Ivo Düntsch

Uncertainly measures of rough set prediction

Abstract The main statistics used in rough set data analysis, the approximation quality, is of limited value when there is a choice of competing models for predicting a decision variable. In keeping within the rough set philosophy of non-invasive data analysis, we present three model selection criteria, using information theoretic entropy in the spirit of the minimum description length principle. Our main procedure is based on the principle of indifference combined with the maximum entropy principle, thus keeping external model assumptions to a minimum. The applicability of the proposed method is demonstrated by a comparison of its error rates with results of C4.5, using 14 published data sets.

Approximation Operators in Qualitative Data Analysis

A large part of qualitative data analysis is concerned with approximations of sets on the basis of relational information. In this paper, we present various forms of set approximations via the unifying concept of modal–style operators. Two examples indicate the usefulness of the approach.

The IsoMetrics usability inventory: An operationalization of ISO 9241-10 supporting summative and formative evaluation of software systems

Aiming at a user-oriented approach in software evaluation on the basis of ISO 9241 Part 10, we present a questionnaire (IsoMetrics) which collects usability data for summative and formative evaluation, and document its construction. The summative version of IsoMetrics shows a high reliability of its subscales and gathers valid information about differences in the usability of different software systems. Moreover, we show that the formative version of IsoMetrics is a powerful tool for supporting the identification of software weaknesses. Finally, we propose a procedure to categorize and prioritize weak points, which subsequently can be used as basic input to usability reviews.

A representation theorem for Boolean contact algebras

We prove a representation theorem for Boolean contact algebras which implies that the axioms for the region connection calculus (RCC) [D.A. Randell, A.G. Cohn, Z. Cui, Computing transitivity tables: a challenge for automated theorem provers, in: D. Kapur (Ed.), Proceedings of the 11th international conference on Automated Deduction (CADE-11), Lecture Notes in Artificial Intelligence, Vol. 607, Springer, Saratoga, Springs, NY, 1992, pp. 786-790] are complete for the class of subalgebras of the algebras of regular closed sets of weakly regular connected T1 spaces.

A relation — algebraic approach to the region connection calculus

Abstract We explore the relation – algebraic aspects of the region connection calculus (RCC) of Randell et al., Proceedings of the CADE, vol 11, pp. 786–790, Springer, Berlin, 1992a. In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras B with the natural ordering as the “part-of” relation, and that the table is closed under first-order definable relations iff B is homogeneous.

Statistical evaluation of rough set dependency analysis

Rough set data analysis (RSDA) has recently become a frequently studied symbolic method in data mining. Among other things, it is being used for the extraction of rules from databases; it is, however, not clear from within the methods of rough set analysis, whether the extracted rules are valid.In this paper, we suggest to enhance RSDA by two simple statistical procedures, both based on randomization techniques, to evaluate the validity of prediction based on the approximation quality of attributes of rough set dependency analysis. The first procedure tests the casualness of a prediction to ensure that the prediction is not based on only a few (casual) observations. The second procedure tests the conditional casualness of an attribute within a prediction rule.The procedures are applied to three data sets, originally published in the context of rough set analysis. We argue that several claims of these analyses need to be modified because of lacking validity, and that other possibly significant results were overlooked.

A logic for rough sets

Abstract The collection of all subsets of a set forms a Boolean algebra under the usual set-theoretic operations, while the collection of rough sets of an approximation space is a regular double Stone algebra (Pomykala and Pomykala, 1988). The appropriate class of algebras for classical propositional logic are Boolean algebras, and it is reasonable to assume that regular double Stone algebras are a class of algebras appropriate for a logic of rough sets. Using the representation theorem for these algebras by Katrinak (1974), we present such a logic for rough sets and its algebraic semantics in the spirit of Andreka and Nemeti (1994).

Relation Algebras and their Application in Temporal and Spatial Reasoning

Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only finitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is “pointless”. Relation algebras were introduced into temporal reasoning by Allen (1983, Communications of the ACM 26(1), 832–843) and into spatial reasoning by Egenhofer and Sharma (1992, Fifth International Symposium on Spatial Data Handling, Charleston, SC). The calculus of relation algebras is also well suited to handle binary constraints as demonstrated e.g. by Ladkin and Maddux (1994, Journal of the ACM 41(3), 435–469). In the present paper I will give an introduction to relation algebras, and an overview of their role in qualitative temporal and spatial reasoning.

A Proximity Approach to Some Region-Based Theories of Space

This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roepers notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roepers paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormanns notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained.

Region---based theory of discrete spaces: A proximity approach

We introduce Boolean proximity algebras as a generalization of Efremovič proximities which are suitable in reasoning about discrete regions. Following Stone’s representation theorem for Boolean algebras, it is shown that each such algebra is isomorphic to a substructure of a complete and atomic Boolean proximity algebra.