Canberra distance

The Canberra distance is a numerical measure of the distance between pairs of points in a vector space, introduced in 1966{{cite journal |last1=Lance |first1=Godfrey N.|last2=Williams |first2=William T. |author2-link=W. T. Williams |title=Computer programs for hierarchical polythetic classification ("similarity analysis") |journal=Computer Journal |year=1966 |volume=9 |issue=1 |pages=60–64 |doi=10.1093/comjnl/9.1.60 |doi-access= }}

and refined in 1967{{cite journal |last1=Lance |first1=Godfrey N. |last2=Williams |first2=William T. |author2-link=W. T. Williams |title=Mixed-data classificatory programs I.) Agglomerative Systems |journal=Australian Computer Journal |year=1967 |pages=15–20 }} by Godfrey N. Lance and William T. Williams. It is a weighted version of L₁ (Manhattan) distance.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

The Canberra distance has been used as a metric for comparing ranked lists and for intrusion detection in computer security.{{cite journal |first1=Syed Masum |last1=Emran |first2=Nong |last2=Ye |year=2002 |title=Robustness of chi-square and Canberra distance metrics for computer intrusion detection |journal=Quality and Reliability Engineering International |volume=18 |issue=1 |pages=19–28 |doi=10.1002/qre.441 |s2cid=122959778 }} It has also been used to analyze the gut microbiome in different disease states.{{cite journal |last1=Hill-Burns |first1=Erin M. |last2=Debelius |first2=Justine W. |last3=Morton |first3=James T. |last4=Wissemann |first4=William T. |last5=Lewis |first5=Matthew R. |last6=Wallen |first6=Zachary D. |last7=Peddada |first7=Shyamal D. |last8=Factor |first8=Stewart A. |last9=Molho |first9=Eric |last10=Zabetian |first10=Cyrus P. |last11=Knight |first11=Rob |last12=Payami |first12=Haydeh |title=Parkinson's disease and Parkinson's disease medications have distinct signatures of the gut microbiome |journal=Movement Disorders |date=May 2017 |volume=32 |issue=5 |pages=739–749 |doi=10.1002/mds.26942 |pmid=28195358 |pmc=5469442 }}

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

: $\mathbf{p}=(p_1,p_2,\dots,p_n)\text{ and }\mathbf{q}=(q_1,q_2,\dots,q_n)$

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.{{cite journal

| last1 = Faith | first1 = Daniel P.

| last2 = Minchin | first2 = Peter R.

| last3 = Belbin | first3 = Lee

| date = April 1987

| doi = 10.1007/bf00038687

| issue = 1–3

| journal = Vegetatio

| pages = 57–68

| title = Compositional dissimilarity as a robust measure of ecological distance

| volume = 69}}

Notes

References

{{cite web |last=Schulz |first=Jan |title=Canberra distance |url=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 |work=Code 10 |accessdate=18 October 2011 }}

Category:Digital geometry

Category:Metric geometry

Category:Distance

Canberra distance

Definition

See also

Notes

References