199 lines
6.0 KiB
Plaintext
199 lines
6.0 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 导入必要的库\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"from sklearn.datasets import make_classification\n",
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"from sklearn.linear_model import LogisticRegression\n",
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"from sklearn.metrics import accuracy_score"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"def generate_and_plot_dataset():\n",
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" # 生成一个逻辑回归的数据集\n",
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" x, y = make_classification(n_samples=100, n_features=2, \n",
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" n_informative=2, n_redundant=0, \n",
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" n_clusters_per_class=1, random_state=42)\n",
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" # 可视化数据集\n",
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" plt.scatter(x[:, 0], x[:, 1], c=y, cmap='viridis')\n",
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" plt.xlabel('Feature 1')\n",
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" plt.ylabel('Feature 2')\n",
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" plt.title('Logistic Regression Dataset')\n",
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" plt.show()\n",
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" return x,y\n",
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"x,y = generate_and_plot_dataset()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"\n",
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"def logistic_regression_analysis(x, y):\n",
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" # 创建逻辑回归模型\n",
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" model = LogisticRegression()\n",
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"\n",
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" # 拟合模型\n",
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" model.fit(x, y)\n",
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"\n",
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" # 预测\n",
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" y_pred = model.predict(x)\n",
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"\n",
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" # 计算准确率\n",
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" accuracy = accuracy_score(y, y_pred)\n",
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" print(f'模型准确率: {accuracy:.2f}')\n",
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"\n",
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" # 绘制决策边界\n",
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" # 创建网格以绘制决策边界\n",
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" xx, yy = np.meshgrid(np.arange(x[:, 0].min() - 1, x[:, 0].max() + 1, 0.01),\n",
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" np.arange(x[:, 1].min() - 1, x[:, 1].max() + 1, 0.01))\n",
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"\n",
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" # 预测网格点的类别\n",
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" Z = model.predict(np.c_[xx.ravel(), yy.ravel()])\n",
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" Z = Z.reshape(xx.shape)\n",
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"\n",
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" # 绘制决策边界\n",
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" plt.contourf(xx, yy, Z, alpha=0.8, cmap='viridis')\n",
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" plt.scatter(x[:, 0], x[:, 1], c=y, edgecolors='k', marker='o', cmap='viridis')\n",
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" plt.xlabel('Feature 1')\n",
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" plt.ylabel('Feature 2')\n",
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" plt.title('Logistic Regression Decision Boundary')\n",
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" plt.show()\n",
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"\n",
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"# 调用函数\n",
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"logistic_regression_analysis(x, y)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 手动实现\n",
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"$$\n",
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"f_{\\vec{w},b}(x) = g(z) = \\frac{1}{1 + e^{-(\\vec{w} \\cdot \\vec{x} + b)}}\n",
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"$$\n",
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"\n",
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"$$\n",
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"J(\\vec{w},b) = -\\frac{1}{m} \\sum_{i=1}^{m} \\left( y^{(i)}\\ln(f_{\\vec{w},b}(x^{(i)})) + (1-y^{(i)})\\ln(1-f_{\\vec{w},b}(x^{(i)})) \\right)\n",
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"$$\n",
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"\n",
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"$$\n",
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"w_j = w_j - \\alpha \\frac{\\partial J(\\vec{w},b)}{\\partial w_j}\n",
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"$$\n",
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"\n",
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"$$\n",
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"b = b - \\alpha \\frac{\\partial J(\\vec{w},b)}{\\partial b}\n",
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"$$\n",
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"\n",
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"$$\n",
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"\\frac{\\partial J(\\vec{w},b)}{\\partial w_j} = \\frac{1}{m} \\sum_{i=1}^{m} (f_{\\vec{w},b}(x^{(i)}) - y^{(i)})x_j^{(i)}\n",
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"$$\n",
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"\n",
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"$$\n",
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"\\frac{\\partial J(\\vec{w},b)}{\\partial b} = \\frac{1}{m} \\sum_{i=1}^{m} (f_{\\vec{w},b}(x^{(i)}) - y^{(i)})\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [],
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"source": [
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"def compute_cost_vectorized(w, b):\n",
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" m = x.shape[0]\n",
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" z = np.dot(x, w) + b\n",
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" f_wb = sigmoid(z)\n",
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" cost = (-1/m) * np.sum(y * np.log(f_wb) + (1 - y) * np.log(1 - f_wb))\n",
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" return cost\n",
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" \n",
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"def sigmoid(z):\n",
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" return 1/(1+np.exp(-z))\n",
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" \n",
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"def gradient_descent(w, b, alpha, num_iterations):\n",
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" m = len(x)\n",
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" for i in range(num_iterations):\n",
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" z = np.dot(w, x.T) + b\n",
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" f_wb = sigmoid(z)\n",
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" w -= alpha * 1/m * np.dot(x.T, (f_wb-y))\n",
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" b -= alpha * 1/m * np.sum(f_wb-y)\n",
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" return w, b"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"if __name__ == \"__main__\":\n",
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" w = np.zeros(x.shape[1])\n",
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" b = 0\n",
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" alpha = 0.04\n",
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" num_iterations = 50000\n",
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" w, b = gradient_descent(w, b, alpha, num_iterations)\n",
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" print(w, b)\n",
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" loss = compute_cost_vectorized(w, b)\n",
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" print(loss)\n",
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" \n",
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" plt.figure(dpi=600)\n",
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" # 绘制数据点\n",
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" plt.scatter(x[:, 0], x[:, 1], c=y, cmap='viridis', edgecolors='k')\n",
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"\n",
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" # 计算决策边界\n",
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" x_min, x_max = x[:, 0].min() - 1, x[:, 0].max() + 1\n",
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" y_min, y_max = x[:, 1].min() - 1, x[:, 1].max() + 1\n",
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" xx = np.linspace(x_min, x_max, 100)\n",
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" # 计算对应的y值: w1*x + w2*y + b = 0 => y = -(w1*x + b)/w2\n",
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" if w[1] != 0:\n",
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" yy = -(w[0] * xx + b) / w[1]\n",
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" plt.plot(xx, yy, color='red', label='Decision Boundary')\n",
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" else:\n",
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" # 当w2=0时，决策边界为垂直线x = -b/w1\n",
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" x_boundary = -b / w[0]\n",
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" plt.axvline(x=x_boundary, color='red', label='Decision Boundary')\n",
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"\n",
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" plt.xlabel('Feature 1')\n",
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" plt.ylabel('Feature 2')\n",
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" plt.title('Logistic Regression Decision Boundary')\n",
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" plt.legend()\n",
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" plt.show()"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "pt",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.10.14"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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