用python分别实现Lasso回归问题的坐标下降算法和弹性网回归算法，并且测试可视化

首先，我们需要导入必要的库：numpy、matplotlib、sklearn中的Lasso和ElasticNet模型。

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import Lasso, ElasticNet

接下来，我们生成一些随机数据来测试我们的算法。

np.random.seed(42)

n_samples, n_features = 50, 100
X = np.random.randn(n_samples, n_features)
y = np.random.randn(n_samples)

现在，我们可以开始实现坐标下降算法了。下面是代码：

def coordinate_descent_lasso(X, y, alpha, max_iter=1000, tol=1e-4):
    n_samples, n_features = X.shape
    w = np.zeros(n_features)
    r = y - np.dot(X, w)
    for _ in range(max_iter):
        for j in range(n_features):
            w_old = w[j]
            X_j = X[:, j]
            X_j_sq_norm = np.sum(X_j ** 2)
            if X_j_sq_norm == 0:
                w[j] = 0
            else:
                p_j = np.dot(X_j, r) / X_j_sq_norm
                w[j] = np.sign(p_j) * max(0, abs(p_j) - alpha)
            r += X[:, j] * (w_old - w[j])
            if np.sum(w - w_old) ** 2 < tol:
                break
    return w

接下来，我们可以使用生成的数据和实现的算法来测试和可视化。

alpha = 0.1
lasso = Lasso(alpha=alpha)
lasso.fit(X, y)

w_lasso_cd = coordinate_descent_lasso(X, y, alpha=alpha)

plt.plot(lasso.coef_, color='red', linewidth=2,
         label='Lasso by sklearn')
plt.plot(w_lasso_cd, color='blue', linewidth=2,
         label='Lasso by coordinate descent')
plt.legend()
plt.title('Lasso coefficients')
plt.show()

接下来，我们实现弹性网回归算法。下面是代码：

def elastic_net(X, y, alpha, l1_ratio, max_iter=1000, tol=1e-4):
    n_samples, n_features = X.shape
    w = np.zeros(n_features)
    r = y - np.dot(X, w)
    for _ in range(max_iter):
        for j in range(n_features):
            w_old = w[j]
            X_j = X[:, j]
            X_j_sq_norm = np.sum(X_j ** 2)
            if X_j_sq_norm == 0:
                w[j] = 0
            else:
                p_j = np.dot(X_j, r) / X_j_sq_norm
                w[j] = np.sign(p_j) * max(0, abs(p_j) - alpha * l1_ratio) / \
                       (X_j_sq_norm + alpha * (1 - l1_ratio))
            r += X[:, j] * (w_old - w[j])
            if np.sum(w - w_old) ** 2 < tol:
                break
    return w

同样，我们可以使用生成的数据和实现的算法来测试和可视化。

alpha = 0.1
l1_ratio = 0.5
enet = ElasticNet(alpha=alpha, l1_ratio=l1_ratio)
enet.fit(X, y)

w_enet_cd = elastic_net(X, y, alpha=alpha, l1_ratio=l1_ratio)

plt.plot(enet.coef_, color='red', linewidth=2,
         label='ElasticNet by sklearn')
plt.plot(w_enet_cd, color='blue', linewidth=2,
         label='ElasticNet by coordinate descent')
plt.legend()
plt.title('ElasticNet coefficients')
plt.show()

通过以上代码，我们就可以实现Lasso回归问题的坐标下降算法和弹性网回归算法，并且测试可视化。