Not able to fit a function with scipy.optimize.curve_fit() - curve-fitting

Question

I would like to fit the following function:
def invlaplace_stehfest2(time,EL,tau):
upsilon=0.25
pmax=6.9
E0=0.0154
M=8
results=[]
for t in time:
func=0
for k in range(1,2*M+1):
SUM=0
for j in range(int(math.floor((k+1)/2)),min(k,M)+1):
dummy=j**(M+1)*scipy.special.binom(M,j)*scipy.special.binom(2*j,j)*scipy.special.binom(j,k-j)/scipy.math.factorial(M)
SUM+=dummy
s=k*math.log(2)/t[enter image description here][1]
func+=(-1)**(M+k)*SUM*pmax*EL/(mp.exp(tau*s)*mp.expint(1,tau*s)*E0+EL)/s
func=func*math.log(2)/t
results.append(func)
return [float(i) for i in results]
To do so I use the following data:
data_time=np.array([69.0,99.0,139.0,179.0,219.0,259.0,295.5,299.03])
data_relax=np.array([6.2536,6.1652,6.0844,6.0253,5.9782,5.9404,5.9104,5.9066])
With the folowing guess:
guess=np.array([0.1,0.05])
And scipy.optimize.curve_fit() as folow:
Parameter,Covariance=scipy.optimize.curve_fit(invlaplace_stehfest2,data_time,data_relax,guess)
For A reason that I don't understand I am not able to fit the data correctly. I get the following graph.
Bad fitting
My function is undoubtedly correct since when I use the correct guess:
guess=np.array([0.33226685047281592707364253044085038793404361200072,8.6682623502960394383501102909774397295654841654769])
I am able to fit correctly my data.
Expected fitting
Any hint on why I am not able to fit correctly? Should I use another method?
Here is the hole program:
##############################################
import matplotlib
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab
import matplotlib.pylab as mplab
import math
from math import *
import numpy as np
import scipy
from scipy.optimize import curve_fit
import mpmath as mp
############################################################################################
def invlaplace_stehfest2(time,EL,tau):
upsilon=0.25
pmax=6.9
E0=0.0154
M=8
results=[]
for t in time:
func=0
for k in range(1,2*M+1):
SUM=0
for j in range(int(math.floor((k+1)/2)),min(k,M)+1):
dummy=j**(M+1)*scipy.special.binom(M,j)*scipy.special.binom(2*j,j)*scipy.special.binom(j,k-j)/scipy.math.factorial(M)
SUM+=dummy
s=k*math.log(2)/t
func+=(-1)**(M+k)*SUM*pmax*EL/(mp.exp(tau*s)*mp.expint(1,tau*s)*E0+EL)/s
func=func*math.log(2)/t
results.append(func)
return [float(i) for i in results]
############################################################################################
###Constant###
####Value####
data_time=np.array([69.0,99.0,139.0,179.0,219.0,259.0,295.5,299.03])
data_relax=np.array([6.2536,6.1652,6.0844,6.0253,5.9782,5.9404,5.9104,5.9066])
###Fitting###
guess=np.array([0.33226685047281592707364253044085038793404361200072,8.6682623502960394383501102909774397295654841654769])
#guess=np.array([0.1,0.05])
Parameter,Covariance=scipy.optimize.curve_fit(invlaplace_stehfest2,data_time,data_relax,guess)
print Parameter
residu=sum(data_relax-invlaplace_stehfest2(data_time,Parameter[0],Parameter[1]))
Graph_Curves=plt.figure()
ax = Graph_Curves.add_subplot(111)
ax.plot(data_time,invlaplace_stehfest2(data_time,Parameter[0],Parameter[1]),"-")
ax.plot(data_time,data_relax,"o")
plt.show()

Answer 1

Non-linear fitters such as the default Levenberg-Marquardt solver used in scipy.optimize.curve_fit(), like most iterative solvers, can stop in a local minimum in error space. If error space is "bumpy" then initial parameter estimates become very important, as in this case.
Scipy has added the Differential Evolution genetic algorithm to the optimize module, which can be used to determine initial parameter estimates for curve_fit(). Scipy's implementation uses the Latin Hypercube algorithm to ensure a thorough search of parameter space, requiring parameter upper and lower bounds within which to search. As you can see below, I have used this scipy module to replace the hard-coded values for the value named "guess" in your code. This does not run quickly, but a somewhat slower correct result is much better than a fast wrong result. Try this code:
import matplotlib
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab
import matplotlib.pylab as mplab
import math
from math import *
import numpy as np
import scipy
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import mpmath as mp
############################################################################################
def invlaplace_stehfest2(time,EL,tau):
upsilon=0.25
pmax=6.9
E0=0.0154
M=8
results=[]
for t in time:
func=0
for k in range(1,2*M+1):
SUM=0
for j in range(int(math.floor((k+1)/2)),min(k,M)+1):
dummy=j**(M+1)*scipy.special.binom(M,j)*scipy.special.binom(2*j,j)*scipy.special.binom(j,k-j)/scipy.math.factorial(M)
SUM+=dummy
s=k*math.log(2)/t
func+=(-1)**(M+k)*SUM*pmax*EL/(mp.exp(tau*s)*mp.expint(1,tau*s)*E0+EL)/s
func=func*math.log(2)/t
results.append(func)
return [float(i) for i in results]
############################################################################################
###Constant###
####Value####
data_time=np.array([69.0,99.0,139.0,179.0,219.0,259.0,295.5,299.03])
data_relax=np.array([6.2536,6.1652,6.0844,6.0253,5.9782,5.9404,5.9104,5.9066])
# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
return np.sum((data_relax - invlaplace_stehfest2(data_time, *parameterTuple)) ** 2)
###Fitting###
#guess=np.array([0.33226685047281592707364253044085038793404361200072,8.6682623502960394383501102909774397295654841654769])
#guess=np.array([0.1,0.05])
parameterBounds = [[0.0, 1.0], [0.0, 10.0]]
# "seed" the numpy random number generator for repeatable results
# note the ".x" here to return only the parameter estimates in this example
guess = differential_evolution(sumOfSquaredError, parameterBounds, seed=3).x
Parameter,Covariance=scipy.optimize.curve_fit(invlaplace_stehfest2,data_time,data_relax,guess)
print Parameter
residu=sum(data_relax-invlaplace_stehfest2(data_time,Parameter[0],Parameter[1]))
Graph_Curves=plt.figure()
ax = Graph_Curves.add_subplot(111)
ax.plot(data_time,invlaplace_stehfest2(data_time,Parameter[0],Parameter[1]),"-")
ax.plot(data_time,data_relax,"o")
plt.show()