261 lines
7.0 KiB
Plaintext
261 lines
7.0 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 94,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 引入库\n",
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import os\n",
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"from sklearn.preprocessing import StandardScaler"
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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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"# 检查os位置\n",
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"print(os.getcwd())"
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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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"生成数据方式\n",
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"$$ x_1 = x, x_2 = x^2, x_3 = \\sqrt{x} $$\n",
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"$$ y = 1.35x_1 + 0.75x_2 + 2.1x_3 + 2.89 $$"
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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": 96,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 生成数据\n",
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"def generate_data():\n",
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" w = np.array([1.35, 0.75, 2.1]) # 权重\n",
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" b = 2.89 # 偏置\n",
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" x_min = 1\n",
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" x_max = 9\n",
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" x = np.linspace(x_min, x_max, 10) # 均匀分布\n",
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" X = np.array([x, x**2, np.sqrt(x)]) # 特征矩阵3x10\n",
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" y = np.dot(w, X) + b # 1x10 一维向量不区分行向量和列向量\n",
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" y += np.random.normal(scale=0.5, size=y.shape)\n",
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" data = np.column_stack((X.T, y)) # 10x4\n",
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" return data\n",
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"\n",
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"# 保存数据\n",
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"def save_data(filename, data):\n",
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" np.savetxt(filename, data, delimiter=',')\n",
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" print(f\"{filename} 已成功创建并写入数据。\")\n",
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"\n",
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"# 生成并保存数据\n",
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"data = generate_data()\n",
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"#save_data('./1_data.txt', data)"
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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": 97,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 读取数据\n",
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"#points = np.genfromtxt(\"./1_data.txt\", delimiter=',')\n",
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"\n",
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"scaler = StandardScaler()\n",
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"points = data\n",
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"\n",
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"m = len(points[:,0])\n",
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"x = points[:, :3] # 10x3\n",
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"y = points[:,3] # 1x10"
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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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"原函数：\n",
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"$$\n",
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"\\vec{w} = {\\begin{bmatrix} w_1 & w_2 & w_3 & \\cdots & w_n \\end{bmatrix}}^T\n",
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"$$\n",
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"\n",
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"$$\n",
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"\\vec{x} = \\begin{bmatrix} x_1 & x_2 & x_3 & \\cdots & x_n \\end{bmatrix}\n",
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"$$\n",
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"\n",
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"$$\n",
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"f_{\\vec{w} \\cdot,b}(\\vec{x}) = \\vec{w} \\cdot \\vec{x} + b\n",
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"$$\n",
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"\n",
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"损失函数: \n",
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"\n",
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"$$\n",
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"\\text{MSE} = \\frac{1}{2m} \\sum_{i=1}^{m} \\left( y^{(i)} - \\hat{y}^{(i)} \\right)^2\n",
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"$$\n",
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"\n",
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"梯度下降：\n",
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"\n",
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"分别对每个w和b求偏导数，然后更新w和b\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": 98,
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"metadata": {},
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"outputs": [],
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"source": [
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"\n",
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"\n",
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"# 定义损失函数\n",
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"def compute_loss(w, b):\n",
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" return np.sum((y - (np.dot(w, x.T) + b)) ** 2) / (2 * m) # w 1x3 x.T 3x10 y 1x10 y-np.dot(w, x.T) 1x10 sum=number\n",
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"\n",
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"# 定义梯度下降\n",
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"def gradient_descent(w, b, alpha, num_iter):\n",
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" loss_history = []\n",
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" for _ in range(num_iter):\n",
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" error = y - np.dot(w, x.T) - b # 1x10\n",
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" # 计算梯度\n",
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" dw = -np.dot(x.T , error) / m # dw 1x3 \n",
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" db = -np.sum(error) / m # db 1x1\n",
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" # 更新w和b\n",
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" w -= alpha * dw\n",
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" b -= alpha * db\n",
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" loss_history.append(compute_loss(w, b))\n",
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" return w, b, loss_history"
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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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" # 初始化w和b\n",
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" w = np.zeros(x.shape[1])\n",
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" b = 0.0\n",
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" # 设置学习率\n",
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" alpha = 0.01\n",
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" # 设置迭代次数\n",
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" num_iter = 10000\n",
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" x = scaler.fit_transform(x) # 特征缩放\n",
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" # 进行梯度下降\n",
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" w, b, loss_history = gradient_descent(w, b, alpha, num_iter)\n",
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" print(\"w:\", w)\n",
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" print(\"b:\", b)\n",
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" \n",
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" # 计算损失\n",
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" loss = compute_loss(w, b)\n",
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" print(\"loss:\", loss)\n",
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"\n",
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" # 绘制\n",
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" plt.figure(dpi=600)\n",
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" plt.plot(range(num_iter), loss_history)\n",
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" plt.xlabel('Iteration')\n",
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" plt.ylabel('Loss')\n",
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" plt.title('Loss History')\n",
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" # 显示学习率\n",
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" plt.text(num_iter * 0.6, max(loss_history) * 0.7, f'Learning Rate: {alpha}', fontsize=12, color='blue')\n",
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"\n",
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" plt.show()"
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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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"# 预测\n",
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"def predict(x, w, b):\n",
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" x_scaled = scaler.transform(x)\n",
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" y = np.dot(w, x_scaled.T) + b\n",
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" return y\n",
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"\n",
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"def original_predict(x):\n",
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" w = np.array([1.35, 0.75, 2.1])\n",
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" b = 2.89\n",
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" return np.dot(w, x.T) + b\n",
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"\n",
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"x_new = np.array([\n",
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" [2, 4, np.sqrt(2)],\n",
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" [3, 9, np.sqrt(3)],\n",
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" [4, 16, np.sqrt(4)]\n",
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"])\n",
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"y_pred = predict(x_new, w, b)\n",
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"print(\"预测值:\", y_pred)\n",
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"y_pred_original = original_predict(x_new)\n",
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"print(\"原始值:\", y_pred_original)"
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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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"# 编码中遇到的错误\n",
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"\n",
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"梯度下降算法中，把x.T和error相乘了，正确应使用矩阵乘法。\n",
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"\n",
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"在特征缩放前，学习率大之后会overflow，导致模型不收敛。\n",
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"\n",
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"特征缩放后，由于代码写错，把结果也缩放了，导致后面预测时，结果不正确。"
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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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"from sklearn.linear_model import LinearRegression\n",
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"from sklearn.pipeline import make_pipeline\n",
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"from sklearn.metrics import mean_squared_error\n",
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"\n",
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"model = make_pipeline(LinearRegression())\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_sklearn = model.predict(scaler.transform(x_new))\n",
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"print(\"使用sklearn预测值:\", y_pred_sklearn)\n",
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"y_pred_original = original_predict(x_new)\n",
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"print(\"原始值:\", y_pred_original)\n",
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"# 计算并打印训练数据的损失（MSE）\n",
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"train_loss = mean_squared_error(y_pred_original, y_pred_sklearn)\n",
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"print(\"训练数据的损失 (MSE):\", train_loss)\n"
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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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