Comparing Objects or Categories Described by Attributes

Abstract

Let us recapitulate briefly the development of the previous chapters. As emphasized above (see Sect. 3.5.4 of Chap. 3), given a set $\mathcal {A} = \{ a^{1}, a^{2}, \ldots , a^{j}, \ldots , a^{p} \}$ of descriptive attributes, description may concern a set $\mathcal {O} = \{ o_{1}, o_{2}, \ldots , o_{i}, \ldots , o_{n} \}$ of elementary objects or a set $\Gamma = \{ C_{1}, C_{2}, \ldots , C_{i}, \ldots , C_{I} \}$ of categories or classes. In Sect. 4.2 of Chap. 4 we studied the problem of the definition of a similarity index between objects or attributes when $\mathcal {A}$ is constituted by Boolean attributes. In this analysis several facets are considered: combinatorial, metrical, ordinal and statistical. We saw in Sect. 4.2.1.3, how to transpose a similarity index between objects defined in the framework of Boolean description, to the case of description by numerical attributes. Conversely, we have transposed a similarity or distance function defined on $\mathcal {O}$ in the case of numerical description to that where $\mathcal {A}$ is a set of Boolean attributes.