Fitting two non-linear models to data

Question

Following the example is given in lmfit, I am trying to set up an example which is similar to my problem. My problem originally is that in my data I can fit two or three models, while my model is highly non-linear but it has for each model just a single free parameter.

My example similar to lmfit documentation:

x = np.linspace(0, 15, 301)
data = (5. * np.sin(2 * x - 0.1) * np.exp(-x*x*0.025) +(-2.6 * np.sin(-0.6 * x + 1.5) * np.exp(-x*x*3.0)+np.random.normal(size=len(x), scale=0.2) ))

def fcn2min(params, x, data):
    model=0
    for i in range(2):
        exec("amp_%d=%s"%(i+1,repr(params['amp_%d'%(i+1)].value)))
        exec("shift_%d=%s"%(i+1,repr(params['shift_%d'%(i+1)].value)))
        exec("omega_%d=%s"%(i+1,repr(params['omega_%d'%(i+1)].value)))
        exec("decay_%d=%s"%(i+1,repr(params['decay_%d'%(i+1)].value)))
        model += eval("amp_%d"%(i+1)) * np.sin(x * eval("omega_%d"%(i+1)) + eval("shift_%d"%(i+1))) * np.exp(-x*x*eval("decay_%d"%(i+1)))
    return (model-data)/data

params=Parameters()
for i in range(2):
    params.add('amp_%d'%(i+1),   value= 10,  vary=True, min=-3, max=3)
    params.add('decay_%d'%(i+1), value= 0.1,vary=True,min=0,max=4.)
    params.add('shift_%d'%(i+1), value= 0.0, vary=True,min=-np.pi, max=np.pi)
    params.add('omega_%d'%(i+1), value= 3.0, vary=True,min=-2.5, max=2.5)

result = minimize(fcn2min, params, args=(x, data),method='nelder')

The obtained rsults:

final = data + result.residual

# write error report
report_fit(params)
[[Variables]]
    amp_1:    -1.74789852 (init= 3)
    decay_1:   0.05493661 (init= 0.1)
    shift_1:   0.07807319 (init= 0)
    omega_1:  -2.00291964 (init= 2.5)
    amp_2:    -1.30857699 (init= 3)
    decay_2:   0.82303744 (init= 0.1)
    shift_2:  -0.04742474 (init= 0)
    omega_2:   2.44085535 (init= 2.5)
[[Correlations]] (unreported correlations are <  0.100)

The free parameters look completely off however on the final results plot it is clear it follows the distribution of data but the amplitudes are not quite right

try:
    import pylab
    pylab.plot(x, data, 'k+')
    pylab.plot(x, final, 'r')
    pylab.show()
except:
    pass

Any suggestion for the modification of the code in order to get the right results?

Answer 1

Ok, I think I found the issue. I am not sure about the purpose of the line

return (model-data)/data

but it should just be

return (model-data)

since that it what you want to minimize.

Furthermore, you should also choose initial values that are in the range. The modified code will result in the following output:

[[Variables]]
    amp_1:     5.23253723 (init= 10)
    decay_1:   0.02762246 (init= 0.1)
    shift_1:  -0.40774606 (init= 0)
    omega_1:   2.06744256 (init= 3)
    amp_2:     2.49467996 (init= 10)
    decay_2:   0.39205207 (init= 0.1)
    shift_2:   0.23347938 (init= 0)
    omega_2:  -0.71995187 (init= 3)
[[Correlations]] (unreported correlations are <  0.100)

Here is the entire code:

from lmfit import minimize, Parameters, Parameter, report_fit
import numpy as np

#http://cars9.uchicago.edu/software/python/lmfit/parameters.html

# create data to be fitted

x = np.linspace(0, 15, 301)
data = (5. * np.sin(2 * x - 0.1) * np.exp(-x*x*0.025) +
(-2.6 * np.sin(-0.6 * x + 1.5) * np.exp(-x*x*3.0)+np.random.normal(size=len(x), scale=0.2) ))

def fcn2min(params, x, data):
    model=0
    for i in range(2):
        exec("amp_%d=%s"%(i+1,repr(params['amp_%d'%(i+1)].value)))
        exec("shift_%d=%s"%(i+1,repr(params['shift_%d'%(i+1)].value)))
        exec("omega_%d=%s"%(i+1,repr(params['omega_%d'%(i+1)].value)))
        exec("decay_%d=%s"%(i+1,repr(params['decay_%d'%(i+1)].value)))
        model += eval("amp_%d"%(i+1)) * np.sin(x * eval("omega_%d"%(i+1)) + eval("shift_%d"%(i+1))) * np.exp(-x*x*eval("decay_%d"%(i+1)))
    return (model-data)#/data

params=Parameters()
for i in range(2):
    params.add('amp_%d'%(i+1),   value= 10,  vary=True, min=0, max=13)
    params.add('decay_%d'%(i+1), value= 0.1,vary=True,min=0,max=1.4)
    params.add('shift_%d'%(i+1), value= 0.0, vary=True,min=-np.pi, max=np.pi)
    params.add('omega_%d'%(i+1), value= 3.0, vary=True,min=-3.5, max=3.5)

result = minimize(fcn2min, params, args=(x, data),method='nelder')

final = data + result.residual
report_fit(params)
try:
    import pylab
    pylab.plot(x, data, 'k+')
    pylab.plot(x, final, 'r')
    pylab.show()
except:
    pass