdata[ymax_index] #Shifting the array as a non-shifted Gausian
xdata=xdata-xmax_val #Shifting the array as a non-shifted Gausian
ydata=ydata/y_maxval
parameters, covariance = fit(Gauss1, xdata, ydata,maxfev=100000)
fit_y = Gauss1(xdata, *parameters)
xdata=xdata+xmax_val
#parameters[1]+=xmax_val
fit_y=np.asarray(fit_y)
fit_y=fit_y*y_maxval # again multiplying the data to get the actual value
string1=r"Fit: $f(x)=Ae^{-frac{(x-x_0)^2}{2b^2}}$;"
string2=rf"with A={Decimal(str(y_maxval)).quantize(Decimal('1.00'))}, b={Decimal(str(parameters[0])).quantize(Decimal('1.00'))}, $x_0$={Decimal(str(parameters[1])).quantize(Decimal('1.00'))}"
string=string1+string2
return fit_y,parameters,string
def Multi_Gaussfit(x,y):
fit_y1,parameters1,string1=Gaussfit(x,y)
y2=y-fit_y1
fit_y2,parameters2,string2=Gaussfit(x,y2)
y3=y-fit_y1-fit_y2
fit_y3,parameters3,string3=Gaussfit(x,y3)
y4=y-fit_y1-fit_y2-fit_y3
fit_y4,parameters4,string4=Gaussfit(x,y4)
y5=y-fit_y1-fit_y2-fit_y3-fit_y4
fit_y5,parameters5,string5=Gaussfit(x,y5)
fit_y=fit_y1+fit_y2+fit_y3+fit_y4+fit_y5
parameters_data=[parameters1[0],parameters1[1],parameters2[0],parameters2[1],parameters3[0],parameters3[1],parameters4[0],parameters4[1],parameters5[0],parameters5[1]]
parameters_name=[r"$sigma_1$",r"$x_{01}$",r"$sigma_2$",r"$x_{02}$",r"$sigma_3$",r"$x_{03}$",r"$sigma_4$",r"$x_{04}$",r"$sigma_5$",r"$x_{05}$"]
parameters=pd.Series(parameters_data,index=parameters_name)
return(fit_y,parameters)
x=np.linspace(-30,30,601) #data along x axis
y=50*np.exp(-(x+20)**2/5)+30*np.exp(-(x+10)**2/7)+10*np.exp(-(x-20)**2/50)+40*np.exp(-(x-10)**2/0.5)+30*np.exp(-(x-0)**2/5) #data along y axis
random_noise=np.random.uniform(low=-2,high=2,size=(len(y)))
y=y+random_noise
fit_y,parameters=Multi_Gaussfit(x, y)
print(parameters)
plt.figure()
plt.plot(x,y,color='k')
#plt.title(string)
plt.plot(x,fit_y,'b-')
plt.figtext(0.9,0.9,str(parameters))
#plt.plot(x,y2,'r-')
plt.show()
#print(*parameters)
#print('FWHM= ', 2.355*parameters[0])