Canberra distance

Short description: Measure of distance between vectors

The Canberra distance is a numerical measure of the distance between pairs of points in a vector space, introduced in 1966[1] and refined in 1967[2] by Godfrey N. Lance and William T. Williams. It is a weighted version of L₁ (Manhattan) distance.[3] The Canberra distance has been used as a metric for comparing ranked lists[3] and for intrusion detection in computer security.[4] It has also been used to analyze the gut microbiome in different disease states.[5]

## Contents

* 1 Definition
* 2 See also
* 3 Notes
* 4 References

## Definition

The Canberra distance d between vectors p and q in an n-dimensional real vector space is given as follows:

d(𝐩,𝐪)=∑i=1n|pi−qi||pi|+|qi|

where

𝐩=(p1,p2,…,pn) and 𝐪=(q1,q2,…,qn)

are vectors.

The Canberra metric, Adkins form, divides the distance d by (n-Z) where Z is the number of attributes that are 0 for p and q.[2][6]

## See also

* Normed vector space
* Metric
* Manhattan distance

## Notes

1. ↑ Lance, Godfrey N.; Williams, William T. (1966). "Computer programs for hierarchical polythetic classification ("similarity analysis")". Computer Journal 9 (1): 60–64. doi:10.1093/comjnl/9.1.60.
2. ↑ 2.0 2.1 Lance, Godfrey N.; Williams, William T. (1967). "Mixed-data classificatory programs I.) Agglomerative Systems". Australian Computer Journal: 15–20.
3. ↑ 3.0 3.1 Giuseppe Jurman; Samantha Riccadonna; Roberto Visintainer; Cesare Furlanello; "Canberra Distance on Ranked Lists", in Shivani Agrawal; Chris Burges; Koby Crammer (editors); Proceedings, Advances in Ranking – NIPS 09 Workshop, 2009, p. 22–27
4. ↑ Emran, Syed Masum; Ye, Nong (2002). "Robustness of chi-square and Canberra distance metrics for computer intrusion detection". Quality and Reliability Engineering International 18 (1): 19–28. doi:10.1002/qre.441.
5. ↑ Hill-Burns, Erin M.; Debelius, Justine W.; Morton, James T.; Wissemann, William T.; Lewis, Matthew R.; Wallen, Zachary D.; Peddada, Shyamal D.; Factor, Stewart A. et al. (May 2017). "Parkinson's disease and Parkinson's disease medications have distinct signatures of the gut microbiome". Movement Disorders 32 (5): 739–749. doi:10.1002/mds.26942. PMID 28195358.
6. ↑ Faith, Daniel P.; Minchin, Peter R.; Belbin, Lee (April 1987). "Compositional dissimilarity as a robust measure of ecological distance". Vegetatio 69 (1–3): 57–68. doi:10.1007/bf00038687.

## References

* Schulz, Jan. "Canberra distance". Code 10. http://www.code10.info/index.php?option=com_content&view=article&id=49:article_canberra-distance&catid=38:cat_coding_algorithms_data-similarity&Itemid=57. Retrieved 18 October 2011.

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