Canberra distance explained

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 and for intrusion detection in computer security.^[4] It has also been used to analyze the gut microbiome in different disease states.^[5]

Definition

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

d( p , q ) = \sum ^{n} _{i =1} (|p _{i-q} _{i|})/(|p _{i|+|q} _{i|})

where

p = (p _{1,p} _{2,...,p} _{n)andq=(q} _{1,q} _{2,...,q} _{n)} 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

References

• Web site: Schulz . Jan . Canberra distance . Code 10 . 18 October 2011 .

Notes and References

1. Lance . Godfrey N.. Williams . William T. . W. T. Williams . Computer programs for hierarchical polythetic classification ("similarity analysis") . Computer Journal . 1966 . 9 . 1 . 60–64 . 10.1093/comjnl/9.1.60 .
2. Lance . Godfrey N. . Williams . William T. . W. T. Williams . Mixed-data classificatory programs I.) Agglomerative Systems . Australian Computer Journal . 1967 . 15–20 .
3. 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. Syed Masum . Emran . Nong . Ye . 2002 . Robustness of chi-square and Canberra distance metrics for computer intrusion detection . . 18 . 1 . 19–28 . 10.1002/qre.441 . 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 . Parkinson's disease and Parkinson's disease medications have distinct signatures of the gut microbiome . Movement Disorders . May 2017 . 32 . 5 . 739–749 . 10.1002/mds.26942 . 28195358 . 5469442 .
6. Faith . Daniel P. . Minchin . Peter R. . Belbin . Lee . April 1987 . 10.1007/bf00038687 . 1–3 . Vegetatio . 57–68 . Compositional dissimilarity as a robust measure of ecological distance . 69.