Minkowski distance
TheMinkowski distanceorMinkowski metricis ametricin anormed vector spacewhich can be considered as a generalization of both theEuclidean distanceand theManhattan distance.It is named after the Polish mathematicianHermann Minkowski.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Minkowski_distance_examples.svg/220px-Minkowski_distance_examples.svg.png)
Definition
[edit]The Minkowski distance of order(whereis an integer) between two points is defined as:
Forthe Minkowski distance is ametricas a result of theMinkowski inequality.[1]Whenthe distance betweenandisbut the pointis at a distancefrom both of these points. Since this violates thetriangle inequality,forit is not a metric. However, a metric can be obtained for these values by simply removing the exponent ofThe resulting metric is also anF-norm.
Minkowski distance is typically used withbeing 1 or 2, which correspond to theManhattan distanceand theEuclidean distance,respectively.[2]In the limiting case ofreaching infinity, we obtain theChebyshev distance:
Similarly, forreaching negative infinity, we have:
The Minkowski distance can also be viewed as a multiple of thepower meanof the component-wise differences betweenand
The following figure shows unit circles (thelevel setof the distance function where all points are at the unit distance from the center) with various values of:
Applications
[edit]The Minkowski metric is very useful in the field ofmachine learningandAI.Many popular machine learning algorithms use specific distance metrics such as the aforementioned to compare the similarity of two data points. Depending on the nature of the data being analyzed, various metrics can be used. The Minkowski metric is most useful for numerical datasets where you want to determine the similarity of size between multiple datapoint vectors.
See also
[edit]- Generalized mean– N-th root of the arithmetic mean of the given numbers raised to the power n
- space– Function spaces generalizing finite-dimensional p norm spaces
- Norm (mathematics)– Length in a vector space
- -norm– Function spaces generalizing finite-dimensional p norm spaces
References
[edit]- ^Şuhubi, Erdoğan S. (2003), "Chapter V: Metric Spaces",Functional Analysis,Springer Netherlands, pp. 261–356,doi:10.1007/978-94-017-0141-9_5,ISBN9789401701419
- ^Zezula, Pavel; Amato, Giuseppe; Dohnal, Vlastislav; Batko, Michal (2006), "Chapter 1, Foundations of Metric Space Searching, Section 3.1, Minkowski Distances",Similarity Search: The Metric Space Approach,Advances in Database Systems, Springer, p. 10,doi:10.1007/0-387-29151-2,ISBN9780387291512