使用三维数据的非线性最小二乘最小化

Question

I have a 3D data and I would like to fit a non-linear model to the data using lmfit . This is the code I have written but it doesn't work.

from lmfit import minimize, Parameters, Parameter, report_fit
import numpy as np

m=np.array([13.8, 14.38, 14.316, 13.799, 14.135, 13.791, 14.531, 14.262, 14.08, 14.267, 14.259, 14.849, 14.396, 13.609])
lower_error_m=np.array([0.221,0.159,0.202,0.276,0.267,0.217,0.109,0.174,0.183,0.317,0.099,0.124,0.300,0.258])
upper_error_m=np.array([0.466,0.988,1.141,0.819,0.944,0.864,1.126,0.837,0.619,0.421,0.692,0.530,0.997,0.655])
asymmetric_error_m = [lower_error_m, upper_error_m]
r=np.array([7.168, 5.99, 6.296, 6.756, 6.297, 6.639, 5.857, 5.94, 7.312, 6.201, 7.191, 5.624, 5.584, 6.701])
lower_error_r=np.array([1.709,1.882,1.038,1.656,1.676,1.593,1.937,1.474,1.511,1.247,1.504,1.386,1.492,1.294])
upper_error_r=np.array([0.223,0.168,0.300,0.399,0.405,0.444,0.201,0.306,0.362,0.058,0.401,0.267,0.471,0.529])
asymmetric_error_r = [lower_error_r, upper_error_r]
q=np.array([0.13, 0.312,0.327,0.322,0.327,0.381,0.357,0.386,0.133,0.354,0.128,0.339,0.479,0.325])
# define objective function: returns the array to be minimized
def fcn2min(params, m, q, data):
    """ model decaying sine wave, subtract data"""
    A0 = params['A0'].value
    B  = params['B'].value
    alpha = params['alpha'].value
    model = 10**( np.log10(A0/(1+q)**B)+alpha*(m-14))
    return model - data

# create a set of Parameters
params = Parameters()
params.add('A0',   value= 6)
params.add('B', value= 0.1)
params.add('alpha', value= -0.1)
# do fit, here with leastsq model
result = minimize(fcn2min, params, args=(m, q, r))

I am getting the following error:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/vol/anaconda/lib/python2.7/site-packages/lmfit-0.9.2_4_gfc81f36-py2.7.egg/lmfit/minimizer.py", line 774, in minimize
    return fitter.minimize(method=method)
  File "/vol/anaconda/lib/python2.7/site-packages/lmfit-0.9.2_4_gfc81f36-py2.7.egg/lmfit/minimizer.py", line 706, in minimize
    return function(**kwargs)
  File "/vol/anaconda/lib/python2.7/site-packages/lmfit-0.9.2_4_gfc81f36-py2.7.egg/lmfit/minimizer.py", line 547, in leastsq
    lsout = scipy_leastsq(self.__residual, vars, **lskws)
  File "/vol/anaconda/lib/python2.7/site-packages/scipy/optimize/minpack.py", line 371, in leastsq
    shape, dtype = _check_func('leastsq', 'func', func, x0, args, n)
  File "/vol/anaconda/lib/python2.7/site-packages/scipy/optimize/minpack.py", line 20, in _check_func
    res = atleast_1d(thefunc(*((x0[:numinputs],) + args)))
  File "/vol/anaconda/lib/python2.7/site-packages/lmfit-0.9.2_4_gfc81f36-py2.7.egg/lmfit/minimizer.py", line 258, in __residual
    out = self.userfcn(params, *self.userargs, **self.userkws)
  File "<stdin>", line 6, in fcn2min
TypeError: unsupported operand type(s) for +: 'int' and 'list'

This is the model I am trying to fit:

where A0 , B and alpha are free parameters of the model. I also have errors on m and r and I am wondering how I could include them in the fitting process? Thanks.

Answer 1

A. Your example runs to completion without such an error for me, with lmfit version 0.9.3.

B. You can include uncertainties in the data by having your objective function return

   return (model - data)/ uncertainty

where uncertainty is the uncertainty in the data.

C. Sorry, but least-squares minimization as with MINPACK does not easily support asymmetric errors or errors in the independent variable.