Curve in forma con limiti dei parametri

Question

ho dati sperimentali:

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] 

E la formula f(x) = m1 + m2/(1 + e^(-m3*(x - m4))). Ho bisogno di trovare m1, m2, m3, m4 con metodo dei minimi quadrati, dove 0,05 < m1 < 0,3 0,3 < m2 < 0,8 0,05 < m3 < 0,5 m4 < 200.

Io uso curve_fit e la mia funzione è:

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 

Ma il programma restituirà l'errore: RuntimeError: Optimal parameters not found: Number of calls to function has reached maxfev = 1000.

Cosa fare?

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

Impostare i parametri iniziali a valori utili:

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

Inoltre, utilizzare xdata invece di x nella funzione f:

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

Questo è il mio programma modificato:

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))) 

Risultato:

(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

In alternativa, è anche possibile utilizzare lmfit che permette di impostare facilmente i confini ed evita il "brutto" if dichiarazione nella funzione. I parametri si ottengono sono i seguenti:

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) 

e l'uscita si ottiene simile a questo:

Ecco l'intero codice con diversi commenti; fatemi sapere se avete ulteriori domande:

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's 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