Canberra distance

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]

Definition

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

d(\mathbf {p} ,\mathbf {q} )=\sum _{i=1}^{n}{\frac {|p_{i}-q_{i}|}{|p_{i}|+|q_{i}|}}

where

\mathbf {p} =(p_{1},p_{2},\dots ,p_{n}){\text{ and }}\mathbf {q} =(q_{1},q_{2},\dots ,q_{n})

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]

^ 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.
^ ^a ^b Lance, Godfrey N.; Williams, William T. (1967). "Mixed-data classificatory programs I.) Agglomerative Systems". Australian Computer Journal: 15–20.
^ ^a ^b 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
^ 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.
^ 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.
^ 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.

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