FANDOM


Definition Edit

The Tversky contrast model similarity is a similarity function $ TverskyModelSim:\{0,1\}^n\times \{0,1\}^n\longrightarrow \mathbb{R} $ defined as

$ TverskyModelSim(u,v) = \alpha f(u\cap v) - \beta f(u-v) - \gamma f(v-u), $

where

  • $ u\cap u $ is the set of positions $ \{i|u_i=v_i=1\} $
  • $ u-v $ is the set of positions $ \{i|u_i=1,v_i=0\} $
  • $ v-u $ is the set of positions $ \{i|u_i=0,v_i=1\} $
  • $ f $ is a monotonic increasing function (usually a weighted sum).
  • $ \alpha $, $ \beta $ and $ \gamma $ are positive reals numbers (used as weights).

Depending of values of $ \alpha $, $ \beta $ and $ \gamma $ this function could not be simmetric.

Examples Edit

  • For $ \alpha=2 $, $ \beta=1 $ and $ \gamma=1 $, and $ f $ the cardinality of sets, $ TverksyModelSim((1,0,1,1,0),(0,1,1,1,1))=2|\{3,4\}|-|\{1\}|-|\{2,5\}| = 4 -1-2 = 1 $.
  • For $ \alpha=1 $, $ \beta=0 $ and $ \gamma=0 $, and $ f $ the cardinality of sets, we have a simmetric function: $ TverskyModelSim((1,0,1,1,0),(0,1,1,1,1))= TverskyModelSim((0,1,1,1,1),(1,0,1,1,0))=|\{3,4\}| = 2 $.
  • For $ \alpha=0 $, $ \beta=0.5 $ and $ \gamma=1 $, and $ f $ the cardinality of sets, we have an asimmetric function: $ TverskyModelSim((1,0,1,1,0),(0,1,1,1,1))=-0.5|\{1\}|-|\{2,5\}| = -2.5 $, but $ TverskyModelSim((0,1,1,1,1),(1,0,1,1,0))|=-0.5|\{2,5\}|-|\{1\}| = -2 $.

Normalization Edit

A possible way of nomalization is the ratio model similarity defined as

$ RatioModelSim(u,v) = \frac{\alpha f(u\cap v)}{\alpha f(u\cap v) + \beta f(u-v) + \gamma f(v-u)} $

Examples Edit

  • For $ \alpha=2 $, $ \beta=1 $ and $ \gamma=1 $, and $ f $ the cardinality of sets, $ RatioModelSim((1,0,1,1,0),(0,1,1,1,1))=\frac{2|\{3,4\}|}{2|\{3,4\}|+|\{1\}|+|\{2,5\}|} = \frac{4}{4+1+2} = 4/7 = 0.57 $.

Variations Edit

When $ \alpha=0 $ instead of a similarity function we get a distance function.


Applications Edit

This function is used to compare entities represented as vectors of boolean features. In contrast with Hamming similarity for vectors, only present features are taken into acount. The weights $ \alpha $, $ \beta $ and $ \gamma $ serves to ponderate de importance of common features of $ u $ and $ v $, features exlusives of $ u $ and features exlusives of $ v $ respectively, providing a wide range of similarity functions.

References Edit