## 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 $[itex]p=1$, $MinkowskiDis((a,b,a,c,d),(b,b,d,c,a))= 1+0+1+0+1 = 3$.
• 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 $[itex]p=1$, $MinkowskiSim((1,0,0,1,0),(0,0,1,1,0))= 1-\frac{1+0+1+0+0}{5} = 3/5 = 0.6$.

## 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 $[itex]p=1$ we have Hamming similarity.

## Aplications Edit

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