GEKKO NL Solver gives fitting values that are not corresponding to the reference values

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I'm trying to find the parameters values for best fitting of the follow model function with a given set of points.

Model:
                 x
  y = ------------------------
       1- (a1*x)^2 + (a2*x)^4

where parameters are:  a1, a2

(1) I used the found optimized parameters values with this equation but it gives around y=~0 for any x (red graph).

enter image description here

(3) Surprisingly, using a model with only one parameter as follow gives much better fitting.

Model:
            x
  y = -------------
       1- (a1*x)^2

where parameters are:  a1

enter image description here

Here is my code (I followed the code for non linear regression as shown here: https://github.com/BYU-PRISM/GEKKO/blob/master/docs/examples.rst)

Note: I played with initial values until got one where the solver didn't fire exception of "Non Solution Found". The values I used are 0.3 for each parameter (I have no idea why this is a good value if at all)

Measure points file can be downloaded from here: https://www.dropbox.com/s/ai5vbd2u5gx8r17/measure.npy?dl=0

import numpy as np
import math
import os.path

import matplotlib.pyplot as plt
from gekko import GEKKO

# Model:
#                x
# y = ------------------------
#      1- (a1*x)^2 + (a2*x)^4
#

# measured data (file exist at https://www.dropbox.com/s/ai5vbd2u5gx8r17/measure.npy?dl=0)
measure_data = np.load("C:/measure.npy")
xm = measure_data[0]
ym = measure_data[1]

# GEKKO model
m = GEKKO()

# parameters
x = m.Param(value=xm)
a = [m.FV(value=0.3) for i in range(2)]
for par in a:
    par.STATUS=1

# variables
y = m.CV(value=ym)
y.FSTATUS=1

m.Equation( y==x/(1-(a[0]*x)**2+(a[1]*x)**4) )
    
# regression mode
m.options.IMODE = 2

# optimize
m.solve(disp=False)


p = [par.value[0] for par in a] 
optimized_y = xm/(1-(p[0]*xm)**2+(p[1]*xm)**4)


plt.figure(1)

plt.plot(xm,ym,'k', label = "measurements")
plt.plot(xm,optimized_y,'r', label = "optimized_y")

plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show()
non-linear-regression gekko
1 Answers

Similar to the suggestions from GEKKO NL solver failed with initial guess of 0, try rearranging the equation to avoid divide-by-zero. An intermediate variable definition (z) can help simplify the expression.

z = m.Intermediate(1-(a[0]*x)**2)
m.Equation(y*z==x + z*(a[1]*x)**4)

Another thing that can help is to define the parameters to reduce the nonlinearity of the problem:

z = m.Intermediate(1-b[0]*(x**2))
m.Equation(y*z==x + b[1]*z*(x**4))

After the problem finishes solving, compute the original values of the a parameters as a[0]=math.sqrt(b[0].value[0]) and a[1]=(b[1].value[0])**(1/4).

It may also be helpful to try a different solver such as m.options.SOLVER=1 (APOPT solver). Nonlinear optimization problems sometimes need a good initial guess or bounds, especially when there are terms such as power of 4. For future questions, please post sample data that reproduces the problem.

user_23669410
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