通过Numpy实现线性回归

通过Numpy实现线性回归，可以分为以下几个步骤：

1. 导入Numpy库以及数据集

   import numpy as np
   data = np.genfromtxt('data.csv', delimiter=',')

2. 划分训练集和测试集

   x_train = data[:-20, 0]
   y_train = data[:-20, 1]
   x_test = data[-20:, 0]
   y_test = data[-20:, 1]

3. 定义模型参数

   a = 0
   b = 0

4. 定义学习率和迭代次数

   learning_rate = 0.0001
   num_iterations = 1000

5. 定义损失函数（均方误差）

   def compute_cost(a, b, x_train, y_train):
       total_cost = 0
       N = len(x_train)
       for i in range(N):
           total_cost += (y_train[i] - (a * x_train[i] + b)) ** 2
       return total_cost / N

6. 定义梯度下降函数

   def gradient_descent(a, b, x_train, y_train, learning_rate, num_iterations):
       N = len(x_train)
       for i in range(num_iterations):
           grad_a = 0
           grad_b = 0
           for j in range(N):
               grad_a += (2/N) * x_train[j] * ((a * x_train[j] + b) - y_train[j])
               grad_b += (2/N) * ((a * x_train[j] + b) - y_train[j])
           a -= learning_rate * grad_a
           b -= learning_rate * grad_b
       return a, b

7. 训练模型并得到最优的模型参数

   a, b = gradient_descent(a, b, x_train, y_train, learning_rate, num_iterations)

8. 评估模型性能并可视化结果

   import matplotlib.pyplot as plt

   plt.scatter(x_train, y_train)
   plt.plot(x_train, a * x_train + b, 'r')
   plt.show()

   print('Train cost:', compute_cost(a, b, x_train, y_train))
   print('Test cost:', compute_cost(a, b, x_test, y_test))