Curve.fit optimization error: 'Covariance of the parameters could not be estimated
Question:
I’m trying to fit some data into a function, but scipy curve_fit gives me the following message: "OptimizeWarning: Covariance of the parameters could not be estimated" and won’t plot or print the parameters of the fit. I believe the power inside the exp function might be the problem, but I don’t know how to work around this error. I tried changing the initial parameters but it won’t help. Here’s the data and the function I want to fit.
Data below as comment format to paste in excel
Thanks in advance.
import pandas as pd #Module for reading data
import numpy as np #Math module
import matplotlib.pyplot as plt #Plot module
from scipy.optimize import curve_fit #Curve fit module
var = pd.read_excel("Data.xlsx") # Read in same directory as file
print(var) #Print File
x1 = list(var['H(T)']) # x values
y1 = list(var['J (A/cm^2)']) # y values
######### FUNCTION ########
def func2(b, Jc1, Jc2, Bl, Bm, y):
return Jc1*np.exp(-b/Bl)+Jc2*(b/Bm)*np.exp((1/y)*(1-(b/Bm)**y))
#p0 = [23987.20423,12345.20423,.34,1,1.5]
popt1, pcov1 = curve_fit(func2, x1, y1, #p0) # Call the module for fit
print(popt1) # Print values in terminal
plt.plot(x1, y1,'b*', label='Data 2-Step') # Plot data
plt.plot(x1, func2(x1, *popt1) , 'k--') # plot fit
plt.xlabel('Field(T)') # name of x label
plt.ylabel('Jc (A/cm^2)') # name of y label
plt.title('Jc at 77K') # name of graph
plt.legend() # legends
plt.grid(True) # grid ?
plt.tight_layout() # to look good
plt.savefig('fitJc77K.png') # saves image
plt.show() # plot
# H(T) J (A/cm^2)
# 0 -0.000625 53831.204230
# 1 0.099721 48796.160850
# 2 0.200126 40844.694180
# 3 0.300065 32798.048680
# 4 0.400528 27049.519580
# 5 0.500758 23531.580950
# 6 0.600639 21316.537570
# 7 0.701246 19782.205290
# 8 0.801331 18735.695240
# 9 0.901387 17920.634920
# 10 1.001922 16973.087830
# 11 1.101876 16800.300530
# 12 1.202004 16390.637040
# 13 1.302481 15986.217990
# 14 1.402464 15600.469840
# 15 1.502709 15291.792590
# 16 1.603156 14915.839150
# 17 1.703227 14638.497350
# 18 1.803399 14316.516400
# 19 1.903890 13946.912170
# 20 2.003945 13532.901590
# 21 2.104074 13162.857140
# 22 2.204638 12655.183070
# 23 2.304548 12171.305820
# 24 2.404604 11015.720630
# 25 2.505168 11324.922750
# 26 2.605005 10736.685710
# 27 2.705279 10095.360850
# 28 2.805771 9444.359790
# 29 2.905536 8200.046140
# 30 3.005882 8263.225400
# 31 3.106156 7593.037460
# 32 3.206066 6908.264970
# 33 3.306484 6322.300950
# 34 3.406685 5758.770370
# 35 3.506523 4535.793440
# 36 3.607015 3777.405290
# 37 3.707288 4103.506880
# 38 3.807199 3611.456510
# 39 3.907544 3186.223490
# 40 4.007819 2081.366770
# 41 4.107584 1669.132320
# 42 4.208147 1527.562750
# 43 4.308421 1374.063490
# 44 4.408404 1500.655660
# 45 4.509114 1260.510480
# 46 4.609096 1081.811560
# 47 4.708935 726.452490
# 48 4.809935 846.458580
# 49 4.910063 410.075915
# 50 5.010046 93.699340
# 51 5.110609 65.085630
# 52 5.210593 33.309590
# 53 5.310866 10.023870
# 54 5.411431 8.439660
# 55 5.511414 0.161990
# 56 5.611542 1.470010
# 57 5.711960 -0.574940
# 58 5.812089 1.173210
# 59 5.912217 -1.641310
Answers:
Part of your problem is that you weren’t able to find good initial parameters. More of your problem is that you didn’t provide sane bounds, and bounds are crucial to prevent divide-by-zero and negative powers. You must disallow negative magnetic field values for the same reason.
