Tversky contrast model similarity

Definition
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

 * 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
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

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

Applications
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.