Definition Edit

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

Examples Edit

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

Normalization Edit

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

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

Examples Edit

• 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 $E=\{0,1\}$, $ManhattanSim((1,0,1,0,1),(1,1,0,0,1)) = 1-\sqrt{\frac{0+1+1+0+0}{5}} = 0.37$.

Applications Edit

• Numeric vectors (codes).
• Vectors of boolean features.