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
field, density = np.array((
[-0.000625, 53831.204230],
[ 0.099721, 48796.160850],
[ 0.200126, 40844.694180],
[ 0.300065, 32798.048680],
[ 0.400528, 27049.519580],
[ 0.500758, 23531.580950],
[ 0.600639, 21316.537570],
[ 0.701246, 19782.205290],
[ 0.801331, 18735.695240],
[ 0.901387, 17920.634920],
[ 1.001922, 16973.087830],
[ 1.101876, 16800.300530],
[ 1.202004, 16390.637040],
[ 1.302481, 15986.217990],
[ 1.402464, 15600.469840],
[ 1.502709, 15291.792590],
[ 1.603156, 14915.839150],
[ 1.703227, 14638.497350],
[ 1.803399, 14316.516400],
[ 1.903890, 13946.912170],
[ 2.003945, 13532.901590],
[ 2.104074, 13162.857140],
[ 2.204638, 12655.183070],
[ 2.304548, 12171.305820],
[ 2.404604, 11015.720630],
[ 2.505168, 11324.922750],
[ 2.605005, 10736.685710],
[ 2.705279, 10095.360850],
[ 2.805771, 9444.359790],
[ 2.905536, 8200.046140],
[ 3.005882, 8263.225400],
[ 3.106156, 7593.037460],
[ 3.206066, 6908.264970],
[ 3.306484, 6322.300950],
[ 3.406685, 5758.770370],
[ 3.506523, 4535.793440],
[ 3.607015, 3777.405290],
[ 3.707288, 4103.506880],
[ 3.807199, 3611.456510],
[ 3.907544, 3186.223490],
[ 4.007819, 2081.366770],
[ 4.107584, 1669.132320],
[ 4.208147, 1527.562750],
[ 4.308421, 1374.063490],
[ 4.408404, 1500.655660],
[ 4.509114, 1260.510480],
[ 4.609096, 1081.811560],
[ 4.708935, 726.452490],
[ 4.809935, 846.458580],
[ 4.910063, 410.075915],
[ 5.010046, 93.699340],
[ 5.110609, 65.085630],
[ 5.210593, 33.309590],
[ 5.310866, 10.023870],
[ 5.411431, 8.439660],
[ 5.511414, 0.161990],
[ 5.611542, 1.470010],
[ 5.711960, -0.574940],
[ 5.812089, 1.173210],
[ 5.912217, -1.641310],
)).T
field = np.clip(field, a_min=0, a_max=None)
def func2(b: np.ndarray,
Jc1: float, Bl: float,
Jc2: float, Bm: float, y: float) -> np.ndarray:
return (
Jc1*np.exp(-b/Bl) +
Jc2*b/Bm*np.exp(
(1 - (b/Bm)**y)/y
)
)
p0 = (5e4, 2, -1e4, 1, 2)
popt1, _ = curve_fit(
f=func2,
xdata=field, ydata=density,
p0=p0,
bounds=(
(-np.inf, 0, -np.inf, 0, 0.1),
( np.inf, 10, np.inf, 10, 10.0),
),
)
print(popt1)
fig, ax = plt.subplots()
ax.scatter(field, density, marker='+', label='Data 2-Step')
ax.plot(field, func2(field, *p0), label='p0')
ax.plot(field, func2(field, *popt1), label='k--')
ax.set_xlabel('Field (T)')
ax.set_ylabel('Jc (A/cm²)')
ax.set_title('Jc at 77°K')
ax.legend()
ax.grid(True)
plt.show()
[ 5.55993112e+04 1.44996434e+00 -1.54454798e+04 5.87121864e-01
1.71151312e+00]
I’m trying to fit some data into a function, but scipy curve_fit gives me the following message: "OptimizeWarning: Covariance of the parameters could not be estimated" and won’t plot or print the parameters of the fit. I believe the power inside the exp function might be the problem, but I don’t know how to work around this error. I tried changing the initial parameters but it won’t help. Here’s the data and the function I want to fit.
