A smoothing spline is found as a solution to the regularized weighted linear regression problem:

\sum\limits_{i=1}^n w_i\lvert y_i - f(x_i) \rvert^2 +
\lambda\int\limits_{x_1}^{x_n} (f^{(2)}(u))^2 d u

where f is a spline function, w is a vector of weights and \lambda is a regularization parameter.

If lam is None, we use the GCV criteria to find an optimal regularization parameter, otherwise we solve the regularized weighted linear regression problem with given parameter. The parameter controls the tradeoff in the following way: the larger the parameter becomes, the smoother the function gets.

This algorithm is a clean room reimplementation of the algorithm introduced by Woltring in FORTRAN [2]. The original version cannot be used in SciPy source code because of the license issues. The details of the reimplementation are discussed here (available only in Russian) [4].

If the vector of weights w is None, we assume that all the points are equal in terms of weights, and vector of weights is vector of ones.

Note that in weighted residual sum of squares, weights are not squared: \sum\limits_{i=1}^n w_i\lvert y_i - f(x_i) \rvert^2 while in splrep the sum is built from the squared weights.

In cases when the initial problem is ill-posed (for example, the product X^T W X where X is a design matrix is not a positive defined matrix) a ValueError is raised.

Compute the (coefficients of) smoothing cubic spline function using lam to control the tradeoff between the amount of smoothness of the curve and its proximity to the data. In case lam is None, using the GCV criteria [1] to find it.

import numpy as np
np.random.seed(1234)
n = 200
def func(x):
   return x**3 + x**2 * np.sin(4 * x)
x = np.sort(np.random.random_sample(n) * 4 - 2)
y = func(x) + np.random.normal(scale=1.5, size=n)

from scipy.interpolate import make_smoothing_spline
spl = make_smoothing_spline(x, y)

import matplotlib.pyplot as plt
grid = np.linspace(x[0], x[-1], 400)
plt.plot(grid, spl(grid), label='Spline')
plt.plot(grid, func(grid), label='Original function')
plt.scatter(x, y, marker='.')
plt.legend(loc='best')
plt.show()