SciPy curve_fit not working when one of the parameters to fit is a power

问题

I'm trying to fit my data to a user defined function using SciPy curve_fit, which works when fitting to a function with a fixed power (func1). But curve_fit does not work when the function contains a power as a parameter to fit to (func2).

Curve_fit still does not work if I provide an initial guess for the parameters usins the keyword p0. I can not use the bounds keyword as the version of SciPy which I have does not have it.

This script illustrates the point:

import scipy
from scipy.optimize import curve_fit
import sys

print 'scipy version: ', scipy.__version__
print 'np.version:    ', np.__version__
print sys.version_info

def func1(x,a):
    return (x-a)**3.0 

def func2(x,a,b):  
    return (x-a)**b

x_train = np.linspace(0, 12, 50)
y       = func2(x_train, 0.5, 3.0)
y_train = y + np.random.normal(size=len(x_train))

print 'dtype of x_train: ', x_train.dtype
print 'dtype of y_train: ', y_train.dtype

popt1, pcov1 = curve_fit( func1, x_train, y_train, p0=[0.6] )
popt2, pcov2 = curve_fit( func2, x_train, y_train, p0=[0.6, 4.0] )

print 'Function 1: ', popt1, pcov1
print 'Function 2: ', popt2, pcov2

Which outputs the following:

scipy version:  0.14.0
np.version:     1.8.2
sys.version_info(major=2, minor=7, micro=6, releaselevel='final', serial=0)
dtype of x_train:  float64
dtype of y_train:  float64
stack_overflow.py:14: RuntimeWarning: invalid value encountered in power
return (x-a)**b
Function 1:  [ 0.50138759] [[  3.90044196e-07]]
Function 2:  [ nan  nan] [[ inf  inf]
 [ inf  inf]]

回答1:

(As @xnx first commented,) the problem with the second formulation (where the exponent b is unknown and considered to be real-valued) is that, in the process of testing potential values for a and b, quantities of the form z**p need to be evaluated, where z is a negative real number and p is a non-integer. This quantity is complex in general, hence the procedure fails. For example, for x=0 and test variables a=0.5, b=4.1, it holds (x-a)**b = (-0.5)**4.1 = 0.0555+0.018j.

来源：https://stackoverflow.com/questions/39046818/scipy-curve-fit-not-working-when-one-of-the-parameters-to-fit-is-a-power