GEKKO Python - simple linear regression from Documentation

Question

By following GEKKO documentation I used the example for the linear and polynomial regression. Here is just the part regarding the simple linear regression.

from gekko import GEKKO
import numpy as np

xm = np.array([0.1,0.3,0.2,0.5,0.8])
ym = np.array([0.52,0.53,0.4,0.6,1.01])
             
#### Solution
m = GEKKO()
m.options.IMODE=2
# coefficients
c = [m.FV(value=0) for i in range(2)]
c[0].STATUS=1
c[1].STATUS=1

x = m.Param(value=xm)
yd = m.Param(value=ym)

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

######### uncomment ############
#y = m.Var()
#m.Minimize((y-yd)**2)
################################
# polynomial model

m.Equation(y==c[0]+c[1]*x)
# linear regression
m.solve(disp=False)
p1 = [c[1].value[0],c[0].value[0]]
p1

I just wonder why the different results are obtained when uncommenting the lines

y = m.Var()
m.Minimize((y-yd)**2)

It seems that the results obtained (linear, quadratic, cubic) in the documentation are not the least squares ones. What the minimizing criteria was used in those cases?

Best Regards, Radovan

Answer 1

Switch to m.options.EV_TYPE=2 to get a squared error objective .

The default in Gekko is the l1-norm objective. Here is a description of the differences:

The l1-norm objective (sum of the absolute error) is less sensitive to outliers and bad data.