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:

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

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.

See also

• Normed vector space
• Metric
• Manhattan distance

References

• Schulz, Jan. "Canberra distance". Code 10. Retrieved 18 October 2011.