目 录
微 积 分 篇
第1 章 函数与极限 ······················································································· 2
1.1 函数 ······························································································· 2
1.1.1 函数的定义 ·············································································· 2
1.1.2 函数的表达形式 ········································································ 3
1.1.3 分段函数 ················································································· 5
1.1.4 函数的运算 ·············································································· 6
1.1.5 基本初等函数与初等函数 ··························································· 7
1.1.6 使用SymPy 进行函数运算 ························································· 12
1.2 极限的概念 ····················································································· 15
1.2.1 数列的极限 ············································································· 15
1.2.2 函数的极限 ············································································· 17
1.3 无穷小量和无穷大量 ········································································· 22
1.3.1 无穷小量的定义 ······································································· 22
1.3.2 无穷小量的性质 ······································································· 23
1.3.3 无穷大量 ················································································ 24
1.3.4 无穷小量与无穷大量的关系 ······················································· 24
1.4 极限的计算 ····················································································· 25
1.4.1 极限的四则运算法则 ································································· 26
1.4.2 复合函数的极限运算法则 ·························································· 28
1.4.3 使用SymPy 求极限 ··································································· 28
习题1 ··································································································· 30
第2 章 导数 ······························································································· 32
2.1 导数的概念 ····················································································· 32
2.1.1 平均变化率 ············································································· 33
2.1.2 瞬时变化率 ············································································· 33
2.1.3 导数的定义 ············································································· 35
2.1.4 导数的几何意义 ······································································· 36
2.1.5 不可导的三种情形 ···································································· 37
2.2 导数的运算 ····················································································· 38
2.2.1 基本导数公式 ·········································································· 38
2.2.2 导数的四则运算法则 ································································· 38
2.2.3 复合函数求导法 ······································································· 39
2.2.4 使用SymPy 求导数 ··································································· 41
2.3 高阶导数 ························································································ 41
2.3.1 高阶导数的定义 ······································································· 41
2.3.2 使用SymPy 求高阶导数 ···························································· 42
习题2 ··································································································· 43
第3 章 极值与最值 ······················································································ 44
3.1 函数的单调性 ·················································································· 44
3.2 函数的极值 ····················································································· 46
3.2.1 极值的定义 ············································································· 46
3.2.2 可能的极值点 ·········································································· 47
3.2.3 极值的判定定理 ······································································· 49
3.2.4 使用SymPy 求函数的极值 ························································· 50
3.3 函数的最值 ····················································································· 51
习题3 ··································································································· 52
第4 章 二元函数的导数与极值 ······································································· 53
4.1 二元函数的概念 ··············································································· 53
4.1.1 二元函数的定义 ······································································· 53
4.1.2 二元函数的定义域 ···································································· 54
4.1.3 二元函数的几何意义 ································································· 55
4.1.4 使用SymPy 求多元函数的函数值 ················································ 55
4.2 二元函数的偏导数 ············································································ 56
4.2.1 偏导数的概念 ·········································································· 56
4.2.2 偏导数的计算 ·········································································· 56
4.2.3 偏导数的几何意义 ···································································· 57
4.2.4 使用SymPy 求偏导数································································ 58
4.3 二元函数的极值 ··············································································· 58
习题4 ··································································································· 60
第5 章 最优化基础：梯度下降法 ···································································· 61
5.1 梯度的定义 ····················································································· 61
5.2 梯度下降法 ····················································································· 62
5.2.1 一元函数的梯度下降法······························································ 62
5.2.2 二元函数的梯度下降法······························································ 63
5.3 使用Python 实现梯度下降法求函数极值 ················································ 66
习题5 ··································································································· 67
线性代数篇
第6 章 向量与编码 ······················································································ 70
6.1 向量的概念与运算 ············································································ 70
6.1.1 向量的概念 ············································································· 70
6.1.2 使用NumPy 建立向量 ······························································· 72
6.1.3 向量的运算 ············································································· 73
6.1.4 使用NumPy 实现向量的运算 ······················································ 74
6.2 向量的范数与相似度 ········································································· 75
6.2.1 范数的定义与NumPy 实现 ························································· 75
6.2.2 向量的相似度 ·········································································· 77
