Euclidean distance

Definition[]

Given a number set $E$ , the Euclidean distance is a function ${\displaystyle EuclideanDis:E^n\times E^n \longrightarrow \mathbb{R}</math< defined as <math> EuclideanDis(u,v)=\sqrt{\sum_{i=1}^{n}(u[i]-v[i])^2}. }$

Examples[]

$EuclideanDis((1,2,3,4),(4,3,2,1)) = \sqrt{3^2+1^2+1^2+3^2} = \sqrt{20} = 4.47$ .
$EuclideanDis((1,0,1,0,1),(1,1,0,0,1)) = \sqrt{0+1+1+0+0} = 1.41$ .

Normalization[]

If $E$ is a bounded set, is possible to normalize the difference dividing by the range of $E$ , then normalization is

${\displaystyle EuclideanSim(u,v)=1-\sqrt{\frac{\sum_{i=1}^{n}(u[i]-v[i])^2}{n}}. }$

Examples[]

If $E=[0,10]$ , $ManhattanSim((1,2,3,4),(4,3,2,1)) = 1-\sqrt{\frac{0.3^2+0.1^2+0.1^2+0.3^2}{4}} = 0.78$ .
If $E=\{0,1\}$ , $ManhattanSim((1,0,1,0,1),(1,1,0,0,1)) = 1-\sqrt{\frac{0+1+1+0+0}{5}} = 0.37$ .