Minkowski distance

Definition
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

 * For $$\delta=1-IdSim$$ and $$p=1$$, $$MinkowskiDis((a,b,a,c,d),(b,b,d,c,a))= 1+0+1+0+1 = 3$$.
 * 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
If $$\delta$$ is normalized, is possible to define the Minkowski similarity as

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

Examples

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

Variations

 * 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 $$p=1$$ we have Hamming similarity.

Aplications

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