使用共享参数拟合多个数据集

Question

我正在尝试使用共享一些参数的不同非线性函数来拟合不同的数据集，它看起来像这样：

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

问题是如何使用 optimize.leastsq 或其他包来使用适当的函数拟合数据，知道它们共享多个参数？

Answer 1

我能够解决（部分是最初的问题）。我发现 symfit 非常全面且易于使用。所以我写了下面的代码

 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)

然而，结果拟合非常糟糕：

 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

知道如何改进贴合度吗？

使用共享参数拟合多个数据集

Fitting multiple datasets with shared paramaters

python

scipy

lmfit

symfit