How to fit a polynomial with some of the coefficients constrained?

Question

Using NumPy's polyfit (or something similar) is there an easy way to get a solution where one or more of the coefficients are constrained to a specific value?

For example, we could find the ordinary polynomial fitting using:

x = np.array([0.0, 1.0, 2.0, 3.0,  4.0,  5.0])
y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
z = np.polyfit(x, y, 3)

yielding

array([ 0.08703704, -0.81349206,  1.69312169, -0.03968254])

But what if I wanted the best fit polynomial where the third coefficient (in the above case z[2]) was required to be 1? Or will I need to write the fitting from scratch?

Answer 1

In this case, I would use curve_fit or lmfit; I quickly show it for the first one.

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

def func(x, a, b, c, d):
  return a + b * x + c * x ** 2 + d * x ** 3

x = np.array([0.0, 1.0, 2.0, 3.0,  4.0,  5.0])
y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])

print(np.polyfit(x, y, 3))

popt, _ = curve_fit(func, x, y)
print(popt)

popt_cons, _ = curve_fit(func, x, y, bounds=([-np.inf, 2, -np.inf, -np.inf], [np.inf, 2.001, np.inf, np.inf]))
print(popt_cons)

xnew = np.linspace(x[0], x[-1], 1000)

plt.plot(x, y, 'bo')
plt.plot(xnew, func(xnew, *popt), 'k-')
plt.plot(xnew, func(xnew, *popt_cons), 'r-')
plt.show()

This will print:

[ 0.08703704 -0.81349206  1.69312169 -0.03968254]
[-0.03968254  1.69312169 -0.81349206  0.08703704]
[-0.14331349  2.         -0.95913556  0.10494372]

So in the unconstrained case, polyfit and curve_fit give identical results (just the order is different), in the constrained case, the fixed parameter is 2, as desired.

The plot looks then as follows:

In lmfit you can also choose whether a parameter should be fitted or not, so you can then also just set it to a desired value (check this answer).