A.2 数学速查
本附录目标
- 复习 高数 / 线代 / 概率 必备公式
- 提供 常用导数 / 积分 表
- 速查 矩阵微积分
A.2.1 极限与连续
| 公式 |
|
| \(\lim_{x \to 0} \dfrac{\sin x}{x} = 1\) |
|
| \(\lim_{x \to \infty} (1 + 1/x)^x = e\) |
|
| \(\lim_{x \to 0} \dfrac{e^x - 1}{x} = 1\) |
|
| \(\lim_{x \to 0} \dfrac{\ln(1+x)}{x} = 1\) |
|
A.2.2 常用导数
| \(f(x)\) |
\(f'(x)\) |
| \(x^n\) |
\(n x^{n-1}\) |
| \(e^x\) |
\(e^x\) |
| \(a^x\) |
\(a^x \ln a\) |
| \(\ln x\) |
\(1/x\) |
| \(\log_a x\) |
\(1/(x \ln a)\) |
| \(\sin x\) |
\(\cos x\) |
| \(\cos x\) |
\(-\sin x\) |
| \(\tan x\) |
\(\sec^2 x\) |
| \(\arcsin x\) |
\(1 / \sqrt{1 - x^2}\) |
| \(\arctan x\) |
\(1 / (1 + x^2)\) |
A.2.3 求导法则
- 链式: \((f(g(x)))' = f'(g) g'\)
- 乘积: \((uv)' = u'v + uv'\)
- 商: \((u/v)' = (u'v - uv') / v^2\)
- 反函数: \(\dfrac{dx}{dy} = 1 / (dy/dx)\)
A.2.4 常用积分
| \(\int f(x) dx\) | |
| ---------------------------------------------------------------------------------- | --- | ---- | --- |
| \(\int x^n dx = \dfrac{x^{n+1}}{n+1} + C\) (\(n \ne -1\)) | |
| \(\int \dfrac{1}{x} dx = \ln | x | + C\) | |
| \(\int e^x dx = e^x + C\) | |
| \(\int \sin x dx = -\cos x + C\) | |
| \(\int \dfrac{1}{1+x^2} dx = \arctan x + C\) | |
| \(\int \dfrac{1}{\sqrt{1-x^2}} dx = \arcsin x + C\) | |
| 高斯 \(\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}\) | |
| 高斯一般 \(\int_{-\infty}^\infty e^{-(x-\mu)^2 / 2\sigma^2} dx = \sigma\sqrt{2\pi}\) | |
| Gamma \(\int_0^\infty x^{n-1} e^{-x} dx = \Gamma(n)\) | |
A.2.5 泰勒展开
\[
f(x) = f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2!}(x-a)^2 + \dots
\]
常见 (在 0 处):
- \(e^x = 1 + x + x^2/2 + x^3/6 + \dots\)
- \(\ln(1+x) = x - x^2/2 + x^3/3 - \dots\)
- \((1+x)^\alpha = 1 + \alpha x + \dfrac{\alpha(\alpha-1)}{2} x^2 + \dots\)
- \(\sin x = x - x^3/6 + x^5/120 - \dots\)
A.2.6 矩阵基本
| 概念 |
定义 |
| 转置 |
\((A^T)_{ij} = A_{ji}\) |
| 迹 |
\(\text{tr}(A) = \sum A_{ii}\) |
| 行列式 |
\(\det A = \prod \lambda_i\) |
| 特征值 |
\(A v = \lambda v\) |
| 正定 |
\(x^T A x > 0\) ∀ \(x \ne 0\) |
| 正交 |
\(A^T A = I\) |
| 对称 |
\(A = A^T\), 实特征值 |
A.2.7 矩阵分解
| 分解 |
形式 |
用途 |
| LU |
\(A = LU\) |
解线性方程 |
| QR |
\(A = QR\) |
OLS |
| Cholesky |
\(A = LL^T\) (对称正定) |
多元高斯 |
| 特征值 |
\(A = V \Lambda V^{-1}\) |
主成分 |
| SVD |
\(A = U \Sigma V^T\) |
任何矩阵, 通用 |
A.2.8 矩阵微积分常用
|
标量对向量 |
向量对向量 |
| \(a^T x\) |
\(a\) |
|
| \(x^T A x\) |
\((A + A^T) x\) |
|
| \(\|x\|^2\) |
\(2 x\) |
|
| \(\log \det A\) (对 \(A\)) |
\(A^{-T}\) |
|
| \(\text{tr}(AB)\) (对 \(A\)) |
\(B^T\) |
|
| \(A x\) (对 \(x\)) |
|
\(A\) |
链式 (Jacobian 乘): \(\dfrac{\partial f}{\partial x} = \dfrac{\partial f}{\partial g} \dfrac{\partial g}{\partial x}\).
