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AI-learning/lab/2_liner-regression-multiply.ipynb
2025-01-04 00:28:28 +08:00

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{
"cells": [
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [],
"source": [
"# 引入库\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import os\n",
"from sklearn.preprocessing import StandardScaler"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"/Users/wolves/Downloads/project/python/pt/lab\n"
]
}
],
"source": [
"# 检查os位置\n",
"print(os.getcwd())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"生成数据方式\n",
"$$ x_1 = x, x_2 = x^2, x_3 = \\sqrt{x} $$\n",
"$$ y = 1.35x_1 + 0.75x_2 + 2.1x_3 + 2.89 $$"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [],
"source": [
"# 生成数据\n",
"def generate_data():\n",
" w = np.array([1.35, 0.75, 2.1]) # 权重\n",
" b = 2.89 # 偏置\n",
" x_min = 1\n",
" x_max = 9\n",
" x = np.linspace(x_min, x_max, 10) # 均匀分布\n",
" X = np.array([x, x**2, np.sqrt(x)]) # 特征矩阵3x10\n",
" y = np.dot(w, X) + b # 1x10 一维向量不区分行向量和列向量\n",
" y += np.random.normal(scale=0.5, size=y.shape)\n",
" data = np.column_stack((X.T, y)) # 10x4\n",
" return data\n",
"\n",
"# 保存数据\n",
"def save_data(filename, data):\n",
" np.savetxt(filename, data, delimiter=',')\n",
" print(f\"{filename} 已成功创建并写入数据。\")\n",
"\n",
"# 生成并保存数据\n",
"data = generate_data()\n",
"#save_data('./1_data.txt', data)"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [],
"source": [
"# 读取数据\n",
"#points = np.genfromtxt(\"./1_data.txt\", delimiter=',')\n",
"\n",
"scaler = StandardScaler()\n",
"points = data\n",
"\n",
"m = len(points[:,0])\n",
"x = points[:, :3] # 10x3\n",
"y = points[:,3] # 1x10"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"原函数:\n",
"$$\n",
"\\vec{w} = {\\begin{bmatrix} w_1 & w_2 & w_3 & \\cdots & w_n \\end{bmatrix}}^T\n",
"$$\n",
"\n",
"$$\n",
"\\vec{x} = \\begin{bmatrix} x_1 & x_2 & x_3 & \\cdots & x_n \\end{bmatrix}\n",
"$$\n",
"\n",
"$$\n",
"f_{\\vec{w} \\cdot,b}(\\vec{x}) = \\vec{w} \\cdot \\vec{x} + b\n",
"$$\n",
"\n",
"损失函数: \n",
"\n",
"$$\n",
"\\text{MSE} = \\frac{1}{2m} \\sum_{i=1}^{m} \\left( y^{(i)} - \\hat{y}^{(i)} \\right)^2\n",
"$$\n",
"\n",
"梯度下降:\n",
"\n",
"分别对每个w和b求偏导数然后更新w和b\n",
" "
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {},
"outputs": [],
"source": [
"\n",
"\n",
"# 定义损失函数\n",
"def compute_loss(w, b):\n",
" 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",
"\n",
"# 定义梯度下降\n",
"def gradient_descent(w, b, alpha, num_iter):\n",
" loss_history = []\n",
" for _ in range(num_iter):\n",
" error = y - np.dot(w, x.T) - b # 1x10\n",
" # 计算梯度\n",
" dw = -np.dot(x.T , error) / m # dw 1x3 \n",
" db = -np.sum(error) / m # db 1x1\n",
" # 更新w和b\n",
" w -= alpha * dw\n",
" b -= alpha * db\n",
" loss_history.append(compute_loss(w, b))\n",
" return w, b, loss_history"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"w: [ 6.74308525 18.58329919 -1.16344888]\n",
"b: 37.79916995636456\n",
"loss: 0.22283060989193695\n"
]
},
{
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"text/plain": [
"<Figure size 3840x2880 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"if __name__ == \"__main__\":\n",
" # 初始化w和b\n",
" w = np.zeros(x.shape[1])\n",
" b = 0.0\n",
" # 设置学习率\n",
" alpha = 0.01\n",
" # 设置迭代次数\n",
" num_iter = 10000\n",
" x = scaler.fit_transform(x) # 特征缩放\n",
" # 进行梯度下降\n",
" w, b, loss_history = gradient_descent(w, b, alpha, num_iter)\n",
" print(\"w:\", w)\n",
" print(\"b:\", b)\n",
" \n",
" # 计算损失\n",
" loss = compute_loss(w, b)\n",
" print(\"loss:\", loss)\n",
"\n",
" # 绘制\n",
" plt.figure(dpi=600)\n",
" plt.plot(range(num_iter), loss_history)\n",
" plt.xlabel('Iteration')\n",
" plt.ylabel('Loss')\n",
" plt.title('Loss History')\n",
" # 显示学习率\n",
" plt.text(num_iter * 0.6, max(loss_history) * 0.7, f'Learning Rate: {alpha}', fontsize=12, color='blue')\n",
"\n",
" plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 100,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"预测值: [11.70402659 17.3020817 24.41357604]\n",
"原始值: [11.55984848 17.3273067 24.49 ]\n"
]
}
],
"source": [
"# 预测\n",
"def predict(x, w, b):\n",
" x_scaled = scaler.transform(x)\n",
" y = np.dot(w, x_scaled.T) + b\n",
" return y\n",
"\n",
"def original_predict(x):\n",
" w = np.array([1.35, 0.75, 2.1])\n",
" b = 2.89\n",
" return np.dot(w, x.T) + b\n",
"\n",
"x_new = np.array([\n",
" [2, 4, np.sqrt(2)],\n",
" [3, 9, np.sqrt(3)],\n",
" [4, 16, np.sqrt(4)]\n",
"])\n",
"y_pred = predict(x_new, w, b)\n",
"print(\"预测值:\", y_pred)\n",
"y_pred_original = original_predict(x_new)\n",
"print(\"原始值:\", y_pred_original)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 编码中遇到的错误\n",
"\n",
"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",
"\n",
"在特征缩放前学习率大之后会overflow导致模型不收敛。\n",
"\n",
"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。"
]
},
{
"cell_type": "code",
"execution_count": 107,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"使用sklearn预测值: [11.64087216 17.23308704 24.38469935]\n",
"原始值: [11.55984848 17.3273067 24.49 ]\n",
"训练数据的损失 (MSE): 0.008843469002214777\n"
]
}
],
"source": [
"from sklearn.linear_model import LinearRegression\n",
"from sklearn.pipeline import make_pipeline\n",
"from sklearn.metrics import mean_squared_error\n",
"\n",
"model = make_pipeline(LinearRegression())\n",
"# 拟合模型\n",
"model.fit(x, y)\n",
"\n",
"# 预测\n",
"y_pred_sklearn = model.predict(scaler.transform(x_new))\n",
"print(\"使用sklearn预测值:\", y_pred_sklearn)\n",
"y_pred_original = original_predict(x_new)\n",
"print(\"原始值:\", y_pred_original)\n",
"# 计算并打印训练数据的损失MSE\n",
"train_loss = mean_squared_error(y_pred_original, y_pred_sklearn)\n",
"print(\"训练数据的损失 (MSE):\", train_loss)\n"
]
}
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