Abstract
This chapter provides a general framework for analyzing and processing relational data in a granular computing environment. It introduces compound information systems for relational data. It also extends an attribute-value language for defining relational patterns.
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Notes
- 1.
The approach introduced in this chapter can also be applied with no changes to a framework that uses a covering of the universe to form granules.
- 2.
In this approach the equality relation in the construction of conditions is used. The approach can easily be extended to a case where the conditions are also constructed by applying equality relations and a membership relation.
- 3.
The notation \(SEM_{IS}((a,v))\) is simplified by writing \(SEM_{IS}(a,v)\).
- 4.
Symbolic values are abbreviated to their first letters. Granules in the table are presented in a simplified form, e.g. the granule \(\left( 30,\{3,4\}\right) \) from column age corresponds to the granule \(\left( (age,30),\{3,4\}\right) \).
- 5.
\(SEM_i\) is the semantics of \(L_i\).
- 6.
The index (i.e. the relation identifier) is omitted if this does not lead to a confusion.
- 7.
It is assumed by default that a condition can be constructed based on two key attributes if they are of the same type.
- 8.
The intersection of \(A_i\) and \(A_j\) is empty because all attributes names are distinct from one another, e.g. \(customer.id\ne purchase.id\).
- 9.
1. The subset of \(A_i\) that consists of all key attributes is denoted by \((A_i)_{key}\). 2. As previously, it is assumed that key attributes are of the same type.
- 10.
\(\pi _{A}(\bullet )\) is understood as a projection over the attributes from A.
- 11.
The rule conclusion is a trivial formula and means that an object which satisfies the formula belongs to the relation.
- 12.
Proofs of the propositions formulated in this chapter can be found in [41].
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Hońko, P. (2017). Compound Information Systems. In: Granular-Relational Data Mining. Studies in Computational Intelligence, vol 702. Springer, Cham. https://doi.org/10.1007/978-3-319-52751-2_6
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DOI: https://doi.org/10.1007/978-3-319-52751-2_6
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