Scipy sigmoid curve fitting

Question

I have some data points and would like to find a fitting function, I guess a cumulative Gaussian sigmoid function would fit, but I don't really know how to realize that.

This is what I have right now:

import numpy as np
import pylab
from scipy.optimize
import curve_fit

def sigmoid(x, a, b):
     y = 1 / (1 + np.exp(-b*(x-a)))
     return y

xdata = np.array([400, 600, 800, 1000, 1200, 1400, 1600])
ydata = np.array([0, 0, 0.13, 0.35, 0.75, 0.89, 0.91])

popt, pcov = curve_fit(sigmoid, xdata, ydata)
print(popt)

x = np.linspace(-1, 2000, 50)
y = sigmoid(x, *popt)

pylab.plot(xdata, ydata, 'o', label='data')
pylab.plot(x,y, label='fit')
pylab.ylim(0, 1.05)
pylab.legend(loc='best')
pylab.show()

But I get the following warning:

.../scipy/optimize/minpack.py:779: OptimizeWarning: Covariance of the parameters could not be estimated category=OptimizeWarning)

Can anyone help? I'm also open for any other possibilities to do it! I just need a curve fit in any way to this data.

Answer 1

You may have noticed the resulting fit is completely incorrect. Try passing some decent initial parameters to curve_fit, with the p0 argument:

popt, pcov = curve_fit(sigmoid, xdata, ydata, p0=[1000, 0.001])

should give a much better fit, and probably no warning either.

(The default starting parameters are [1, 1]; that is too far from the actual parameters to obtain a good fit.)

Answer 2

You could set some reasonable bounds for parameters, for example, doing

def fsigmoid(x, a, b):
    return 1.0 / (1.0 + np.exp(-a*(x-b)))

popt, pcov = curve_fit(fsigmoid, xdata, ydata, method='dogbox', bounds=([0., 600.],[0.01, 1200.]))

I've got output

[7.27380294e-03 1.07431197e+03]

and curve looks like

First point at (400,0) was removed as useless. You could add it, though result won't change much...

UPDATE

Note, that bounds are set as ([low_a,low_b],[high_a,high_b]), so I asked for scale to be within [0...0.01] and location to be within [600...1200]