Data below as comment format to paste in excel
Thanks in advance.
import pandas as pd #Module for reading data
import numpy as np #Math module
import matplotlib.pyplot as plt #Plot module
from scipy.optimize import curve_fit #Curve fit module
var = pd.read_excel("Data.xlsx") # Read in same directory as file
print(var) #Print File
x1 = list(var['H(T)']) # x values
y1 = list(var['J (A/cm^2)']) # y values
######### FUNCTION ########
def func2(b, Jc1, Jc2, Bl, Bm, y):
return Jc1*np.exp(-b/Bl)+Jc2*(b/Bm)*np.exp((1/y)*(1-(b/Bm)**y))
#p0 = [23987.20423,12345.20423,.34,1,1.5]
popt1, pcov1 = curve_fit(func2, x1, y1, #p0) # Call the module for fit
print(popt1) # Print values in terminal
plt.plot(x1, y1,'b*', label='Data 2-Step') # Plot data
plt.plot(x1, func2(x1, *popt1) , 'k--') # plot fit
plt.xlabel('Field(T)') # name of x label
plt.ylabel('Jc (A/cm^2)') # name of y label
plt.title('Jc at 77K') # name of graph
plt.legend() # legends
plt.grid(True) # grid ?
plt.tight_layout() # to look good
plt.savefig('fitJc77K.png') # saves image
plt.show() # plot
# H(T) J (A/cm^2)
# 0 -0.000625 53831.204230
# 1 0.099721 48796.160850
# 2 0.200126 40844.694180
# 3 0.300065 32798.048680
# 4 0.400528 27049.519580
# 5 0.500758 23531.580950
# 6 0.600639 21316.537570
# 7 0.701246 19782.205290
# 8 0.801331 18735.695240
# 9 0.901387 17920.634920
# 10 1.001922 16973.087830
# 11 1.101876 16800.300530
# 12 1.202004 16390.637040
# 13 1.302481 15986.217990
# 14 1.402464 15600.469840
# 15 1.502709 15291.792590
# 16 1.603156 14915.839150
# 17 1.703227 14638.497350
# 18 1.803399 14316.516400
# 19 1.903890 13946.912170
# 20 2.003945 13532.901590
# 21 2.104074 13162.857140
# 22 2.204638 12655.183070
# 23 2.304548 12171.305820
# 24 2.404604 11015.720630
# 25 2.505168 11324.922750
# 26 2.605005 10736.685710
# 27 2.705279 10095.360850
# 28 2.805771 9444.359790
# 29 2.905536 8200.046140
# 30 3.005882 8263.225400
# 31 3.106156 7593.037460
# 32 3.206066 6908.264970
# 33 3.306484 6322.300950
# 34 3.406685 5758.770370
# 35 3.506523 4535.793440
# 36 3.607015 3777.405290
# 37 3.707288 4103.506880
# 38 3.807199 3611.456510
# 39 3.907544 3186.223490
# 40 4.007819 2081.366770
# 41 4.107584 1669.132320
# 42 4.208147 1527.562750
# 43 4.308421 1374.063490
# 44 4.408404 1500.655660
# 45 4.509114 1260.510480
# 46 4.609096 1081.811560
# 47 4.708935 726.452490
# 48 4.809935 846.458580
# 49 4.910063 410.075915
# 50 5.010046 93.699340
# 51 5.110609 65.085630
# 52 5.210593 33.309590
# 53 5.310866 10.023870
# 54 5.411431 8.439660
# 55 5.511414 0.161990
# 56 5.611542 1.470010
# 57 5.711960 -0.574940
# 58 5.812089 1.173210
# 59 5.912217 -1.641310
Part of your problem is that you weren’t able to find good initial parameters. More of your problem is that you didn’t provide sane bounds, and bounds are crucial to prevent divide-by-zero and negative powers. You must disallow negative magnetic field values for the same reason.