6.2.3 使用NumPy 计算向量相似性 ······················································ 80
6.3 向量间的线性关系 ············································································ 81
6.3.1 线性组合 ················································································ 81
6.3.2 线性相关与线性无关 ································································· 81
6.4 实战案例：K-means 聚类算法解决鸢尾花归类问题 ··································· 83
6.4.1 鸢尾花数据集Iris ····································································· 83
6.4.2 K-means 聚类算法 ···································································· 84
6.4.3 使用K-means 聚类算法求解Iris 分类问题 ······································ 85
习题6 ··································································································· 87
第7 章 矩阵与数字图像处理 ·········································································· 88
7.1 矩阵的基本知识 ··············································································· 88
7.1.1 矩阵的概念 ············································································· 88
7.1.2 几种特殊矩阵 ·········································································· 92
7.1.3 使用NumPy 建立矩阵 ······························································· 93
7.2 矩阵的运算 ··················································································· 100
7.2.1 矩阵的基本运算 ····································································· 100
7.2.2 使用NumPy 进行矩阵运算 ······················································· 106
7.3 实战案例：矩阵在数字图像处理中的应用 ············································ 109
7.3.1 图像基础 ·············································································· 109
7.3.2 数字图像的矩阵表示 ······························································· 111
7.3.3 矩阵运算实现图像处理···························································· 112
7.4 矩阵的初等变换 ············································································· 116
7.5 阶梯形矩阵与矩阵的秩 ···································································· 117
7.5.1 阶梯形矩阵 ··········································································· 117
7.5.2 矩阵的秩 ·············································································· 119
7.5.3 使用NumPy 和SymPy 求行最简阶梯形矩阵及矩阵的秩 ·················· 120
习题7 ································································································· 121
第8 章 行列式 ·························································································· 123
8.1 行列式的概念 ················································································ 123
8.1.1 二阶与三阶行列式 ·································································· 123
8.1.2 n 阶行列式 ··········································································· 126
8.2 方阵的行列式 ················································································ 128
8.3 使用NumPy 求行列式 ······································································ 129
习题8 ································································································· 130
第9 章 线性方程组 ···················································································· 132
9.1 线性方程组的概念 ·········································································· 132
9.2 消元法解线性方程组 ······································································· 133
9.3 齐次线性方程组 ············································································· 140
9.4 非齐次线性方程组 ·········································································· 144
9.5 使用NumPy 和SymPy 求解线性方程组 ··············································· 146
9.5.1 使用numpy.linalg.solve()求解线性方程组 ····································· 146
9.5.2 使用NumPy 和SymPy 求解一般线性方程组 ································· 147
习题9 ································································································· 148
第10 章 矩阵的特征值与特征向量 ································································ 150
10.1 特征值与特征向量的概念 ································································ 150
10.2 使用NumPy 求特征值与特征向量 ····················································· 153
习题10 ······························································································· 153
概率统计篇
第11 章 Pandas 基础 ················································································· 156
11.1 建立DataFrame 对象 ······································································ 156
11.2 打开CSV 文件 ············································································· 158
11.3 查看DataFrame 对象的属性 ····························································· 159
11.4 选择数据 ····················································································· 161
11.4.1 使用df[]运算符选择某列数据 ·················································· 161
11.4.2 使用df.iloc[]选择数据 ···························································· 164
习题11 ······························································································· 165
第12 章 数据的整理与展示 ········································································· 167
12.1 数据的属性 ·················································································· 168
12.2 数据的预处理 ··············································································· 169
12.2.1 缺失值处理 ········································································· 169
12.2.2 归一化 ··············································································· 171
12.2.3 规范化 ··············································································· 172
12.3 数据整理与展示 ············································································ 172
12.3.1 分布数列 ············································································ 172
12.3.2 数据可视化 ········································································· 174
习题12 ······························································································· 177
第13 章 描述统计 ····················································································· 178
13.1 数据位置的描述 ············································································ 179
13.2 数据集中趋势的度量 ······································································ 179
13.3 数据离散趋势的度量 ······································································ 181
13.4 数据分布形态的度量 ······································································ 184
习题13 ······························································································· 185
第14 章 概率的定义与运算 ········································································· 186
14.1 随机事件 ····················································································· 186
14.1.1 随机现象 ············································································ 186
14.1.2 随机事件 ············································································ 187
