# 用Python实现机器学习算法——线性回归算法

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Python 被称为是最接近 AI 的语言。最近一位名叫Anna-Lena Popkes（德国波恩大学计算机科学专业的研究生，主要关注机器学习和神经网络。）的小姐姐在GitHub上分享了自己如何使用Python（3.6及以上版本）实现7种机器学习算法的笔记，并附有完整代码。所有这些算法的实现都没有使用其他机器学习库。这份笔记可以帮大家对算法以及其底层结构有个基本的了解，但并不是提供最有效的实现。

• 数据集
• 是d-维向量
• 是一个目标变量，它是一个标量

• 它有一个实值加权向量
• 它有一个实值偏置量 b
• 它使用恒等函数作为其激活函数

a) 梯度下降法

b) 正态方程(封闭形式解)：

In [4]:

import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
np.random.seed(123)


In [5]:

# We will use a simple training set
X = 2 * np.random.rand(500, 1)
y = 5 + 3 * X + np.random.randn(500, 1)
fig = plt.figure(figsize=(8,6))
plt.scatter(X, y)
plt.title("Dataset")
plt.xlabel("First feature")
plt.ylabel("Second feature")
plt.show()


In [6]:

# Split the data into a training and test set
X_train, X_test, y_train, y_test = train_test_split(X, y)
print(f'Shape X_train: {X_train.shape}')
print(f'Shape y_train: {y_train.shape}')
print(f'Shape X_test: {X_test.shape}')
print(f'Shape y_test: {y_test.shape}')


Shape X_train: (375, 1)

Shape y_train: (375, 1)

Shape X_test: (125, 1)

Shape y_test: (125, 1)

In [23]:

class LinearRegression:

def __init__(self):
pass

def train_gradient_descent(self, X, y, learning_rate=0.01, n_iters=100):
"""
Trains a linear regression model using gradient descent
"""
# Step 0: Initialize the parameters
n_samples, n_features = X.shape
self.weights = np.zeros(shape=(n_features,1))
self.bias = 0
costs = []

for i in range(n_iters):
# Step 1: Compute a linear combination of the input features and weights
y_predict = np.dot(X, self.weights) + self.bias

# Step 2: Compute cost over training set
cost = (1 / n_samples) * np.sum((y_predict - y)**2)
costs.append(cost)

if i % 100 == 0:
print(f"Cost at iteration {i}: {cost}")

# Step 3: Compute the gradients
dJ_dw = (2 / n_samples) * np.dot(X.T, (y_predict - y))
dJ_db = (2 / n_samples) * np.sum((y_predict - y))

# Step 4: Update the parameters
self.weights = self.weights - learning_rate * dJ_dw
self.bias = self.bias - learning_rate * dJ_db

return self.weights, self.bias, costs

def train_normal_equation(self, X, y):
"""
Trains a linear regression model using the normal equation
"""
self.weights = np.dot(np.dot(np.linalg.inv(np.dot(X.T, X)), X.T), y)
self.bias = 0

return self.weights, self.bias

def predict(self, X):
return np.dot(X, self.weights) + self.bias


﻿

In [24]:

regressor = LinearRegression()
w_trained, b_trained, costs = regressor.train_gradient_descent(X_train, y_train, learning_rate=0.005, n_iters=600)
fig = plt.figure(figsize=(8,6))
plt.plot(np.arange(n_iters), costs)
plt.title("Development of cost during training")
plt.xlabel("Number of iterations")
plt.ylabel("Cost")
plt.show()


Cost at iteration 0: 66.45256981003433
Cost at iteration 100: 2.2084346146095934
Cost at iteration 200: 1.2797812854182806
Cost at iteration 300: 1.2042189195356685
Cost at iteration 400: 1.1564867816573
Cost at iteration 500: 1.121391041394467


In [28]:

n_samples, _ = X_train.shape
n_samples_test, _ = X_test.shape

y_p_train = regressor.predict(X_train)
y_p_test = regressor.predict(X_test)

error_train =  (1 / n_samples) * np.sum((y_p_train - y_train) ** 2)
error_test =  (1 / n_samples_test) * np.sum((y_p_test - y_test) ** 2)

print(f"Error on training set: {np.round(error_train, 4)}")
print(f"Error on test set: {np.round(error_test)}")


Error on training set: 1.0955

Error on test set: 1.0

# To compute the parameters using the normal equation, we add a bias value of 1 to each input example
X_b_train = np.c_[np.ones((n_samples)), X_train]
X_b_test = np.c_[np.ones((n_samples_test)), X_test]

reg_normal = LinearRegression()
w_trained = reg_normal.train_normal_equation(X_b_train, y_train)


y_p_train = reg_normal.predict(X_b_train)
y_p_test = reg_normal.predict(X_b_test)

error_train =  (1 / n_samples) * np.sum((y_p_train - y_train) ** 2)
error_test =  (1 / n_samples_test) * np.sum((y_p_test - y_test) ** 2)

print(f"Error on training set: {np.round(error_train, 4)}")
print(f"Error on test set: {np.round(error_test, 4)}")


Error on training set: 1.0228

Error on test set: 1.0432

# Plot the test predictions

fig = plt.figure(figsize=(8,6))
plt.scatter(X_train, y_train)
plt.scatter(X_test, y_p_test)
plt.xlabel("First feature")
plt.ylabel("Second feature")
plt.show()