A.2.9 概率基础
| 公式 |
|
| 全概率 |
\(P(B) = \sum_i P(B \mid A_i) P(A_i)\) |
| 贝叶斯 |
\(P(A \mid B) = \dfrac{P(B \mid A) P(A)}{P(B)}\) |
| 独立 |
\(P(AB) = P(A) P(B)\) |
| 期望线性 |
\(E[aX + bY] = a E[X] + b E[Y]\) |
| 方差 |
\(V(X) = E[X^2] - (E[X])^2\) |
| 协方差 |
\(\text{Cov}(X, Y) = E[XY] - E[X] E[Y]\) |
| 和方差 |
\(V(X+Y) = V(X) + V(Y) + 2\text{Cov}\) |
| 矩母函数 |
\(M_X(t) = E[e^{tX}]\) |
A.2.10 常见分布速查
| 分布 |
期望 |
方差 |
| Bernoulli(\(p\)) |
\(p\) |
\(p(1-p)\) |
| Binomial(\(n,p\)) |
\(np\) |
\(np(1-p)\) |
| Poisson(\(\lambda\)) |
\(\lambda\) |
\(\lambda\) |
| Geometric(\(p\)) |
\(1/p\) |
\((1-p)/p^2\) |
| Uniform(\(a,b\)) |
\((a+b)/2\) |
\((b-a)^2/12\) |
| Exponential(\(\lambda\)) |
\(1/\lambda\) |
\(1/\lambda^2\) |
| Normal(\(\mu, \sigma^2\)) |
\(\mu\) |
\(\sigma^2\) |
| Gamma(\(\alpha, \beta\)) |
\(\alpha/\beta\) |
\(\alpha/\beta^2\) |
| Beta(\(\alpha, \beta\)) |
\(\dfrac{\alpha}{\alpha+\beta}\) |
\(\dfrac{\alpha\beta}{(\alpha+\beta)^2 (\alpha+\beta+1)}\) |
| \(\chi^2_k\) |
\(k\) |
\(2k\) |
| \(t_k\) (\(k>2\)) |
\(0\) |
\(k/(k-2)\) |
A.2.11 重要不等式
- Markov: \(P(X \geq a) \leq E[X]/a\) (\(X \geq 0\))
- Chebyshev: \(P(|X - \mu| \geq k\sigma) \leq 1/k^2\)
- Jensen: 凸函数 \(E[\phi(X)] \geq \phi(E[X])\)
- Cauchy-Schwarz: \(|E[XY]| \leq \sqrt{E[X^2] E[Y^2]}\)
- Hoeffding: 有界变量样本均值的指数集中
- 大数定律 / 中心极限定理 (§5)
A.2.12 优化常用
| 概念 |
|
| 梯度 \(\nabla f\) |
上升最快方向 |
| Hessian \(\nabla^2 f\) |
凸性: 半正定 |
| 凸函数 |
局部最小 = 全局最小 |
| 拉格朗日 |
\(L(x, \lambda) = f(x) + \lambda g(x)\) |
| KKT |
约束优化最优条件 |
A.2.13 下一步
- 下节 §A.3 概率分布表 (z, t, χ², F)。