Least squares method

Question

I have a set of data that follows a double gaussian distribution, whose equation depends on 4 parameters. What I'd like to do is:

• starting from a value for each of the 4 parameters (that are the ones I know that are closer to the parameters of the double gaussian that I obtain from the set of data)
• increasing them step by step
• plot the double gaussian with these parameters
• and check, using the least squares method, which are the better values to plot a double gaussian that is the best fit for the double gaussian that I obtain with the set of data.

I've tried to do it, below is my code:

import matplotlib.pyplot as plt
import numpy as np

def DGauss(x,I1,I2,sigma1,sigma2):
    return (I1/np.sqrt(2*np.pi*sigma1))*np.exp(-x*x/(2*sigma1*sigma1)) + (I2/np.sqrt(2*np.pi*sigma2))*np.exp(-x*x/(2*sigma2*sigma2))

def Inte(x0,xf,npoints,I1,I2,sigma1,sigma2):
     delta = (xf-x0)/npoints
     h = 0
     for i in range(npoints):
         x = x0 + i*delta
         if (i % 2) == 0:
             fact = 4.0
         else:
             fact = 2.0 

         h += fact*DGauss(x,I1,I2,sigma1,sigma2)

     return (abs(delta)/3.0)*(h + DGauss(x0,I1,I2,sigma1,sigma2) + DGauss(xf,I1,I2,sigma1,sigma2))

def IntBLM(start,end):
    delta = 1
    VarIntBLM = (1.0/3.0)*(BLM[start]+4.0*BLM[start+delta]+BLM[end])
    return VarIntBLM

Position = [9.88, 8.68, 7.48, 6.28, 5.08, 3.88, 3.28, 3.13, 3.08, 3.03, 2.98, 2.93, 2.88, 2.83, 2.78, 2.73, 2.68, 2.63, 2.58, 2.53, 2.48, 2.43, 2.38, 2.33, 2.28, 2.23, 2.18, 2.13, 2.08, 2.03, 1.98, 1.93, 1.88, 1.83, 1.78, 1.73, 1.68, 1.63, 1.58, 1.53, 1.48, 1.43, 1.38, 1.33, 1.28, 1.23, 1.18, 1.13, 1.08, 1.03, 0.98, 0.93, 0.88, 0.83, 0.78, 0.73, 0.68, 0.63, 0.58, 0.53, 0.48, 0.43, 0.38, 0.33, 0.28, 0.23, 0.18, 0.13, 0.08, 0.03]

yVal = [0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 1.05493778e-04, 3.68936803e-04, 1.73977308e-03, 9.86279398e-03, 1.52954321e-02, 2.42624032e-02, 2.87455974e-02, 3.23848413e-02, 3.28592719e-02, 3.94523416e-02, 4.61509051e-02, 5.70161813e-02, 6.37672003e-02, 7.19426767e-02, 7.76393407e-02, 8.56568678e-02, 9.61526244e-02, 1.04328101e-01, 1.13506059e-01, 1.19940597e-01, 1.26006198e-01, 1.40933276e-01, 1.50796653e-01, 1.66514643e-01, 1.80650226e-01, 1.93889404e-01, 2.04754098e-01, 2.17940237e-01, 2.28067057e-01, 2.37930434e-01, 2.51644042e-01, 2.63511800e-01, 2.80759742e-01, 2.95686820e-01, 3.08715010e-01, 3.31184602e-01, 3.46480617e-01, 3.69846614e-01, 3.85406655e-01, 4.06188347e-01, 4.28394495e-01, 4.50020137e-01, 4.83039107e-01, 5.07460625e-01, 5.31670573e-01, 5.54879204e-01, 5.78351278e-01, 6.02561809e-01, 6.25664363e-01, 6.57048471e-01, 6.82893863e-01, 7.13327944e-01, 7.32580267e-01, 7.69608000e-01, 7.87699892e-01, 8.14072753e-01, 8.33588519e-01, 8.52102386e-01, 8.71090683e-01, 8.94562175e-01, 9.16187816e-01, 9.37602470e-01, 9.56802338e-01, 9.69197565e-01, 9.78321903e-01, 9.84861934e-01, 9.93142904e-01,]

BLM = [0., 0., 0., 0.000181, 0.000452, 0.002352, 0.000724, 0.000633, 0.001267, 0.000633, 0.006244, 0.000814, 0.002805, 0.000633, 0.002443, 0.000814,0.001629, 0.001086, 0.001448, 0.001357, 0.002443, 0.001991, 0.002805, 0.010407,0.001991, 0.000814, 0.002352, 0.002714, 0.002081, 0.002714, 0.002352, 0.00181, 0.004253, 0.001719, 0.003077, 0.001448, 0.002805, 0.002896, 0.002986, 0.0019, 0.002262, 0.002262, 0.004072, 0.004434, 0.006425, 0.002986, 0.005701, 0.002352,0.004163, 0.004615, 0.001991, 0.001538, 0.005158, 0.0019,   0.003981, 0.004615,0.004615, 0.003167, 0.005339, 0.003619, 0.005339, 0.00181,  0.002624, 0.001267, 0.0038, 0.001629, 0.002986, 0.000633, 0.001719, 0.000543]

npoints = 13 
npairs = 13


hMin = 1
h=0

for k in range(13):
    sigma1 = 0.01*(k+1) + 1
    for l in range(13):
        sigma2 = 0.01*(l+1) + 1
        for m in range(13):
            I1 = 0.1*(m+1)
            for j in range(13):
                I2 = 0.1*(j+1)
                for i in range(13):
                    startPos = 50+i 
                    endPos = 52+i 
                    startX = Position[startPos]
                    endX = Position[endPos]

                    h += (IntBLM(startPos,endPos)-Inte(startX,endX,npoints,I1,I2,sigma1,sigma2))**2

                if h < hMin:
                    hMin = h
                    I1min = I1
                    I2min = I2
                    sigma1min = sigma1
                    sigma2min = sigma2

plt.figure(1)
plt.plot(Position,yVal,'o')
plt.plot(Position,DGauss(Posi,I1min,I2min,sigma1min,sigma2min), label='Least Square')
plt.show()

I've done different tests, printing the values of the gaussian parameters, the value returned by the least squares method, but the problem is that the values of I1min, I2min, sigma1min, sigma2min are always the ones obtained from the first cycle of the loop, and the value of I1min is always equal to I2min , link the value of sigma1min is equal to sigma2min , and I know that this isn't right. Could anyone help me please? Thanks!

Answer 1

Ok, the logic is thus:

1. hMin = 1 , h=0
   2. h += (IntBLM(startPos,endPos)-Int(startX,endX,npoints,I1,I2,sigma1,sigma2))**2
      a. if h is greater than or equal to 1, the conditional never executes
      b. else if h is now between 0 and 1, the conditional executes
      1. hMin = h
         Now, h will never be less than hMin , since it always increases by a positive amount (a squared value). Thus the conditional never executes again, and I1min, I2min, sigma1min, sigma2min maintain their instantiated values