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。
在线性回归中，我们想要建立一个模型，来拟合一个因变量 y 与一个或多个独立自变量(预测变量) x 之间的关系。
给定：
线性回归模型可以使用以下方法进行训练
a) 梯度下降法
线性回归模型的训练过程有不同的步骤。首先(在步骤 0 中)，模型的参数将被初始化。在达到指定训练次数或参数收敛前，重复以下其他步骤。
第 0 步：
用0 (或小的随机值)来初始化权重向量和偏置量，或者直接使用正态方程计算模型参数
第 1 步(只有在使用梯度下降法训练时需要)：
计算输入的特征与权重值的线性组合，这可以通过矢量化和矢量传播来对所有训练样本进行处理：
In [4]:
import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
np.random.seed(123)
数据集
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)
线性回归分类
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
使用梯度下降进行训练
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
测试（梯度下降模型）
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
使用正规方程（normal equation）训练
# 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)
测试（正规方程模型）
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
可视化测试预测
# 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()