Curve fitting - monotonically increasing derivative

Question

I am trying to get a physical significant fit to some experimental data. I know that not only will the y values increase monotonically with x but also that dy/dx will also increase monotonically. I have tried a number of fitting functions including a polynomial fit and a univariate spline but neither of these has allowed me to produce the fit I am looking for.

So, I'm looking for a curve fitting function (in scipy?) that will allow me to define the known constraints of the final curve. Below is an example of my data, with a fitted line that does not display a monotonically increasing derivative.

import numpy as np
import matplotlib.pyplot as plt

data =  np.array([[  6.30991828, -10.22329935],
                  [  6.30991828, -10.2127338 ],
                  [  6.47697236, -10.01359361],
                  [  6.47697236,  -9.89353722],
                  [  6.47697236,  -9.81708052],
                  [  6.55108034,  -9.42113403],
                  [  6.55108034,  -9.21932801],
                  [  6.58617165,  -8.40428977],
                  [  6.62007321,  -7.6500927 ]])

interp = np.linspace(min(data[:,0]), max(data[:,0]), 20)
f = np.polyfit(data[:,0], data[:,-1], 3)
data_interp = np.polyval(f, interp)
plt.plot(data[:,0], data[:,1], 'x', interp, data_interp, '-')

EDIT: I believe that you can do this in MATLAB with the slmengine.

Martin Martin · Accepted Answer · 2014-03-06T15:31:28

Edit: Another attempt. I posted half-baked answer before. And I failed in reading too. I hope this is better.

from scipy.optimize import minimize
import numpy as np
import matplotlib.pyplot as plt

data = np.array([[  6.30991828, -10.22329935],
                  [  6.30991828, -10.2127338 ],
                  [  6.47697236, -10.01359361],
                  [  6.47697236,  -9.89353722],
                  [  6.47697236,  -9.81708052],
                  [  6.55108034,  -9.42113403],
                  [  6.55108034,  -9.21932801],
                  [  6.58617165,  -8.40428977],
                  [  6.62007321,  -7.6500927 ]])

x = data[:, 0]

def polynomial(p, x):
    return p[0]+p[1]*x+p[2]*x**2+p[3]*x**3

def constraint_2nd_der(p):
    return 2*p[2]+6*p[3]*x

def constraint_1st_der(p):
    return p[1]+2*p[2]*x+3*p[3]*x**2

def objective(p):
    return ((polynomial(p, x)-data[:, 1])**2).sum()


cons = (dict(type='ineq', fun=constraint_1st_der), dict(type='ineq', fun=constraint_2nd_der))
res = minimize(objective, x0=np.array([0., 0., 0., 0.]), method='SLSQP', constraints=cons)
if res.success:
    pars = res.x
    x = np.linspace(data[:, 0].min(), data[:, 0].max(), 100)
    pol = polynomial(pars, x)
    plt.plot(data[:, 0], data[:, 1], 'x', x, pol, '-')
    plt.show()
else:
    print 'Failed'

Curve fitting - monotonically increasing derivative

2 Answers