scipy.optimize.curve_fit producing nonsensical curve fits

Question

I am trying to fit a curve to some generated data that resemble an exponential function when plotted. I am using scipy.optimize.curve_fit as it seems like it is the best (and best documented) for the job. The actual data are newly generated each time I run the code, but here is an example set:

import pandas
import scipy.optimize as opt

x1 = [0.4145392937447818, 0.7807888116968482, 0.7903528929788539, 
1.5081613036989836, -0.295895237606155, -0.0855307279546107, 
1.0523973736479486, -0.6967509832843239, -0.30499200990688413, 
1.1990545631966807, -1.270460772249312, 0.9531042718153095, 1.5747175535222993, 
-0.6483709650867473, 0.47820180254528477, 1.14266851615097, 0.6237953640100202, 
0.0664027559951128, 0.877280002485417, 0.9432317053343211, 1.0367424879878504, 
-0.6410400513164749, 1.667835241401498, -0.20484029870424125, 
2.887026948755316]

y1 = [0.718716626591187, 0.579938466590508, 0.722005637974309, 
1.61842778379047, 0.331301712743162, 0.342649242449043, 1.14950611092907, 
0.299221762023701, 0.345063839940754, 1.08398125906313, 0.315433168226251, 
1.3343730617376, 1.32514210008176, 0.308702648499771, 0.495749985226691, 
0.406025683910759, 0.445087968405107, 0.423578575247177, 0.816264419038205, 
1.16110461165631, 1.81572974380867, 0.420890068255763, 0.821468286117842, 
0.416275933630732, 4.7877353794036]

data = pandas.DataFrame({"Pi_values": x1, 
                         "CO2_at_solubility": y1})

Then, I do the curve fitting business...

##Define curve fitting
def func(x, m, c, c0):
    return c0 + m**x * c

#draw the figure
fig, ax1 = plt.subplots()
plt.xlabel('Pi Parameter')
plt.ylabel('CO2 wt%')

#plot generated data
#tried converting pandas columns to np arrays based on an issue another user was having, but it does not help
x1 = data["Pi_values"].values
y1 = data["CO2_at_solubility"].values

# Curve fitting with scipy.optimize.curve_fit
popt, pcov = opt.curve_fit(func, x1, y1)
# Use the optimized parameters to plot the best fit
plt.plot(x1, y1, 'o', x1, func(x1, *popt))

And here is the very weird result. No matter what form of the equation I try in fun, if it is able to fit any "curve" it looks like this mess:

Or this mess...

Any idea what could be going on here? I've not been able to find any other examples like this. I'm running python3.5 in a jupyter notebook.

Other things I tried that didn't work: other forms of the equation; other equations; changing initial guess values; scaling values in case y values were too small.

Answer 1

You just need to sort the x values with

data.sort_values(by='Pi_values', ascending=True, inplace=True)

before curve_fit:

x1 = data["Pi_values"].values
y1 = data["CO2_at_solubility"].values
# Curve fitting with scipy.optimize.curve_fit
popt, pcov = opt.curve_fit(func, x1, y1)
# Use the optimized parameters to plot the best fit
plt.plot(x1, y1, 'o', x1, func(x1, *popt))

Answer 2

The elements of the x axis need to be sorted when plotting.

Example:

x1, y1 = zip(*sorted(zip(x1, y1)))
# Curve fitting with scipy.optimize.curve_fit
popt, pcov = opt.curve_fit(func, x1, y1)
# Use the optimized parameters to plot the best fit
plt.plot(x1, y1, 'o', x1, func(x1, *popt))

which results in: