Python code

# This source code is public domain
import numpy
import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
G=1
k=1
f0=0.5
x = numpy.linspace(-8,8,16)[1:-1]
numpy.random.seed(50)
y =numpy.random.normal(G/(1+numpy.exp(-k*G*x)*(G/f0-1)),0.1)
err=1E8
err=numpy.mean(numpy.square(y-G/(1+numpy.exp(-k*G*x)*(G/f0-1))))
print(err,G,k,f0)
numpy.random.seed(2)
for i in range(5000):
 faktor=1+0.01*(numpy.random.rand()-.5)
 for ivar in 1,2,3:
 backup=[err,G,k,f0]
 var=backup[:]
 var[ivar]=var[ivar]*faktor
 err,G,k,f0=var
 err_neu = numpy.mean(numpy.square(y-G/(1+numpy.exp(-k*G*x)*(G/f0-1))))
 if err_neu<err:
 err=err_neu
 else:
 var[ivar]=backup[ivar]
 err,G,k,f0=var
print(err,G,k,f0,"(Fehlerquadrat minimiert)")
 
xneu = numpy.linspace(-8,8,50)
yneu = G/(1+numpy.exp(-k*G*xneu)*(G/f0-1))
xr = x
yr = G/(1+numpy.exp(-k*G*xr)*(G/f0-1))
residuen = []
for i in range(len(x)): residuen +=[ [x[i]+8, x[i]+8],[y[i]*10, yr[i]*10], 'g-']
fig = plt.figure(figsize=(4.2, 3.2))
y0 = plt.plot(*residuen[:-3], color='#60c060', linewidth=1.5)
y0, = plt.plot(*residuen[-3:],label='Residuum', color='#60c060', linewidth=1.5)
y2, = plt.plot(xneu+8,yneu*10,'r-',label='Modellfunktion')
plt.setp(y2, linewidth=1.5)
y1, = plt.plot(x+8,y*10,'o',label='Messpunkte')
plt.xlabel('x')
plt.ylabel('y')
order = y1,y2,y0
plt.legend(order,[p.get_label() for p in order],frameon=True, loc='lower right')
plt.gca().xaxis.set_major_locator(ticker.MultipleLocator(2))
plt.grid(True, alpha=0.7)
plt.tight_layout()
plt.savefig('MDKQ1.svg')