Fitting multiple datasets with shared paramaters

Question

I am trying to fit different data set with different non-linear function that shared some parameters and it look something like this:

import matplotlib
from matplotlib import pyplot as plt
from scipy import optimize
import numpy as np

#some non-linear function
def Sigma1x(x,C11,C111,C1111,C11111):
    return C11*x+1/2*C111*pow(x,2)+1/6*C1111*pow(x,3)+1/24*C11111*pow(x,4)

def Sigma2x(x,C12,C112,C1112,C11112):
    return C12*x+1/2*C112*pow(x,2)+1/6*C1112*pow(x,3)+1/24*C11112*pow(x,4)

def Sigma1y(y,C12,C111,C222,C112,C1111,C1112,C2222,C12222):
    return C12*y+1/2*(C111-C222+C112)*pow(y,2)+1/12*(C111+2*C1112-C2222)*pow(y,3)+1/24*C12222*pow(y,4)

def Sigma2y(y,C11,C222,C222,C2222):
    return C11*y+1/2*C222*pow(y,2)+1/6*C2222*pow(y,3)+1/24*C22222*pow(y,4)

def Sigmaz(z,C11,C12,C111,C222,C112,C1111,C1112,C2222,C1122,C11111,C11112,C122222,C11122,C22222):
    return (C11+C12)*z+1/2*(2*C111-C222+3*C112)*pow(z,2)+1/6*(3/2*C1111+4*C1112-1/2*C222+3*C1122)*pow(z,3)+\
                    1/24*(3*C11111+10*C11112-5*C12222+10*C11122-2*C22222)*pow(z,4)

# Experimental datasets

Xdata=np.loadtxt('x-direction.txt') #This contain x axis and two other dataset, should be fitted with Sigma1x and Sigma2x
Ydata=np.loadtxt('y-direction.txt') #his contain yaxis and two other dataset, should be fitted with Sigma1yand Sigma2y
Zdata=nploadtxt('z-direction.txt')#This contain z axis and one dataset  fitted with Sigmaz

The question is how to use optimize.leastsq or other packages to fit the data with the appropriate function, knowing that they share multiple paramaters?

Answer 1

I was able to solve ( partially the initial question). I found symfit a very comprehensive and easy to use. So i wrote the following code

 import matplotlib.pyplot as plt

 from symfit import *
 import numpy as np
 from symfit.core.minimizers import DifferentialEvolution, BFGS
 Y_strain = np.genfromtxt('Y_strain.csv', delimiter=',')
 X_strain=np.genfromtxt('X_strain.csv', delimiter=',')
 xmax=max(X_strain[:,0])
 xmin=min(X_strain[:,0])
 xdata = np.linspace(xmin, xmax, 50)
 ymax=max(Y_strain[:,0])
 ymin=max(Y_strain[:,0])
 ydata=np.linspace(ymin, ymax, 50)

 x,y,Sigma1x,Sigma2x,Sigma1y,Sigma2y=  variables('x,y,Sigma1x,Sigma2x,Sigma1y,Sigma2y')
 C11,C111,C1111,C11111,C12,C112,C1112,C11112,C222,C2222,C12222,C22222 =  parameters('C11,C111,C1111,C11111,C12,C112,C1112,C11112,C222,C2222,C12222,C22222')

model =Model({
     Sigma1x:C11*x+1/2*C111*pow(x,2)+1/6*C1111*pow(x,3)+1/24*C11111*pow(x,4),
     Sigma2x:C12*x+1/2*C112*pow(x,2)+1/6*C1112*pow(x,3)+1/24*C11112*pow(x,4),
     #Sigma1y:C12*y+1/2*(C111-C222+C112)*pow(y,2)+1/12*(C111+2*C1112-C2222)*pow(y,3)+1/24*C12222*pow(y,4),
     #Sigma2y:C11*y+1/2*C222*pow(y,2)+1/6*C2222*pow(y,3)+1/24*C22222*pow(y,4),  
})
  fit = Fit(model, x=X_strain[:,0], Sigma1x=X_strain[:,1],Sigma2x=X_strain[:,2])
  fit_result = fit.execute()
  print(fit_result)
  plt.scatter(Y_strain[:,0],Y_strain[:,2])
  plt.scatter(Y_strain[:,0],Y_strain[:,1])
  plt.plot(xdata, model(x=xdata, **fit_result.params).Sigma1x)
  plt.plot(xdata, model(x=xdata, **fit_result.params).Sigma2x)

However, The resulting fit is very bad :

 Parameter Value        Standard Deviation
 C11       1.203919e+02 3.988977e+00
 C111      -6.541505e+02 5.643111e+01
 C1111     1.520749e+03 3.713742e+02
 C11111    -7.824107e+02 1.015887e+03
 C11112    4.451211e+03 1.015887e+03
 C1112     -1.435071e+03 3.713742e+02
 C112      9.207923e+01 5.643111e+01
 C12       3.272248e+01 3.988977e+00
 Status message         Desired error not necessarily achieved due to precision loss.
 Number of iterations   59
 Objective              <symfit.core.objectives.LeastSquares object at 0x000001CC00C0A508>
 Minimizer              <symfit.core.minimizers.BFGS object at 0x000001CC7F84A548>

 Goodness of fit qualifiers:
 chi_squared            6.230510793023184
 objective_value        3.115255396511592
 r_squared              0.991979767376565

Any idea's how to improve the fit?