FANDOM


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

  • If $ E $ is a number set, $ \delta $ is the difference and $ p=1 $ we have the Manhattan distance.
  • If $ E $ is a number set, $ \delta $ is the difference and $ p=2 $ we have the Euclidean distance.
  • If $ E $ is a number set, $ \delta $ is the difference and $ p=\infty $ we have the Tchebychev distance.
  • If $ E=\{0,1\} $, $ \delta=1-IdSim $ and $ <math>p=1 $ we have Hamming similarity.

Aplications Edit

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

References Edit