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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

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