Calculating the centroids of two superposed gaussian functions - statistics

Question

I am trying to find a solution to the following problem. I have a set of points which should model a sum of 2 Gaussian functions centered at different points. I need to find these two points. Up to now my approach has been to find the centroid of the whole set and cut the set of date below and above it. Then I calculate the centroid of each piece and those are my centers. This approach however cuts the information of, say, the left Gaussian which leaks into the right half of the data. This makes the procedure fail when the Gaussians are close together. Is there way to do this more intelligently? Due to the computational difficulty I would prefer if the solution didn't involve curve fitting.

Answer 1

As the OP does not show any data, it is not clear how noisy the data is. Furthermore, it is not clear how "close together" is defined here. In the following I have a simple approximation that works with low noise and the assumption that the left hand side data is dominated by the left Gaussian, while the right hand side is dominated by the right Gaussian. This gives some restrictions to position, height, and especially standard deviation.
It surely works for a single peak, but is quite OK for mixed double peaks ( within the above mentioned restrictions )
#!/usr/bin/python
import matplotlib.pyplot as plt
import numpy as np
def gaussian( x, x0, s, a):
return a * np.exp( -( x - x0 )**2 / ( 2 * s**2 ) ) / np.sqrt( 2 * np.pi * s**2 )
def get_x0( x1, x2, x3, y1, y2, y3 ):
l12= np.log( y1 / y2 )
l13= np.log( y1 / y3 )
return ( ( x2 + x1 )/2. - ( x3 + x1 )/2. * l12/l13 * ( x3 - x1 ) / ( x2 - x1 ) ) / ( 1 - l12 / l13 * (x3 - x1 ) / ( x2 - x1 ) )
fig = plt.figure( )
ax = fig.add_subplot( 2, 1, 1 )
xL = np.linspace(-8, 8, 150 )
yL = np.fromiter( ( gaussian( x,-2.1, 1.2, 8 ) for x in xL ), np.float )
marker=[10,15,20]
x1 = xL[ marker[0] ]
x2 = xL[ marker[1] ]
x3 = xL[ marker[2] ]
y1 = yL[ marker[0] ]
y2 = yL[ marker[1] ]
y3 = yL[ marker[2] ]
print get_x0( x1, x2, x3, y1, y2, y3 )
ax.plot( xL, yL )
ax.scatter( [ x1, x2, x3 ],[ y1, y2, y3 ])
bx = fig.add_subplot( 2, 1, 2 )
yL = np.fromiter( ( gaussian( x,-2.1, 1.2, 8) + gaussian( x,0.7, 1.4, 6 ) for x in xL ), np.float )
marker=[10,15,20]
x1 = xL[ marker[0] ]
x2 = xL[ marker[1] ]
x3 = xL[ marker[2] ]
y1 = yL[ marker[0] ]
y2 = yL[ marker[1] ]
y3 = yL[ marker[2] ]
bx.scatter( [ x1, x2, x3 ],[ y1, y2, y3 ])
print get_x0( x1, x2, x3, y1, y2, y3 )
marker=[-20,-25,-30]
x1 = xL[ marker[0] ]
x2 = xL[ marker[1] ]
x3 = xL[ marker[2] ]
y1 = yL[ marker[0] ]
y2 = yL[ marker[1] ]
y3 = yL[ marker[2] ]
bx.scatter( [ x1, x2, x3 ],[ y1, y2, y3 ])
print get_x0( x1, x2, x3, y1, y2, y3 )
bx.plot( xL, yL )
plt.show()
Shows:
#Single
-2.0999999999999455
#Double
-2.0951188129317813
0.6998760921436634
which is pretty close to -2.1 and 0.7
In case of noise some averaging might be required.