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Definition Edit

Given \delta: E\times E \longrightarrow \mathbb{R} a distance function between elements of a universe set E, the Minkowski distance is a function MinkowskiDis:E^n\times E^n \longrightarrow \mathbb{R} defined as


MinkowskiDis(u,v)=\left(\sum_{i=1}^{n}\delta'(u[i],v[i])^p\right)^{1/p},

where p is a positive integer.

Examples Edit

  • For \delta=1-IdSim and <math>p=1, MinkowskiDis((a,b,a,c,d),(b,b,d,c,a))= 1+0+1+0+1 = 3.
  • For \delta the difference and p=1 (Manhattan distance), MinkowskiDis((1,4,2,3),(1,3,4,1))= 0+1+2+2 =5 .
  • For \delta the difference and p=1 (Euclidean distance), MinkowskiDis((1,4,2,3),(1,3,4,1))= \sqrt{0^2+1^2+2^2+2^2} = 3 .

Normalization Edit

If \delta is normalized, is possible to define the Minkowski similarity as


MinkowskiSim(u,v)=1-\frac{MinkowskiDis(u,v)}{\sqrt[p]{n}}.

Examples Edit

  • If E=\{0,1\}, \delta=1-IdSim and <math>p=1, MinkowskiSim((1,0,0,1,0),(0,0,1,1,0))= 1-\frac{1+0+1+0+0}{5} = 3/5 = 0.6 .

Variations Edit

Aplications Edit

  • Comparing vectors in a metric space (usually vectors in \mathbb{R}^n).
  • Comparing vectors of boolean features.

References Edit

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