Distances for NN Methods

Mikovski Metric

$$\delta ({\mathbf{x}}_k,{\mathbf{x}}_l)={\left[\sum _{i=1}^d|x_{ki}-k_{li}{|}^s\right]}^{1/s}$$

Euclidean Distance

$${\delta }_E({\mathbf{x}}_k,{\mathbf{x}}_l)=[({\mathbf{x}}_k-{\mathbf{x}}_l{)}^T({\mathbf{x}}_k-{\mathbf{x}}_l){]}^{1/2}$$

City Block Distance / Manhattan Distance

$$\delta ({\mathbf{x}}_k,{\mathbf{x}}_l)=\sum _{i=1}^d|x_{ki}-x_{li}|$$

Chobychev Distance

$$\delta ({\mathbf{x}}_k,{\mathbf{x}}_l)=\underset{i}{\max }|x_{ki}-x_{li}|$$

Squared Distance

$$\delta ({\mathbf{x}}_k,{\mathbf{x}}_l)=({\mathbf{x}}_k-{\mathbf{x}}_l{)}^{T}Q({\mathbf{x}}_k-{\mathbf{x}}_l)$$

Nonlinear Distances

$$\delta ({\mathbf{x}}_k,{\mathbf{x}}_l)=\{\begin{array}{ll}H& \text{if δE(xk,xl)≥T}\\ 0& \text{ if δE(xk,xl)<T}\end{array}$$