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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, 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 aritmetic 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

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