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Delete math.ipynb

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3-Python编程基础/3.4-AI数学基础与应用/math.ipynb deleted 100644 → 0
{ ++ /dev/null
{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"id": "f72cecef",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"导数值: 4.999999999810711\n"
]
}
],
"source": [
"import numpy as np # 导入numpy库，主要用于科学计算\n",
"\n",
"# 定义一个函数f(x)，计算表达式 x^2 + 3x + 2 的值\n",
"def f(x):\n",
" return x**2 + 3*x + 2\n",
"\n",
"# 定义一个数值导数函数 numerical_derivative\n",
"# 输入：\n",
"# f ：目标函数\n",
"# x ：求导点\n",
"# eps ：一个非常小的数，用来计算导数的差分间隔，默认是1e-6\n",
"def numerical_derivative(f, x, eps=1e-6):\n",
" # 利用中心差分公式近似导数：\n",
" # f'(x) ≈ [f(x + eps) - f(x - eps)] / (2 * eps)\n",
" return (f(x + eps) - f(x - eps)) / (2 * eps)\n",
"\n",
"x = 1.0 # 设定求导点x=1.0\n",
"\n",
"# 打印在x=1.0处的导数近似值 \n",
"print(\"导数值:\", numerical_derivative(f, x))\n"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "8449ad63",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dy/dx = 42.0\n"
]
}
],
"source": [
"# 复合函数的例子：y = f(g(x))\n",
"# 其中 f(x) = x^2，g(x) = 3x + 1\n",
"x = 2.0 # 给定自变量x的值\n",
"\n",
"g = 3 * x + 1 # 计算内部函数g(x)的值，g(2) = 3*2 + 1 = 7\n",
"f = g**2 # 计算外部函数f(g)的值，f(7) = 7^2 = 49\n",
"\n",
"# 链式法则求导：\n",
"# dy/dx = df/dg * dg/dx\n",
"# df/dg 是 f 对 g 的导数，f(g) = g^2，因此 df/dg = 2g\n",
"df_dg = 2 * g # 计算 df/dg，此处为 2 * 7 = 14\n",
"\n",
"# dg/dx 是 g 对 x 的导数，g(x) = 3x + 1，导数为3\n",
"dg_dx = 3 # 计算 dg/dx，值为3\n",
"\n",
"# 计算 dy/dx = df/dg * dg/dx = 14 * 3 = 42\n",
"dy_dx = df_dg * dg_dx\n",
"\n",
"print(\"dy/dx =\", dy_dx) # 输出导数值\n"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "02622489",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"优化后的 w: 3.0000000004074074\n"
]
}
],
"source": [
"# 初始化参数 w，目标是通过优化让 w 接近 3\n",
"w = 5.0 # 初始参数值为 5，距离目标值 3 有一定的偏差\n",
"\n",
"# 设置学习率（learning rate），控制每次参数更新的步长\n",
"lr = 0.1 # 学习率是 0.1，不能太大也不能太小\n",
"\n",
"# 梯度下降的迭代过程\n",
"for i in range(100): # 执行 100 次迭代更新\n",
" grad = 2 * (w - 3) # 计算损失函数 (w - 3)^2 的梯度，公式推导如下：\n",
" # 假设损失函数 f(w) = (w - 3)^2\n",
" # 则 f'(w) = 2*(w - 3)\n",
" \n",
" w -= lr * grad # 使用梯度下降法更新参数 w\n",
" # w = w - 学习率 × 梯度\n",
" # 目的是逐步减小损失函数的值，向最小值（最优 w=3）靠近\n",
"\n",
"# 输出最终优化后的参数 w，应该非常接近 3\n",
"print(\"优化后的 w:\", w)\n"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "7988f5f1",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"输出结果: [[1.2 1.6]]\n"
]
}
],
"source": [
"import numpy as np # 导入NumPy库，用于处理数组和矩阵运算\n",
"\n",
"x = np.array([[1, 2]]) # 定义输入数据x，是一个形状为(1, 2)的二维数组，表示有两个特征\n",
"\n",
"W = np.array([[0.1, 0.2], # 定义权重矩阵W，形状为(2, 2)，表示有2个输入和2个输出神经元\n",
" [0.3, 0.4]])\n",
"\n",
"b = np.array([[0.5, 0.6]]) # 定义偏置项b，形状为(1, 2)，每个输出神经元对应一个偏置值\n",
"\n",
"output = np.dot(x, W) + b # 先进行矩阵乘法x·W（形状变为1x2），再加上偏置b（逐元素加法）\n",
" # 即 output = xW + b，表示一层神经网络的线性变换\n",
"\n",
"print(\"输出结果:\", output) # 打印输出结果，形如 [[1.2 1.6]]\n"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "f345a811",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2]]\n",
"[[0.1 0.2]\n",
" [0.3 0.4]]\n"
]
}
],
"source": [
"print(x)\n",
"print(W)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "6972756f",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"降维后的数据: [[-2.68412563 0.31939725]\n",
" [-2.71414169 -0.17700123]\n",
" [-2.88899057 -0.14494943]\n",
" [-2.74534286 -0.31829898]\n",
" [-2.72871654 0.32675451]]\n"
]
}
],
"source": [
"from sklearn.decomposition import PCA # 从sklearn库中导入PCA类，用于主成分分析降维\n",
"from sklearn.datasets import load_iris # 导入load_iris函数，用来加载鸢尾花数据集\n",
"\n",
"X = load_iris().data # 加载鸢尾花数据集的特征数据，X是一个形状为(150, 4)的数组，表示150个样本，每个样本4个特征\n",
"\n",
"pca = PCA(n_components=2) # 创建一个PCA对象，指定降维到2个主成分（把4维数据降到2维）\n",
"\n",
"X_reduced = pca.fit_transform(X) \n",
"# 先用X训练PCA模型（fit），计算主成分方向\n",
"# 然后将原始数据X映射到这两个主成分构成的新空间中（transform）\n",
"# fit_transform是fit和transform的合并操作，输出降维后的数据，形状为(150, 2)\n",
"\n",
"print(\"降维后的数据:\", X_reduced[:5]) \n",
"# 打印降维后数据的前5条样本，每条样本现在只有两个特征（两个主成分的值）\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0f8108c5",
"metadata": {},
"outputs": [],
"source": [
"import pandas as pd\n",
"from sklearn.datasets import load_iris\n",
"\n",
"iris = load_iris() \n",
"df = pd.DataFrame(iris.data, columns=iris.feature_names)\n",
"df['label'] = iris.target\n",
"print(df.head()) # 打印前5行\n"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "pytorch_gpu",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.3"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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