Fitting data with an exponential law

Question

I'd like to fit some data with an exponential function. I used scipy.optimize.curve_fit because I already used it for other fits. This time, there is an issue and I can't figure out what's wrong.

Here is what the data looks like when plotted : data.png

as you see it seems to follow an exponential law.

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

data = np.array([ 
    0.,  1.93468444,  3.69735865,  5.38185988,  6.02549022,
    6.69199075,  7.72316694,  8.08913061,  8.84570241,  8.69711608,
    8.80038144,  9.78951087,  9.68486674, 10.06175145, 10.44039495,
   10.0481156 ,  9.76656204,  9.88581457,  9.81805445, 10.42432252,
   10.41102239, 11.2911395 ,  9.64866184,  9.98072231, 10.83644694,
   10.24748571, 10.81333209, 10.75949899, 10.90367328, 10.42446764,
   10.51441017, 10.73047737, 10.8159758 , 10.51013538, 10.02862504,
    9.76352112, 10.64829309, 10.6293347 , 10.67752596, 10.34801542,
   10.53158576, 10.92883362, 10.67002314, 10.37015825, 10.74876349,
   10.12821343, 10.8974205 , 10.1591103 , 10.588377  , 11.92134556,
   10.309095  , 11.1174362 , 10.72654524, 10.60890374, 10.37456491,
   10.05935346, 11.21295863, 11.09013951, 10.60862773, 11.2558922 ,
   11.24660234, 10.35981557, 10.81284365, 10.96113067, 10.22716439,
    9.8394873 , 10.01892084, 10.38237311, 10.04920671, 10.87782442,
   10.42438756, 10.05614503, 10.5446946 ,  9.99974368, 10.76930547,
   10.22164072, 10.36942999, 10.89888302, 10.47035428, 10.58157374,
   11.12615892, 11.30866718, 10.33215937, 10.46723351, 10.54072701,
   11.45027197, 10.45895588, 10.34176601, 10.78405493, 10.43964778,
   10.34047484, 10.25099046, 11.05847515, 10.27408195, 10.27529163,
   10.16568845, 10.86451738, 10.73205291, 10.73300649, 10.49463959,
   10.03729782
])

t = np.linspace(0, 100, len(data)) #time array

def expo(x, a, b, c): #exponential function for fitting
   return a * np.exp(b * x) + c

fig1, ax1 = plt.subplots()
ax1.plot(t, data, ".", label="data")
coefs = curve_fit(expo, t, data)[0] # fitting
ax1.plot(t, expo(t, coefs[0], coefs[1], coefs[2]), "-", label="fit")
ax1.legend()
plt.show()

The problem is that curve_fit() returns very big or very small coefficients a,b and c while it should return something more like a = -10.5, b = -0.2, c = 10.5

Answer 1

The fitting process works by finding a local minimum of a loss function. If the problem is unconstrained, there may be several such local minima, each giving different values of parameters, and you may get a different one than the one that you are expecting.

If you have a guess what the parameters should be, you can provide it to narrow the search:

# with an initial guess for values of a, b, c 
coefs = curve_fit(expo, t, data, p0=[-10, -1, 10])[0]

The coefficients it produces are:

array([-10.48815244,  -0.2091102 ,  10.56699883])

Alternatively, you can specify bonds for the parameters:

# with lower and upper bounds for a, b, c
coefs = curve_fit(expo, t, data, bounds=([-20, -2, 0], [-10, 2, 20]))[0]

This gives the same results as above.