给定样本集合 ${\displaystyle X\subset {\mathbb{R}}^m}$, ${\displaystyle x_i=(x_{1i},\cdots ,x_{mi}{)}^{\mathrm{T}}\in X,x_j=(x_{1j},\cdots ,x_{mj}{)}^{\mathrm{T}}\in X}$. 定义 ${\displaystyle x_i,x_j}$ 的Minkowski 距离为 $$d_{ij}={(\sum _{k=1}^m|x_{ki}-x_{kj}{|}^p)}^{\frac{1}{p}},p\ge 1.$$

• ${\displaystyle p=2}$ 时, 称为 Euclide 距离: ${\displaystyle d_{ij}={(\sum _{k=1}^m|x_{ki}-x_{kj}{|}^2)}^{\frac{1}{2}}}$;
• ${\displaystyle p=1}$ 时, 称为 Manhattan 距离: ${\displaystyle d_{ij}=\sum _{k=1}^m|x_{ki}-x_{kj}|}$;
• ${\displaystyle p=\mathrm{\infty }}$ 时, 称为 Chebyshev 距离: ${\displaystyle d_{ij}=\underset{k}{max}|x_{ki}-x_{kj}|}$.

给定样本集合 ${\displaystyle X=[x_{ij}{]}_{m\times n}}$, 协方差矩阵 为 ${\displaystyle S}$. 则 ${\displaystyle x_i,x_j}$ 的Mahalanobis距离定义为 $$d_{ij}=[(x_i-x_j{)}^{\mathrm{T}}{S}^{-1}(x_i-x_j){]}^{\frac{1}{2}}.$$
${\displaystyle x_i,x_j}$ 是 ${\displaystyle X}$ 的第 ${\displaystyle i,j}$ 列.

当 ${\displaystyle S=I}$, 马氏距离就是欧式距离.

${\displaystyle x_i,x_j}$ 的相关系数定义为 $$r_{ij}=\frac{\sum _{k=1}^m(x_{ki}-{\bar{x}}_i)(x_{kj}-{\bar{x}}_j)}{{[\sum _{k=1}^m(x_{ki}-{\bar{x}}_i{)}^2\sum _{k=1}^m(x_{kj}-{\bar{x}}_j{)}^2]}^{\frac{1}{2}}},$$ 其中 ${\displaystyle {\bar{x}}_i=\frac{1}{m}\sum _{k=1}^m{x}_{ki},{\bar{x}}_j=\frac{1}{m}\sum _{k=1}^m{x}_{kj}}$.

给定正数 ${\displaystyle T}$.

1. ${\displaystyle \mathrm{\forall }{x}_i,x_j\in G:d_{ij}\le T}$, 则 ${\displaystyle G}$ 是一个类.
2. ${\displaystyle \mathrm{\forall }{x}_i,\mathrm{\exists }{x}_j\in G:d_{ij}\le T}$, 则 ${\displaystyle G}$ 是一个类.
3. ${\displaystyle \mathrm{\forall }{x}_i\in G,\frac{1}{n_G-1}\sum _{x_j\in G}{d}_{ij}\le T}$, 则 ${\displaystyle G}$ 是一个类.
4. 给定正数 ${\displaystyle V}$. ${\displaystyle \mathrm{\forall }{x}_i,x_j}$, ${\displaystyle d_{ij}}$ 满足 $$\frac{1}{n_G(n_G-1)}\sum _{x_i\in G}\sum _{x_j\in G}{d}_{ij}\le T,d_{ij}\le V,$$ 则 ${\displaystyle G}$ 是一个类.

1. 均值/中心 ${\displaystyle {\bar{x}}_G=\frac{1}{n_G}\sum _{i=1}^{n_G}{x}_i}$.
2. 直径 ${\displaystyle D_G=\underset{x_i,x_j\in G}{max}{d}_{ij}}$.
3. 样本散布矩阵 ${\displaystyle A_G=\sum _{i=1}^{n_G}(x_i-{\bar{x}}_G)(x_i-{\bar{x}}_G{)}^{\mathrm{T}}}$;
   样本协方差矩阵 ${\displaystyle S_G=\frac{1}{m-1}{A}_G=\frac{1}{m-1}\sum _{i=1}^{n_G}(x_i-{\bar{x}}_G)(x_i-{\bar{x}}_G{)}^{\mathrm{T}}}$.

1. 最短距离(single linkage) ${\displaystyle D_{pq}=min\{d_{ij}|x_i\in G_p,x_j\in G_q\}}$.
2. 最长距离/完全连接(complete linkage) ${\displaystyle D_{pq}=max\{d_{ij}|x_i\in G_p,x_j\in G_q\}}$.
3. 中心距离 ${\displaystyle D_{pq}=d_{{\bar{x}}_p{\bar{x}}_q}}$.
4. 平均距离 ${\displaystyle D_{pq}=\frac{1}{n_p{n}_q}\sum _{x_i\in G_p}\sum _{x_j\in G_q}{d}_{ij}}$.