机器学习模型及其算法（线性回归）

1.线性回归模型

最小二乘法代码实现：

import numpy as np
import matplotlib.pyplot as plt

1.导入数据

points = np.genfromtxt('data.csv', delimiter=',')
points[0,0]
# 提取points中的两列数据，分别作为x，y
x = points[:, 0]
y = points[:, 1]
# 用plt画出散点图
plt.scatter(x, y)
plt.show()

2.定义损失函数

# 损失函数是系数的函数，另外还要传入数据的x，y
def compute_cost(w, b, points):
    total_cost = 0
    M = len(points)
    # 逐点计算平方损失误差，然后求平均数
    for i in range(M):
        x = points[i, 0]
        y = points[i, 1]
        total_cost += ( y - w * x - b ) ** 2 
    return total_cost/M

3.定义算法拟合函数

# 先定义一个求均值的函数
def average(data):
    sum = 0
    num = len(data)
    for i in range(num):
        sum += data[i]
    return sum/num

# 定义核心拟合函数


def fit(points):
    M = len(points)
    x_bar = average(points[:, 0])
    
    sum_yx = 0
    sum_x2 = 0
    sum_delta = 0
    
    for i in range(M):
        x = points[i, 0]
        y = points[i, 1]
        sum_yx += y * ( x - x_bar )
        sum_x2 += x ** 2
    # 根据公式计算w
    w = sum_yx / ( sum_x2 - M * (x_bar**2) )
    
    for i in range(M):
        x = points[i, 0]
        y = points[i, 1]
        sum_delta += ( y - w * x )
    b = sum_delta / M
    
    return w, b

4.测试

w, b = fit(points)
print("w is: ", w)
print("b is: ", b)
cost = compute_cost(w, b, points)
print("cost is: ", cost)

w is: 1.3224310227553846
b is: 7.991020982269173
cost is: 110.25738346621313

5.画出拟合曲线

plt.scatter(x, y)
# 针对每一个x，计算出预测的y值
pred_y = w * x + b

plt.plot(x, pred_y, c='r')
plt.show()

使用梯度下降法实现线性回归

4.定义模型超参数

alpha = 0.0001
initial_w = 0
initial_b = 0
num_iter = 10

5.定义核心梯度下降算法

def grad_desc(points, initial_w, initial_b, alpha, num_iter):
    w = initial_w
    b = initial_b
    # 定义一个list保存所有的损失函数值，用来显示下降的过程
    cost_list = []
    
    for i in range(num_iter):
        cost_list.append( compute_cost(w, b, points) )
        w, b = step_grad_desc( w, b, alpha, points )
    
    return [w, b, cost_list]

def step_grad_desc( current_w, current_b, alpha, points ):
    sum_grad_w = 0
    sum_grad_b = 0
    M = len(points)
    
    # 对每个点，代入公式求和
    for i in range(M):
        x = points[i, 0]
        y = points[i, 1]
        sum_grad_w += ( current_w * x + current_b - y ) * x
        sum_grad_b += current_w * x + current_b - y
    
    # 用公式求当前梯度
    grad_w = 2/M * sum_grad_w
    grad_b = 2/M * sum_grad_b
    
    # 梯度下降，更新当前的w和b
    updated_w = current_w - alpha * grad_w
    updated_b = current_b - alpha * grad_b
    
    return updated_w, updated_b

6.测试

w, b, cost_list = grad_desc( points, initial_w, initial_b, alpha, num_iter )
print("w is: ", w)
print("b is: ", b)
cost = compute_cost(w, b, points)
print("cost is: ", cost)
plt.plot(cost_list)#默认使用轮次的索引作为x轴的表示
plt.show()

调用sklearn库完成线性回归

from sklearn.linear_model import LinearRegression
lr = LinearRegression()

x_new = x.reshape(-1, 1)#-1是无所谓多少行，1是表示1列，变成矩阵的形式
y_new = y.reshape(-1, 1)
lr.fit(x_new, y_new)

# 从训练好的模型中提取系数和截距
w = lr.coef_[0][0]
b = lr.intercept_[0]
print("w is: ", w)
print("b is: ", b)
cost = compute_cost(w, b, points)
print("cost is: ", cost)