scipy curve_fit doesn't work well

Question

I am trying to fit some data using the following code:

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

def fseries(x, a0, a1, b1, w):
    f = a0 + (a1 * np.cos(w * x)) + (b1 * np.sin(w * x))
    return f

x = np.arange(0, 10)
y = [-45.0, -17.0, -33.0,  50.0, 48.0,  -3.0,  -1.0,   2.0,  84.0, 71.0]

res = scipy.optimize.curve_fit(fseries, x, y, maxfev=10000)

xt = np.linspace(0, 10, 100)
yt = fseries(xt, res[0][0], res[0][1], res[0][2], res[0][3])

plt.plot(x,y)
plt.plot(xt, yt, 'r')
plt.show()

Which makes this plot:

Any thoughts on what I am not understanding or doing wrong?

Answer 1

First of all, curve fitting is not a magical device that creates a good curve for any given data set. You can't fit an exponential curve well to a logarithmic data set. If you look at your data, does it look like it is well described by the function you define? Doesn't it rather look like an overlay of a linear and a sine function?
Then curve fitting is an iterative process, that is highly dependent on start values. From the scipy manual :

p0 : None, scalar, or N-length sequence, optional Initial guess for the parameters. If None, then the initial values will all be 1

Why not provide a better guess for p0 ?
Last but not least, you get back two arrays. I would read out both, even if you only need one. It simplifies your code. Try

p0 = (10, 20, 20, 1.5)
res, _popcv = scipy.optimize.curve_fit(fseries, x, y, p0, maxfev=10000)
xt = np.linspace(0, 10, 100)
yt = fseries(xt, *res)

and you get already a better fit.
You can improve the fit further, when you define a better fit function with

def fseries(x, a0, a1, b1, w):
    f = a0 * x + (a1 * np.cos(w * x)) + (b1 * np.sin(w * x))
    return f

Whether this function is useful, you have to decide. Just because it fits better the data set, doesn't mean it is the right descriptor in your situation.