梯度下降法/正规方程求解线性回归问题Python实现

算法逻辑请参考吴恩达机器学习相关视频

梯度下降法

import numpy as np
import matplotlib.pyplot as plt


def compute_cost(theta0, theta1, X, y):
    total_cost = 0
    M = X.shape[0]
    for i in range(M):
        x_i = X[i, 0]
        y_i = y[i, 0]
        total_cost += (theta0 + theta1 * x_i - y_i) ** 2
    return total_cost / (2 * M)


def gradient_descent(X, y, init_theta0, init_theta1, alpha, num_iterations):
    cost_list = []
    theta0 = init_theta0
    theta1 = init_theta1
    for i in range(num_iterations):
        cost_list.append(compute_cost(theta0, theta1, X, y))
        theta0, theta1 = step_gradient_descent(theta0, theta1, alpha, X)
    return theta0, theta1, cost_list


def step_gradient_descent(theta0, theta1, alpha, X):
    grad_theta0 = 0
    grad_theta1 = 0
    M = X.shape[0]
    for i in range(M):
        x_i = X[i, 0]
        y_i = y[i, 0]
        grad_theta0 += (theta0 + theta1 * x_i - y_i)
        grad_theta1 += (theta0 + theta1 * x_i - y_i) * x_i
    grad_theta0 = grad_theta0 / M
    grad_theta1 = grad_theta1 / M

    updated_theta0 = theta0 - alpha * grad_theta0
    updated_theta1 = theta1 - alpha * grad_theta1
    return updated_theta0, updated_theta1


X = 2 * np.random.rand(100, 1)
y = 4 + 4 * X + np.random.randn(100, 1)

theta0, theta1, cost_list = gradient_descent(X, y, init_theta0=0, init_theta1=0, alpha=0.005, num_iterations=1000)
print("theta0 is :", theta0)
print("theta1 is :", theta1)

cost = compute_cost(theta0, theta1, X, y)
print("cost_list:", cost_list)
print("cost is:", cost)
plt.plot(cost_list)
plt.show()

plt.subplots(figsize=(12, 8))
plt.plot(X, y, "b.")
reg_model = theta0 + theta1 * X
plt.plot(X, reg_model, c='r')
plt.show()

theta0 is : 4.74244227311996
theta1 is : 3.1105226654110716
cost is: 0.08444660160107187

正规方程

import numpy as np
import matplotlib.pyplot as plt

X = 2 * np.random.rand(100, 1)
X_prev = X
y = 4 + 4 * X + np.random.randn(100, 1)
X0 = np.ones((100, 1))

X = np.c_[X0, np.matrix(X)]
Y = np.matrix(y)
A = np.dot(X.T, X)
theta = np.linalg.inv(A) @ X.T @ Y

print(theta)
theta0 = np.float_(theta[0][0])
theta1 = np.float_(theta[1][0])

plt.subplots(figsize=(12, 8))
plt.plot(X_prev, y, "b.")
reg_model = theta0 + theta1 * X_prev
plt.plot(X_prev, reg_model, c='r')
plt.show()