14.1.3 样本空间 ············································································ 188
14.1.4 随机事件的关系与运算 ·························································· 188
14.1.5 使用NumPy 模拟随机事件 ····················································· 191
14.2 概率的定义 ·················································································· 192
14.2.1 概率的统计定义 ··································································· 192
14.2.2 概率的古典定义 ··································································· 193
14.2.3 使用NumPy 模拟计算概率 ····················································· 195
14.3 概率的加法公式 ············································································ 197
14.3.1 互斥事件概率的加法公式 ······················································· 197
14.3.2 任意事件概率的加法公式 ······················································· 199
14.4 概率的乘法公式 ············································································ 199
14.4.1 条件概率 ············································································ 199
14.4.2 概率的乘法公式 ··································································· 202
14.4.3 独立事件的概率乘法公式 ······················································· 203
14.5 全概率公式 ·················································································· 203
14.6 贝叶斯公式 ·················································································· 205
习题14 ······························································································· 206
第15 章 随机变量 ····················································································· 208
15.1 随机变量的概念 ············································································ 208
15.2 离散型随机变量概率分布 ································································ 209
15.2.1 分布列 ··············································································· 209
15.2.2 两点分布 ············································································ 211
15.2.3 二项分布 ············································································ 211
15.3 连续型随机变量及其分布 ································································ 212
15.3.1 概率密度函数 ······································································ 212
15.3.2 均匀分布 ············································································ 213
15.3.3 正态分布 ············································································ 213
15.4 使用NumPy 生成指定分布的随机数 ·················································· 217
习题15 ······························································································· 219
第16 章 随机变量的数字特征 ······································································ 220
16.1 数学期望 ····················································································· 221
16.1.1 离散型随机变量的数学期望 ···················································· 221
16.1.2 连续型随机变量的数学期望 ···················································· 223
16.1.3 数学期望的性质 ··································································· 223
16.1.4 使用NumPy 计算均值与期望 ·················································· 224
16.2 方差 ··························································································· 225
16.2.1 离散型随机变量的方差 ·························································· 226
16.2.2 连续型随机变量的方差 ·························································· 226
16.2.3 方差的性质 ········································································· 227
16.2.4 使用NumPy 计算方差和标准差 ··············································· 228
16.3 常见分布的数学期望与方差 ····························································· 229
16.4 使用Pandas 进行描述统计 ······························································· 229
习题16 ······························································································· 232
第17 章 相关分析与回归分析 ······································································ 233
17.1 散点图 ························································································ 233
17.2 相关关系 ····················································································· 234
17.3 线性相关及其度量 ········································································· 235
17.4 回归分析 ····················································································· 237
17.4.1 回归分析的概念 ··································································· 237
17.4.2 回归分析的分类 ··································································· 237
17.4.3 一元线性回归分析 ································································ 238
17.4.4 多元线性回归分析 ································································ 242
17.5 实战案例：建立线性回归模型求解波士顿房价问题 ······························ 243
习题17 ······························································································· 246
应 用 篇
第18 章 神经网络 ····················································································· 248
18.1 神经元模型 ·················································································· 249
18.2 神经网络结构 ··············································································· 252
18.2.1 网络结构 ············································································ 252
18.2.2 前向传播 ············································································ 252
18.2.3 损失函数 ············································································ 254
18.2.4 反向传播 ············································································ 254
18.3 神经网络的数学公式推导 ································································ 254
18.4 使用Keras 实现神经网络求解波士顿房价预测问题 ······························· 256
习题18 ······························································································· 258
第19 章 卷积神经网络 ··············································································· 259
19.1 AlexNet 卷积神经网络简介 ······························································ 260
19.2 AlexNet 卷积神经网络技术详解 ························································ 261
19.2.1 卷积 ·················································································· 261
19.2.2 池化 ·················································································· 273
19.2.3 全连接层与Dropout 技术 ························································ 275
19.3 AlexNet 网络的结构分析 ································································· 277
19.4 AlexNet 网络的Keras 实现 ······························································ 279
19.5 实战案例：使用AlexNet 求解猫狗图片分类问题 ·································· 280
习题19 ······························································································· 284
参考文献 ··································································································· 286
附录A 标准正态分布函数数值表 ·································································· 287
