常见的距离函数总结
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1.距离函数总结
- 闵可夫斯基距离：给定样本 x → i = ( x i , 1 , x i , 2 , ⋯   , x i , n ) T \overrightarrow{\mathbf{x}}_{i}=\left(x_{i, 1}, x_{i, 2}, \cdots, x_{i, n}\right)^{T} x i=(xi,1,xi,2,⋯,xi,n)T, x → j = ( x j , 1 , x j , 2 , ⋯   , x j , n ) T \overrightarrow{\mathbf{x}}_{j}=\left(x_{j, 1}, x_{j, 2}, \cdots, x_{j, n}\right)^{T} x j=(xj,1,xj,2,⋯,xj,n)T，则闵可夫斯基距离定义为：
distance( ( x → i , x → j ) = ( ∑ d = 1 n ∣ x i , d − x j , d ∣ p ) 1 / p \text { distance( }\left(\overrightarrow{\mathbf{x}}_{i}, \overrightarrow{\mathbf{x}}_{j}\right)=\left(\sum_{d=1}^{n}\left|x_{i, d}-x_{j, d}\right|^{p}\right)^{1 / p} distance( (x i,x j)=(d=1∑n∣xi,d−xj,d∣p)1/p
- 当p=2时，闵可夫斯基距离就是欧式距离：
distance ( x → i , x → j ) = ∥ x → i − x → j ∥ 2 = ∑ d = 1 n ∣ x i , d − x j , d ∣ 2 \operatorname{distance}\left(\overrightarrow{\mathbf{x}}_{i}, \overrightarrow{\mathbf{x}}_{j}\right)=\left\|\overrightarrow{\mathbf{x}}_{i}-\overrightarrow{\mathbf{x}}_{j}\right\|_{2}=\sqrt{\sum_{d=1}^{n}\left|x_{i, d}-x_{j, d}\right|^{2}} distance(x i,x j)=∥∥∥x i−x j∥∥∥2=d=1∑n∣xi,d−xj,d∣2
- 当p=1时，闵可夫斯基距离就是曼哈顿距离：
distance ( x → i , x → j ) = ∥ x → i − x → j ∥ 1 = ∑ d = 1 n ∣ x i , d − x j , d ∣ \text { distance }\left(\overrightarrow{\mathbf{x}}_{i}, \overrightarrow{\mathbf{x}}_{j}\right)=\left\|\overrightarrow{\mathbf{x}}_{i}-\overrightarrow{\mathbf{x}}_{j}\right\|_{1}=\sum_{d=1}^{n}\left|x_{i, d}-x_{j, d}\right| distance (x i,x j)=∥∥∥x i−x j∥∥∥1=d=1∑n∣xi,d−xj,d∣
