I would like to fit ellipsometric data to complex model using LMFit. Two measured parameters, psi
and delta
, are variables in a complex function rho
.
I could try with separating problem to real and imaginary part with shared parameters or piecewise approach, but is there any way to do it directly with complex function? Fitting only real part of function works beautifully, but when I define complex residual function I get:
TypeError: no ordering relation is defined for complex numbers.
Below is my code for real function fitting and my attempt at tackling complex fit problem:
from __future__ import division
from __future__ import print_function
import numpy as np
from pylab import *
from lmfit import minimize, Parameters, Parameter, report_errors
#=================================================================
# MODEL
def r01_p(eps2, th):
c=cos(th)
s=(sin(th))**2
stev= sqrt(eps2) * c - sqrt(1-(s / eps2))
imen= sqrt(eps2) * c + sqrt(1-(s / eps2))
return stev/imen
def r01_s(eps2, th):
c=cos(th)
s=(sin(th))**2
stev= c - sqrt(eps2) * sqrt(1-(s/eps2))
imen= c + sqrt(eps2) * sqrt(1-(s/eps2))
return stev/imen
def rho(eps2, th):
return r01_p(eps2, th)/r01_s(eps2, th)
def psi(eps2, th):
x1=abs(r01_p(eps2, th))
x2=abs(r01_s(eps2, th))
return np.arctan2(x1,x2)
#=================================================================
# REAL FIT
#
#%%
# generate data from model
th=linspace(deg2rad(45),deg2rad(70),70-45)
error=0.01
var_re=np.random.normal(size=len(th), scale=error)
data = psi(2,th) + var_re
# residual function
def residuals(params, th, data):
eps2 = params['eps2'].value
diff = psi(eps2, th) - data
return diff
# create a set of Parameters
params = Parameters()
params.add('eps2', value= 1.0, min=1.5, max=3.0)
# do fit, here with leastsq model
result = minimize(residuals, params, args=(th, data),method="leastsq")
# calculate final result
final = data + result.residual
# write error report
report_errors(params)
# try to plot results
th, data, final=rad2deg([th, data, final])
try:
import pylab
clf()
fig=plot(th, data, 'r o',
th, final, 'b')
setp(fig,lw=2.)
xlabel(r'$\theta$ $(^{\circ})$', size=20)
ylabel(r'$\psi$ $(^{\circ})$',size=20)
show()
except:
pass
#%%
#=================================================================
# COMPLEX FIT
# TypeError: no ordering relation is defined for complex numbers
"""
# data from model with added noise
th=linspace(deg2rad(45),deg2rad(70),70-45)
error=0.001
var_re=np.random.normal(size=len(th), scale=error)
var_im=np.random.normal(size=len(th), scale=error) * 1j
data = rho(4-1j,th) + var_re + var_im
# residual function
def residuals(params, th, data):
eps2 = params['eps2'].value
diff = rho(eps2, th) - data
return np.abs(diff)
# create a set of Parameters
params = Parameters()
params.add('eps2', value= 1.5+1j, min=1+1j, max=3+3j)
# do fit, here with leastsq model
result = minimize(residuals, params, args=(th, data),method="leastsq")
# calculate final result
final = data + result.residual
# write error report
report_errors(params)
"""
#=================================================================
Edit: I solved problem with separated variables for imaginary and real part. Data should be shaped as [[imaginary_data],[real_data]], objective function must return 1D array.
def objective(params, th_data, data):
eps_re = params['eps_re'].value
eps_im = params['eps_im'].value
d = params['d'].value
residual_delta = data[0,:] - delta(eps_re - eps_im*1j, d, frac, lambd, th_data)
residual_psi = data[1,:] - psi(eps_re - eps_im*1j, d, frac, lambd, th_data)
return np.append(residual_delta,residual_psi)
# create a set of Parameters
params = Parameters()
params.add('eps_re', value= 1.5, min=1.0, max=5 )
params.add('eps_im', value= 1.0, min=0.0, max=5 )
params.add('d', value= 10.0, min=5.0, max=100.0 )
# All available methods
methods=['leastsq','nelder','lbfgsb','anneal','powell','cobyla','slsqp']
# Chosen method
#metoda='leastsq'
# run the global fit to all the data sets
result = minimize(objective, params, args=(th_data,data),method=metoda))
....
return ...
The lmfit FAQ suggests simply taking both real and imaginary parts by using numpy.ndarray.view
, which means you don't need to go through the separation of the real and imaginary parts manually.
def residuals(params, th, data):
eps2 = params['eps2'].value
diff = rho(eps2, th) - data
# The only change required is to use view instead of abs.
return diff.view()
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