The curve corresponds to the parameters

Question

I have experimental data:

xdata = [85,86,87,88,89,90,91,91.75,93,96,100,101,102,103,104,105,106,107.25,108.25,109,109.75,111,112,112.75,114,115.25,116,116.75,118,119.25,120,121,122,122.5,123.5,125.25,126,126.75,127.75,129.25,130.25,131,131.75,133,134.25,135,136,137,138,139,140,141,142,143,144,144.75,146,146.75,148,149.25,150,150.5,152,153.25,154,155,156.75,158,159,159.75,161,162,162.5,164,165,166] ydata = [0.2,0.21,0.18,0.21,0.19,0.2,0.21,0.2,0.18,0.204,0.208,0.2,0.21,0.25,0.2,0.19,0.216,0.22,0.224,0.26,0.229,0.237,0.22,0.246,0.25,0.264,0.29,0.274,0.29,0.3,0.27,0.32,0.38,0.348,0.372,0.398,0.35,0.42,0.444,0.48,0.496,0.55,0.51,0.54,0.57,0.51,0.605,0.57,0.65,0.642,0.6,0.66,0.7,0.688,0.69,0.705,0.67,0.717,0.69,0.728,0.75,0.736,0.73,0.744,0.72,0.76,0.752,0.74,0.76,0.7546,0.77,0.74,0.758,0.74,0.78,0.76] 

And the formula f(x) = m1 + m2 / (1 + e ^ (-m3*(x - m4))) . I need to find m1, m2, m3, m4 by the least squares method, where 0.05 m1 <0.3 0.3 <m2 <0.8 0.05 m3 <0.5 100 <m4 <200.

I am using curve_fit and my function is:

 def f(xdata, m1, m2, m3, m4): if m1 > 0.05 and m1 < 0.3 and \ m2 > 0.3 and m2 < 0.8 and \ m3 > 0.05 and m3 < 0.5 and \ m4 > 100 and m4 < 200: return m1 + (m2 * 1. / (1 + e ** (-m3 * (x - m4)))) return (abs(m1) + abs(m2) + abs(m3) + abs(m4)) * 1e14 # some large number 

But the program will return an error: RuntimeError: Optimal parameters not found: Number of calls to function has reached maxfev = 1000.

What to do?

 import numpy as np from scipy.optimize import curve_fit from math import e xdata = np.array([85,86,87,88,89,90,91,91.75,93,96,100,101,102,103,104,105,106,107.25,108.25,109,109.75,111,112,112.75,114,115.25,116,116.75,118,119.25,120,121,122,122.5,123.5,125.25,126,126.75,127.75,129.25,130.25,131,131.75,133,134.25,135,136,137,138,139,140,141,142,143,144,144.75,146,146.75,148,149.25,150,150.5,152,153.25,154,155,156.75,158,159,159.75,161,162,162.5,164,165,166])` ydata = np.array([0.2,0.21,0.18,0.21,0.19,0.2,0.21,0.2,0.18,0.204,0.208,0.2,0.21,0.25,0.2,0.19,0.216,0.22,0.224,0.26,0.229,0.237,0.22,0.246,0.25,0.264,0.29,0.274,0.29,0.3,0.27,0.32,0.38,0.348,0.372,0.398,0.35,0.42,0.444,0.48,0.496,0.55,0.51,0.54,0.57,0.51,0.605,0.57,0.65,0.642,0.6,0.66,0.7,0.688,0.69,0.705,0.67,0.717,0.69,0.728,0.75,0.736,0.73,0.744,0.72,0.76,0.752,0.74,0.76,0.7546,0.77,0.74,0.758,0.74,0.78,0.76]) def f(xdata, m1, m2, m3, m4): if m1 > 0.05 and m1 < 0.3 and \ m2 > 0.3 and m2 < 0.8 and \ m3 > 0.05 and m3 < 0.5 and \ m4 > 100 and m4 < 200: return m1 + (m2 * 1. / (1 + e ** (-m3 * (x - m4)))) return (abs(m1) + abs(m2) + abs(m3) + abs(m4)) * 1e14 print curve_fit(f, xdata, ydata) 

