A Set Theory Justification of Garuti’s Compatibility Index: Generalization of Jaccard Index Working within Weighted Environment

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Author: Claudio Garuti
Page Range: 195-214
Published in: International Journal of Mathematics, Game Theory, and Algebra, 30#4
ISSN: 1060-9881

Table of Contents

ABSTRACT

This chapter is aimed to increase the certainty and reliability of G index through different examples and showing that G index can be reached by two different paths, first using vector algebra as a weighted vector projection applying the concept of inner vector product and next, applying main concepts of set theory. Finally, comparing G index with others known compatibility indices. And, using set theory, showing that the G index is a sort of generalization of the Jaccard (J) index that considers the weights of the elements of two different sets. Making it applicable in the decision-making domain to measure similarity (compatibility) between priority vectors, system values and recognition of behavioral patterns among others features.
There is huge number of applications of G index in different fields, such as:

• On Group Decision Making; measuring how close are the value systems that belong to different people.
• On Quality Tests, measuring what MCDM decision method is building a better metric.
• On Agricultural; measuring the proximity between the cultivate plants against a healthy plant (based on its micro and macronutrients) and selecting the best nutrient seller.
• On Shiftwork Prioritization; measuring how close are the different views among the different stakeholders (Worker’s view, Company view, Community view).
• On Company Social Responsibility (CSR); measuring how close are the different views among the different stakeholders (Economic, Environmental and Social view).
• On Natural Risk Disaster & Management; to enhance the resource allocation process (aligning resources with priority hazards and social vulnerabilities).

But, one application that G has made an important contribution is the medical diagnose field. In this case G index is located in the core of the system (the AI engine) for pattern recognition. The engine allows to compare weighted profiles to calculate the similarity (compatibility) between the patient’s profile (the set of patient’s symptoms and signs) against a set of weighted diseases profiles located in the data base. The result of this process is a prioritized list of possible diseases (with a degree of certainty from 0% – 100%) for the patient.

Keywords: Saaty’s Compatibility index, Jaccard’s Similitude Index, Garuti’s Compatibility Index, Weighted environments, Set Theory, Order Topology

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