17-1 神经网络基础：逻辑回归与激活函数

1线性回归（连续预测）
2        ↓
3逻辑回归（二分类） ← 本次课核心
4        ↓
5激活函数（非线性的来源）
6        ↓
7神经网络（多层堆叠）

 1import numpy as np
 2import matplotlib.pyplot as plt
 3
 4x = np.linspace(-5, 5, 200)
 5
 6# 定义激活函数
 7def sigmoid(x):
 8    return 1 / (1 + np.exp(-x))
 9
10def tanh(x):
11    return np.tanh(x)
12
13def relu(x):
14    return np.maximum(0, x)
15
16def leaky_relu(x, alpha=0.01):
17    return np.where(x > 0, x, alpha * x)
18
19# 绘制对比图
20fig, axes = plt.subplots(2, 2, figsize=(12, 8))
21
22functions = [sigmoid, tanh, relu, leaky_relu]
23titles = ['Sigmoid', 'Tanh', 'ReLU', 'Leaky ReLU']
24
25for ax, func, title in zip(axes.flat, functions, titles):
26    ax.plot(x, func(x), linewidth=2)
27    ax.axhline(y=0, color='k', linestyle='-', linewidth=0.5)
28    ax.axvline(x=0, color='k', linestyle='-', linewidth=0.5)
29    ax.set_title(title)
30    ax.grid(True, alpha=0.3)
31    ax.set_xlim(-5, 5)
32    ax.set_ylim(-2, 2)
33
34plt.tight_layout()
35plt.show()

 1import numpy as np
 2from sklearn.datasets import make_classification
 3from sklearn.model_selection import train_test_split
 4
 5# 生成二分类数据集（线性可分）
 6X, y = make_classification(n_samples=500, n_features=2, n_redundant=0,
 7                           n_informative=2, n_clusters_per_class=1,
 8                           random_state=42)
 9
10X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
11
12class LogisticRegression:
13    def __init__(self, learning_rate=0.1, n_iterations=1000):
14        self.lr = learning_rate
15        self.n_iterations = n_iterations
16        self.weights = None
17        self.bias = None
18
19    def _sigmoid(self, z):
20        return 1 / (1 + np.exp(-z))
21
22    def fit(self, X, y):
23        n_samples, n_features = X.shape
24        self.weights = np.zeros(n_features)
25        self.bias = 0
26
27        for _ in range(self.n_iterations):
28            linear_output = np.dot(X, self.weights) + self.bias
29            y_pred = self._sigmoid(linear_output)
30
31            dw = (1 / n_samples) * np.dot(X.T, (y_pred - y))
32            db = (1 / n_samples) * np.sum(y_pred - y)
33
34            self.weights -= self.lr * dw
35            self.bias -= self.lr * db
36
37    def predict_proba(self, X):
38        return self._sigmoid(np.dot(X, self.weights) + self.bias)
39
40    def predict(self, X, threshold=0.5):
41        return (self.predict_proba(X) >= threshold).astype(int)
42
43    def score(self, X, y):
44        return np.mean(self.predict(X) == y)
45
46# 训练与评估
47model = LogisticRegression(learning_rate=0.1, n_iterations=2000)
48model.fit(X_train, y_train)
49print(f"训练集准确率: {model.score(X_train, y_train):.4f}")
50print(f"测试集准确率: {model.score(X_test, y_test):.4f}")