Scipy.optimize.curve_fit will not comply with the cosine power law

Question

For several hours, I have been trying to fit the model to a (generated) dataset like casus for the problem I was struggling with. I generated basic functions for the function f (x) = A * cos ^ n (x) + b and added some noise. When I try to set a dataset using this function and curve_fit, I get an error

./tester.py:10: RuntimeWarning: invalid value encountered in power
return Amp*(np.cos(x))**n + b
/usr/lib/python2.7/dist-packages/scipy/optimize/minpack.py:690: OptimizeWarning: Covariance of the parameters could not be estimated  category=OptimizeWarning)

The code that I use to create datapoints and fit the model is as follows:

#!/usr/bin/env python

from __future__ import print_function
import numpy as np
from scipy.optimize import curve_fit
from matplotlib.pyplot import figure, show, rc, plot

def f(x, Amp, n, b):

    return np.real(Amp*(np.cos(x))**n + b)

x = np.arange(0, 6.28, 0.01)
randomPart = np.random.rand(len(x))-0.5
fig = figure()
sample = f(x, 5, 2, 5)+randomPart
frame = fig.add_subplot(1,1,1)

frame.plot(x, sample, label="Sample measurements")

popt, pcov = curve_fit(f, x, sample, p0=(1,1,1))

modeldata = f(x, popt[0], popt[1], popt[2])
print(modeldata)
frame.plot(x, modeldata, label="Best fit")

frame.legend()
frame.set_xlabel("x")
frame.set_ylabel("y")

show()

Noisy data is displayed - see image below.

- , ? , - , , . , curve_fit numpy python p0 . scipy, , scipy 0.17.0-1.

Answer 1

:

>>> (-2)**1.1
(-2.0386342710747223-0.6623924280875919j)
>>> np.array(-2)**1.1
__main__:1: RuntimeWarning: invalid value encountered in power
nan

python, numpy doubles , :

>>> np.sqrt(-1)
__main__:1: RuntimeWarning: invalid value encountered in sqrt
nan

np.abs , , . , ( ) , -op (: , , . ).

def f(x, Amp, n, b):

    return Amp*(np.abs(np.cos(x)))**n + b # only change here

:

(4.96482314, 2.03690954, 5.03709923]) (5,2,5).

---

, , (duh). , , , , , . , , cos(x)^n, cos(x), , n - , . Diophantine, .

( ) : , , :

def f(x, Amp, n, b):

    return Amp*np.abs(np.cos(x.astype(np.complex128))**n) + b

, , , , , . :

(5.02849409, 1.97655728, 4.96529108). . , ( np.abs), -0.37, , .

, - , . 2, , . , - . popt, , . , currying, .

from __future__ import print_function
import numpy as np
from scipy.optimize import curve_fit
from matplotlib.pyplot import subplots, show

def f_aux(x, Amp, n, b):
    return Amp*np.abs(np.cos(x.astype(np.complex128))**n) + b

def f_real(x, Amp, n, b):
    return Amp*np.cos(x)**n + b


x = np.arange(0, 2*np.pi, 0.01)  # pi
randomPart = np.random.rand(len(x)) - 0.5
sample = f(x, 5, 2, 5) + randomPart

fig,(frame_aux,frame) = subplots(ncols=2)
for fr in frame_aux,frame:
    fr.plot(x, sample, label="Sample measurements")
    fr.legend()
    fr.set_xlabel("x")
    fr.set_ylabel("y")

# auxiliary fit for n value
popt_aux, pcov_aux = curve_fit(f_aux, x, sample, p0=(1,1,1))

modeldata = f(x, *popt_aux)
#print(modeldata)
print('Auxiliary fit parameters: {}'.format(popt_aux))
frame_aux.plot(x, modeldata, label="Auxiliary fit")

# check visually, test if it close to an integer, but otherwise
n = np.round(popt_aux[1])

# actual fit with integral exponent
popt, pcov = curve_fit(lambda x,Amp,b,n=n: f_real(x,Amp,n,b), x, sample, p0=(popt_aux[0],popt_aux[2]))

modeldata = f(x, popt[0], n, popt[1])
#print(modeldata)
print('Final fit parameters: {}'.format([popt[0],n,popt[1]])) 
frame.plot(x, modeldata, label="Best fit")

frame_aux.legend()
frame.legend()

show()

, , . , , , :

:

Auxiliary fit parameters: [ 5.02628994  2.00886409  5.00652371]
Final fit parameters: [5.0288141074549699, 2.0, 5.0009730316739462]

: , .