Python - 最小化卡方

Question

I have been trying to fit a linear model to a set of stress/strain data by minimizing chi-squared. Unfortunately using the code below is not correctly minimizing the chisqfunc function. It is finding the minimum at the initial conditions, x0 , which is not correct. I have looked through the scipy.optimize documentation and tested minimizing other functions which has worked correctly. Could you please suggest how to fix the code below or suggest another method I can use to fit a linear model to data by minimizing chi-squared?

import numpy
import scipy.optimize as opt

filename = 'data.csv'

data = numpy.loadtxt(open(filename,"r"),delimiter=",")

stress = data[:,0]
strain = data[:,1]
err_stress = data[:,2]

def chisqfunc((a, b)):
    model = a + b*strain
    chisq = numpy.sum(((stress - model)/err_stress)**2)
    return chisq

x0 = numpy.array([0,0])

result =  opt.minimize(chisqfunc, x0)
print result

Thank you for reading my question and any help would be greatly appreciated.

Cheers, Will

EDIT: Data set I am currently using: Link to data

Answer 1

The problem is that your initial guess is very far from the actual solution. If you add a print statement inside chisqfunc() like print (a,b) , and rerun your code, you'll get something like:

(0, 0)
(1.4901161193847656e-08, 0.0)
(0.0, 1.4901161193847656e-08)

This means that minimize evaluates the function only at these points.

if you now try to evaluate chisqfunc() at these 3 pairs of values, you'll see that they EXACTLY match, for example

print chisqfunc((0,0))==chisqfunc((1.4901161193847656e-08,0))
True

This happens because of rounding floating points arithmetics. In other words, when evaluating stress - model , the var stress is too many order of magnitude larger than model , and the result is truncated.

One could then just try bruteforcing it, increasing floating point precision, with writing data=data.astype(np.float128) just after loading the data with loadtxt . minimize fails, with result.success=False , but with a helpful message

Desired error not necessarily achieved due to precision loss.

One possibility is then to provide a better initial guess, so that in the subtraction stress - model the model part is of the same order of magnitude, the other to rescale the data, so that the solution will be closer to your initial guess (0,0) .

It is MUCH better if you just rescale the data, making for example nondimensional with respect to a certain stress value (like the yelding/cracking of this material)

This is an example of the fitting, using as a stress scale the maximum measured stress. There are very few changes from your code:

import numpy
import scipy.optimize as opt

filename = 'data.csv'

data = numpy.loadtxt(open(filename,"r"),delimiter=",")

stress = data[:,0]
strain = data[:,1]
err_stress = data[:,2]


smax = stress.max()
stress = stress/smax
#I am assuming the errors err_stress are in the same units of stress.
err_stress = err_stress/smax

def chisqfunc((a, b)):
    model = a + b*strain
    chisq = numpy.sum(((stress - model)/err_stress)**2)
    return chisq

x0 = numpy.array([0,0])

result =  opt.minimize(chisqfunc, x0)
print result
assert result.success==True
a,b=result.x*smax
plot(strain,stress*smax)
plot(strain,a+b*strain)

Your linear model is quite good, ie your material has a very linear behaviour for this range of deformation (what material is it anyway?):