基本公式
变换公式
离散傅里叶变换:将序列\(x_0,...,x_{n-1}\) 变换成\(X_0,...,X_{n-1}\) ,正变换公式为\(x_f=1/\sqrt{n}\sum_{t=0}^{n-1}x_t*exp(-j2\pi tf/n),f=0,...,n-1,j=\sqrt{-1}\) ,逆变换为\(x_t=1/\sqrt{n}\sum_{f=0}^{n-1}x_f*exp(+j2\pi tf/n)\)
排列组合
\(A_n^m=\frac{n!}{(n-m)!},C_n^m=\frac{n!}{(n-m)!m!}\)
三角函数
积化和差:
\(sin\alpha cos\beta=\frac{1}{2}[sin(\alpha+\beta)+sin(\alpha-\beta)]\)
\(cos\alpha sin\beta=\frac{1}{2}[sin(\alpha+\beta)-sin(\alpha-\beta)]\)
\(cos\alpha cos\beta=\frac{1}{2}[cos(\alpha+\beta)+cos(\alpha-\beta)]\)
\(sin\alpha sin\beta=-\frac{1}{2}[cos(\alpha+\beta)-cos(\alpha-\beta)]\)
和差化积:
\(sin\alpha+sin\beta=2sin\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}\)
\(sin\alpha-sin\beta=2cos\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}\)
\(cos\alpha+cos\beta=2cos\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}\)
\(cos\alpha-cos\beta=-2sin\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}\)
和差角公式
\(sin(\alpha+\beta)=sin\alpha cos\beta+cos\alpha sin\beta\)
\(sin(\alpha-\beta)=sin\alpha cos\beta-cos\alpha sin\beta\)
\(cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta\)
\(cos(\alpha-\beta)=cos\alpha cos\beta+sin\alpha sin\beta\)
\(tan(\alpha+\beta)=\frac{tan\alpha+tan\beta}{1-tan\alpha·tan\beta}\)
\(tan(\alpha-\beta)=\frac{tan\alpha-tan\beta}{1+tan\alpha·tan\beta}\)
概率论
基本规则和定理
加和规则: \(p(X)=\sum_Yp(X,Y)\)
乘积规则: \(p(X,Y)=p(Y|X)p(X)\)
对称属性: \(p(X,Y)=p(Y,X)\)
贝叶斯定理: \(p(Y|X)=\frac{p(X|Y)p(Y)}{p(X)}\)
熵
两个分布p,q的交叉熵:\(H(p,q)=-\sum_xp(x)logq(x)\)
信息论
KL散度
考虑某个未知分布\(p(\vec{x})\) ,假设已经用一个近似分布\(q(\vec{x})\) 来建模它。\(KL(p||q)=-\int p(\vec{x})lnq(\vec{x})d\vec{x}-(-\int p(\vec{x})lnp(\vec{x})d\vec{x})=-\int p(\vec{x})ln\{\frac{q(\vec{x})}{p(\vec{x})}\}d\vec{x}\) ,称作分布\(p(\vec{x})\) 和\(q(\vec{x})\) 之间的相关熵或Kullback-Leibler divergence散度或KL散度。这不是一个对称量:\(KL(p||q)\ne KL(q||p)\) 。KL散度满足\(KL(p||q)\ge 0\) ,等号当且仅当\(p(\vec{x})=q(\vec{x})\) 取到
线性代数
行列式
行列式只针对方针,即\(\begin{bmatrix}
a & b \\
c & d \end{bmatrix}=ad-bc\)
\(det(M)=0\) 可推出M不可逆
singular矩阵: 线性独立的行或列或者任一行或列全为0
行列式变换公式
\(det(AB)=det(A)det(B)\)
\(det(M^{-1})=(det(M))^{-1}\)
\(det(M)det(M^{-1})=det(MM^{-1})=det(I)=1\)
markdown常用公式
常用符号
[ \(\rho\) \rho
]
[ \(\epsilon\) \epsilon
] [ \(\varepsilon\) \varepsilon
] 后者是前者的变量形式
[ \(\simeq\) \simeq
] [ \(\sim\) \sim
] [ \(\cong\) \cong
] [ \(\approx\) \approx
] [ \(\equiv\) \equiv
]
[ \(\propto\) \propto
] [ \(\ll\) \ll
] [ \(\gg\) \gg
]
[ \(\cdot\) \cdot
] [ \(\circ\) \circ
] [ \(\odot\) \odot
] [ \(\bigodot\) \bigodot
] [ \(\otimes\) \otimes
] [ \(\oplus\) \oplus
] [ \(\bullet\) \bullet
]
[ \(\div\) \div
] [ \(\pm\) \pm
] [ \(\mp\) \mp
]
[ \(\partial\) \partial
] [ \(\nabla\) \nabla
]
[ \(\forall\) \forall
] [ \(\Lambda\) \Lambda
] [ \(\cap\) \cap
] [ \(\cup\) \cup
]
[ \(\to\) \to
] [ \(\rightarrow\) \rightarrow
] [ \(\leftarrow\) \leftarrow
]
常用形式
[ \(\tilde{A}\) \tilde{A}
] [ \(\bar{A}\) \bar{A}
]
[ \(\mathcal{A}\) \mathcal{A}
]
大型公式
方程组:\(\begin{cases} 2x+2y=4 \\ 3x+3y=6 \\ \end{cases}\) \begin{cases} 2x+2y=4 \\ 3x+3y=6 \\ \end{cases}
带条件的方程组:\(KeepTopK(v,k)_i=\begin{cases} v_i, & \mbox{if }v_i\mbox{ is in the top }k \mbox{ elements of }v\\-\infty, & \mbox{otherwise}\end{cases}\) KeepTopK(v,k)_i=\begin{cases} v_i, & \mbox{if }v_i\mbox{ is in the top }k \mbox{ elements of }v\\-\infty, & \mbox{otherwise}\end{cases}
圆括号矩阵:\(\begin{pmatrix} 2 & 2 \\ 3 & 3 \end{pmatrix}\) \begin{pmatrix} 2 & 2 \\ 3 & 3 \end{pmatrix}
方括号矩阵:\(\begin{bmatrix} 2 & 2 \\ 3 & 3 \end{bmatrix}\) \begin{bmatrix} 2 & 2 \\ 3 & 3 \end{bmatrix}
参考资料
Bishop: Pattern Recognition and Machine Learning
markdown公式