写一个适用于高维特征的spline regression，带有l2惩罚项，要求给出所有的权重，只能使用numpy库，并且写一个示例代码，输入为13维的特征

Spline Regression with L2 Regularization

Spline regression is a popular method for modelling non-linear relationships between an independent variable and a dependent variable. In this method, the independent variable is divided into intervals, and a polynomial equation is fitted to each interval. The polynomials are then joined together smoothly at the interval boundaries to create a continuous function.

L2 regularization, also known as Ridge regression, is a technique for preventing overfitting in regression models. It adds a penalty term to the loss function that favors smaller weights, effectively shrinking the coefficients towards zero.

To implement spline regression with L2 regularization in high-dimensional feature space, we can use a basis function approach. We first define a set of basis functions that span the feature space, such as cubic splines or B-splines. We then fit a linear regression model using these basis functions as predictors, and add an L2 penalty term to the loss function.

The weight vector w can be found by minimizing the following objective function:

minimize ||y - Xw||^2 + alpha * ||w||^2

where y is the target variable, X is the design matrix with basis functions as columns, w is the weight vector, and alpha is the regularization parameter.

To solve for w, we can use the closed-form solution:

w = (X^T X + alpha * I)^-1 X^T y

where I is the identity matrix.

Example code:

import numpy as np

# generate random data with 13 features
np.random.seed(0)
X = np.random.randn(100, 13)
y = np.random.randn(100)

# define basis functions
def cubic_spline(x):
return np.hstack([x**3, x**2, x, np.ones_like(x)])

# create design matrix with cubic spline basis functions
X_basis = np.hstack([cubic_spline(X[:, i]) for i in range(X.shape[1])])

# set regularization parameter
alpha = 0.1

# solve for weight vector using closed-form solution
w = np.linalg.inv(X_basis.T.dot(X_basis) + alpha * np.eye(X_basis.shape[1])).dot(X_basis.T).dot(y)

# print weight vector
print(w)