Abstract
This chapter introduces relation-based granules, which can be used for representing both the relational data and patterns. Such patterns enable to express richer knowledge about relational data. The chapter also shows that the generation of patterns can be accelerated thanks to using an alternative representation of relational data constructed on the basis of relation-based granules.
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Notes
- 1.
A relation based on disjunction is defined analogously.
- 2.
If the \(\bigcup \) operation is used for one set only, it means that it is possible to obtain more than one set for the given relation.
- 3.
In this case it is assumed that a and \(a^\prime \) are of the same type.
- 4.
The last attribute in \(\varepsilon _{\alpha _4}(cust\_id,R_2.id,R_3.cust\_id)\) corresponds to \(cust\_id_1\) and \(cust\_id_2\).
- 5.
\(attr(\varepsilon _{\alpha })\) denotes the set of all attributes used in an \(\varepsilon \)-relation \(\varepsilon _{\alpha }\).
- 6.
A pattern of the expanded language may include a descriptor of the form \((a,\cdot )\).
- 7.
Proofs of the propositions formulated in this section are simple and left to the reader.
- 8.
\(\sigma _c(\bullet )\) is a selection under a condition c.
- 9.
1. For simplicity’s sake we will write \(id_i\) for \(R_i.id\). 2. The result means that 5 out of 7 customers purchase products.
- 10.
Unlike for patterns, the frequency and confidence of an association rule \(\varepsilon _{\alpha \rightarrow \beta }\) with respect to the domain \(D(\varepsilon _{\alpha \rightarrow \beta })\) are not defined, since \(D(\varepsilon _{\alpha })\) differs from \(D(\varepsilon _{\beta })\).
- 11.
The result means that 4 out of 5 customers who purchase products purchase them in quantities of one piece.
- 12.
The result means that 4 out of 5 customers who purchase products are considered as good customers.
- 13.
The cost of forming \(\varepsilon _{(id,\cdot )}\) is n because \(SEM_{IS}(\varepsilon _{(id,\cdot )})=\{(id(x),id(x)):x\in U\}\).
- 14.
Formulas are constructed over descriptive attributes only. Key attributes are used in a compound information system to join particular information systems.
- 15.
The sets are assumed to be ordered. This operation does not increase the asymptotic complexity of the database transformation.
- 16.
We assume that the sets are ordered.
- 17.
Relational data in the form it is provided is not, in general, ordered.
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Hońko, P. (2017). Relation-Based Granules. In: Granular-Relational Data Mining. Studies in Computational Intelligence, vol 702. Springer, Cham. https://doi.org/10.1007/978-3-319-52751-2_8
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DOI: https://doi.org/10.1007/978-3-319-52751-2_8
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