Extend Tversky’s Ratio Model to an Asymmetric Similarity Measurement Model with Three Conditional Parameters: 3p-ASM Model

Abstract

Abstract Generally, the similarity between objects is often measured by symmetric operators, such as Cosine, Dice, and Jaccard similarity. However, the ratio model originally proposed by Tversky pointed out that the similarity using the feature matching method tends to be asymmetric. Furthermore, in many practical situations, the existing similarity measures using the feature matching method have some limitations: the calculation formulas are symmetrical, it is not intuitive based on binary features, and it is not easy to calculate based on fuzzy sets. To overcome such limitations, some other asymmetries have proposed to directly combine Tversky’s ratio model with the classical symmetric similarity metric, which in turn leads to the inability to identify different features between the compared objects and affects their similarity accuracy. Therefore, aiming to avoid these, this paper will focus on extending Tversky’s ratio model to a series of 3-parameter asymmetric similarity metrics (3p-ASM), using three conditional parameters to describe both common and different features. First, the set-based 3p-ASM is achieved due to the general and fuzzy set-theoretic operations, when estimating features in $$\left\{ 0,1\right\} $$ 0 , 1 and $$\left[ 0,1\right] $$ 0 , 1 , respectively. Then, considering that the estimated values of features can also be expressed as vectors, it will be extended to the vector-based 3p-ASM. Finally, a vector form of 3p-ASM is compared with existing classical methods and a comparative analysis is performed to demonstrate its effectiveness and validity. It is then applied to the KNN model in order to select the most similar items.

Keywords

• Similarity
• Asymmetric
• Set-theoretic operations
• Tversky’s ratio model
• 3p-ASM