FANDOM


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 <math> 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 $.

Variations Edit

Applications Edit

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


References Edit