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
field, density = np.array((
[-0.000625, 53831.204230],
[ 0.099721, 48796.160850],
[ 0.200126, 40844.694180],
[ 0.300065, 32798.048680],
[ 0.400528, 27049.519580],
[ 0.500758, 23531.580950],
[ 0.600639, 21316.537570],
[ 0.701246, 19782.205290],
[ 0.801331, 18735.695240],
[ 0.901387, 17920.634920],
[ 1.001922, 16973.087830],
[ 1.101876, 16800.300530],
[ 1.202004, 16390.637040],
[ 1.302481, 15986.217990],
[ 1.402464, 15600.469840],
[ 1.502709, 15291.792590],
[ 1.603156, 14915.839150],
[ 1.703227, 14638.497350],
[ 1.803399, 14316.516400],
[ 1.903890, 13946.912170],
[ 2.003945, 13532.901590],
[ 2.104074, 13162.857140],
[ 2.204638, 12655.183070],
[ 2.304548, 12171.305820],
[ 2.404604, 11015.720630],
[ 2.505168, 11324.922750],
[ 2.605005, 10736.685710],
[ 2.705279, 10095.360850],
[ 2.805771, 9444.359790],
[ 2.905536, 8200.046140],
[ 3.005882, 8263.225400],
[ 3.106156, 7593.037460],
[ 3.206066, 6908.264970],
[ 3.306484, 6322.300950],
[ 3.406685, 5758.770370],
[ 3.506523, 4535.793440],
[ 3.607015, 3777.405290],
[ 3.707288, 4103.506880],
[ 3.807199, 3611.456510],
[ 3.907544, 3186.223490],
[ 4.007819, 2081.366770],
[ 4.107584, 1669.132320],
[ 4.208147, 1527.562750],
[ 4.308421, 1374.063490],
[ 4.408404, 1500.655660],
[ 4.509114, 1260.510480],
[ 4.609096, 1081.811560],
[ 4.708935, 726.452490],
[ 4.809935, 846.458580],
[ 4.910063, 410.075915],
[ 5.010046, 93.699340],
[ 5.110609, 65.085630],
[ 5.210593, 33.309590],
[ 5.310866, 10.023870],
[ 5.411431, 8.439660],
[ 5.511414, 0.161990],
[ 5.611542, 1.470010],
[ 5.711960, -0.574940],
[ 5.812089, 1.173210],
[ 5.912217, -1.641310],
)).T
field = np.clip(field, a_min=0, a_max=None)
def func2(b: np.ndarray,
Jc1: float, Bl: float,
Jc2: float, Bm: float, y: float) -> np.ndarray:
return (
Jc1*np.exp(-b/Bl) +
Jc2*b/Bm*np.exp(
(1 - (b/Bm)**y)/y
)
)
p0 = (5e4, 2, -1e4, 1, 2)
popt1, _ = curve_fit(
f=func2,
xdata=field, ydata=density,
p0=p0,
bounds=(
(-np.inf, 0, -np.inf, 0, 0.1),
( np.inf, 10, np.inf, 10, 10.0),
),
)
print(popt1)
fig, ax = plt.subplots()
ax.scatter(field, density, marker='+', label='Data 2-Step')
ax.plot(field, func2(field, *p0), label='p0')
ax.plot(field, func2(field, *popt1), label='k--')
ax.set_xlabel('Field (T)')
ax.set_ylabel('Jc (A/cm²)')
ax.set_title('Jc at 77°K')
ax.legend()
ax.grid(True)
plt.show()
[ 5.55993112e+04 1.44996434e+00 -1.54454798e+04 5.87121864e-01
1.71151312e+00]