World Library  
Flag as Inappropriate
Email this Article

Minkowski distance

Article Id: WHEBN0020326689
Reproduction Date:

Title: Minkowski distance  
Author: World Heritage Encyclopedia
Language: English
Subject: Distance, Lp space, Time series
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Minkowski distance

The Minkowski distance is a metric on Euclidean space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.

Definition

The Minkowski distance of order p between two points

P=(x_1,x_2,\ldots,x_n)\text{ and }Q=(y_1,y_2,\ldots,y_n) \in \mathbb{R}^n

is defined as:

\left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}

For p\geq1, the Minkowski distance is a metric as a result of the Minkowski inequality. When p<1, the distance between (0,0) and (1,1) is 2^{1/p}>2, but the point (0,1) is at a distance 1 from both of these points. Since this violates the triangle inequality, for p<1 it is not a metric.

Minkowski distance is typically used with p being 1 or 2. The latter is the Euclidean distance, while the former is sometimes known as the Manhattan distance. In the limiting case of p reaching infinity, we obtain the Chebyshev distance:

\lim_{p\to\infty}{\left(\sum_{i=1}^n |x_i-y_i|^p\right)^\frac{1}{p}} = \max_{i=1}^n |x_i-y_i|. \,

Similarly, for p reaching negative infinity, we have:

\lim_{p\to-\infty}{\left(\sum_{i=1}^n |x_i-y_i|^p\right)^\frac{1}{p}} = \min_{i=1}^n |x_i-y_i|. \,

The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between P and Q.

The following figure shows unit circles with various values of p:

Unit circles using different Minkowski distance metrics.

See also

External links

Simple IEEE 754 implementation in C++

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.