Answer 1

Set initial parameters for useful values:

 curve_fit(f, xdata, ydata, p0=(0.1, 0.5, 0.1, 150))) 

Also, use xdata instead of x in your f function:

 return m1 + (m2 * 1. / (1 + e ** (-m3 * (xdata - m4)))) 

This is my modified program:

 def f(xdata, m1, m2, m3, m4): if (0.05 < m1 < 0.3 and 0.3 < m2 < 0.8 and 0.05 < m3 < 0.5 and 100 < m4 < 200): return m1 + (m2 * 1. / (1 + e ** (-m3 * (xdata - m4)))) return 1e38 print(curve_fit(f, xdata, ydata, p0=(0.1, 0.5, 0.1, 150))) 

Result:

 (array([ 0.19567035, 0.56792559, 0.13434829, 129.98915877]), array([[ 2.94622909e-05, -3.96126279e-05, 1.99236054e-05, 7.48438125e-04], [ -3.96126279e-05, 9.24145662e-05, -4.62302643e-05, 5.04671621e-04], [ 1.99236054e-05, -4.62302643e-05, 3.77364832e-05, -2.43866126e-04], [ 7.48438125e-04, 5.04671621e-04, -2.43866126e-04, 1.34700612e-01]])) 

Answer 2

Alternatively, you can also use lmfit , which makes it easy to set boundaries and avoid the "ugly" if statement in your function. You get the following options:

 m1: 0.19567033 +/- 0.005427 (2.77%) (init= 0.1) m2: 0.56792558 +/- 0.009613 (1.69%) (init= 0.5) m3: 0.13434829 +/- 0.006143 (4.57%) (init= 0.2) m4: 129.989156 +/- 0.367009 (0.28%) (init= 150) 

and the result is as follows:

Here is the whole code with a few comments; let me know if you have further questions:

 from lmfit import minimize, Parameters, Parameter, report_fit import numpy as np xdata = np.array([85,86,87,88,89,90,91,91.75,93,96,100,101,102,103,104,105,106,107.25,108.25,109,109.75,111,112,112.75,114,115.25,116,116.75,118,119.25,120,121,122,122.5,123.5,125.25,126,126.75,127.75,129.25,130.25,131,131.75,133,134.25,135,136,137,138,139,140,141,142,143,144,144.75,146,146.75,148,149.25,150,150.5,152,153.25,154,155,156.75,158,159,159.75,161,162,162.5,164,165,166]) ydata = np.array([0.2,0.21,0.18,0.21,0.19,0.2,0.21,0.2,0.18,0.204,0.208,0.2,0.21,0.25,0.2,0.19,0.216,0.22,0.224,0.26,0.229,0.237,0.22,0.246,0.25,0.264,0.29,0.274,0.29,0.3,0.27,0.32,0.38,0.348,0.372,0.398,0.35,0.42,0.444,0.48,0.496,0.55,0.51,0.54,0.57,0.51,0.605,0.57,0.65,0.642,0.6,0.66,0.7,0.688,0.69,0.705,0.67,0.717,0.69,0.728,0.75,0.736,0.73,0.744,0.72,0.76,0.752,0.74,0.76,0.7546,0.77,0.74,0.758,0.74,0.78,0.76]) def fit_fc(params, x, data): m1 = params['m1'].value m2 = params['m2'].value m3 = params['m3'].value m4 = params['m4'].value model = m1 + (m2 * 1. / (1 + np.exp(-m3 * (x - m4)))) return model - data #that what you want to minimize # create a set of Parameters # 'value' is the initial condition # 'min' and 'max' define your boundaries params = Parameters() params.add('m1', value= 0.1, min=0.05, max=0.3) params.add('m2', value= 0.5, min=0.3, max=0.8) params.add('m3', value= 0.2, min=0.05, max=0.5) params.add('m4', value= 150.0, min=100, max=200) # do fit, here with leastsq model result = minimize(fit_fc, params, args=(xdata, ydata)) # calculate final result final = ydata + result.residual # write error report report_fit(params) #plot results try: import pylab pylab.plot(xdata, ydata, 'k+') pylab.plot(xdata, final, 'r') pylab.show() except: pass