FANDOM


Definition Edit

Given a number set $ E $, the Manhattan distance is a function $ ManhattanDis:E^n\times E^n \longrightarrow \mathbb{R} $ defined as

$ ManhattanDis(u,v)=\sum_{i=1}^{n}|u[i]-v[i]|. $

Examples Edit

  • $ ManhattanDis((1,2,3,4),(4,3,2,1)) = 3+1+1+3 = 8 $.
  • $ ManhattanDis((1,0,1,0,1),(1,1,0,0,1)) = 0+1+1+0+0 = 2 $.

Normalization Edit

If $ E $ is a bounded set, it is possible to normalize the difference dividing by the range of $ E $, then normalization is

$ ManhattanSim(u,v)=1-\frac{\sum_{i=1}^{n}|u[i]-v[i]|}{n}. $

that is the arithmetic mean of the normalized differences.

Examples Edit

  • If $ E=[0,10] $, $ ManhattanSim((1,2,3,4),(4,3,2,1)) = 1-\frac{0.3+0.1+0.1+0.3}{4} = 0.8 $.
  • If $ E=\{0,1\} $, $ ManhattanSim((1,0,1,0,1),(1,1,0,0,1)) = 1-\frac{0+1+1+0+0}{5} = 0.6 $.

Variations Edit

Applications Edit

  • Numeric vectors (codes).
  • Vectors of boolean features.

References Edit