Canberra distance

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

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.

See also

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. 1 2 Lance, Godfrey N.; Williams, William T. (1967). "Mixed-data classificatory programs I.) Agglomerative Systems". Australian Computer Journal: 15–20.
  3. 1 2 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. S2CID 122959778.
  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.; Molho, Eric; Zabetian, Cyrus P.; Knight, Rob; Payami, Haydeh (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. PMC 5469442. 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