{"id":{"repo_id":"regina","oai_identifier":"oai:uregina.scholaris.ca:10294/5848"},"canonical_url":"https://search.dev.ndltd.org/etd/regina/oai:uregina.scholaris.ca:10294/5848","repository":{"repo_id":"regina","name":"University of Regina","base_url":"https://uregina.scholaris.ca/server/oai/request"},"display":{"title":"Structured Rough Set Approximations","abstract":"Rough set theory is widely used in many areas, such as artificial intelligence, machine learning and data mining. Lower and upper approximations are two fundamental notions for concept analysis with rough set theory. In rough set theory, one can obtain two kinds of sets in an information table, namely, definable and undefinable sets. Intuitively, a definable set represents some- thing we can describe precisely. On the other hand, for an undefinable set, one cannot describe it precisely due to limited available information. One of the main issues in rough set theory is to approximate an undefinable set by a pair of definable sets, called the lower and upper approximations. By introducing these two approxima- tions, approximate inferences can be made about an undefinable set. There are several formulations of rough set approximations. Pawlak proposed to construct the two approximations as unions of equivalence classes, which is now used in main stream research in rough set theory. By explicitly expressing the individual equivalence classes in Pawlak approximations, Bryniarski used a pair of families of equivalence classes as rough set approximations, which are also known as structured Pawlak approximations. Moreover, Deng et al. proposed the adaptive approximations by using a sequence of equivalence relations with different granularities. Although the latter two formulations take the structure and semantics of the approximations into consideration, they have not received their due attention. The main objective of this thesis is to present a further exploration of structured and adaptive approximations. We propose a generalized definition of structured rough set approximations from the view of semantics. It can be verified that the proposed structured approximations cover the same sets of objects as Pawlak, Bryniarski and adaptive approximations. In this sense, they are consistent and mathematically equiv- alent. However, the constituents of these approximations are quite different. The new formulation highlights the semantics of approximations and displays a well-defined in- ternal structure, which will benefit the rule learning process in concept analysis with rough set theory. The comparisons between the proposed structured rough set ap- proximations and Pawlak, Bryniarski, and adaptive approximations are investigated. We also analyze the relationships between the new framework and Grzyma la-Busse's LERS systems which are complementary to the new framework.","abstract_html":"Rough set theory is widely used in many areas, such as artificial intelligence, machine learning and data mining. Lower and upper approximations are two fundamental notions for concept analysis with rough set theory. In rough set theory, one can obtain two kinds of sets in an information table, namely, definable and undefinable sets. Intuitively, a definable set represents some- thing we can describe precisely. On the other hand, for an undefinable set, one cannot describe it precisely due to limited available information. One of the main issues in rough set theory is to approximate an undefinable set by a pair of definable sets, called the lower and upper approximations. By introducing these two approxima- tions, approximate inferences can be made about an undefinable set. There are several formulations of rough set approximations. Pawlak proposed to construct the two approximations as unions of equivalence classes, which is now used in main stream research in rough set theory. By explicitly expressing the individual equivalence classes in Pawlak approximations, Bryniarski used a pair of families of equivalence classes as rough set approximations, which are also known as structured Pawlak approximations. Moreover, Deng et al. proposed the adaptive approximations by using a sequence of equivalence relations with different granularities. Although the latter two formulations take the structure and semantics of the approximations into consideration, they have not received their due attention. The main objective of this thesis is to present a further exploration of structured and adaptive approximations. We propose a generalized definition of structured rough set approximations from the view of semantics. It can be verified that the proposed structured approximations cover the same sets of objects as Pawlak, Bryniarski and adaptive approximations. In this sense, they are consistent and mathematically equiv- alent. However, the constituents of these approximations are quite different. The new formulation highlights the semantics of approximations and displays a well-defined in- ternal structure, which will benefit the rule learning process in concept analysis with rough set theory. The comparisons between the proposed structured rough set ap- proximations and Pawlak, Bryniarski, and adaptive approximations are investigated. We also analyze the relationships between the new framework and Grzyma la-Busse's LERS systems which are complementary to the new framework.","abstract_has_math":false,"creators":["Hu, Mengjun"],"institution":"Faculty of Graduate Studies and Research, University of Regina","degree_name":"Master of Science (MSc)","degree_level":"Master's","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":["Yao, Yiyu"],"committee_chairs":[],"committee_members":["Yao, JingTao","Butz, Cortney J."],"year":2014,"date_issued":"2014-10","date_published":"2014-10","updated_at":"2026-07-24T04:03:39Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.82465/4557"],"render_values":[{"text":"https://doi.org/10.82465/4557","href":"https://doi.org/10.82465/4557","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/10294/5848","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Yao, Yiyu"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Yao, JingTao","Butz, Cortney J."]},{"key":"dc:creator","label":"Author","values":["Hu, Mengjun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-07-22T19:18:19Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-07-22T19:18:19Z"]},{"key":"dc:date.issued","label":"Date","values":["2014-10"]},{"key":"dc:publisher","label":"Institution","values":["Faculty of Graduate Studies and Research, University of Regina"]},{"key":"dc:type","label":"Dc Type","values":["master thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Master's"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MSc)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Faculty of Graduate Studies and Research, University of Regina"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.82465/4557"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10294/5848"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Science, University of Regina. x, 103 p."]},{"key":"dc:description.abstract","label":"Abstract","values":["Rough set theory is widely used in many areas, such as artificial intelligence, machine learning and data mining. Lower and upper approximations are two fundamental notions for concept analysis with rough set theory. In rough set theory, one can obtain two kinds of sets in an information table, namely, definable and undefinable sets. Intuitively, a definable set represents some- thing we can describe precisely. On the other hand, for an undefinable set, one cannot describe it precisely due to limited available information. One of the main issues in rough set theory is to approximate an undefinable set by a pair of definable sets, called the lower and upper approximations. By introducing these two approxima- tions, approximate inferences can be made about an undefinable set. There are several formulations of rough set approximations. Pawlak proposed to construct the two approximations as unions of equivalence classes, which is now used in main stream research in rough set theory. By explicitly expressing the individual equivalence classes in Pawlak approximations, Bryniarski used a pair of families of equivalence classes as rough set approximations, which are also known as structured Pawlak approximations. Moreover, Deng et al. proposed the adaptive approximations by using a sequence of equivalence relations with different granularities. Although the latter two formulations take the structure and semantics of the approximations into consideration, they have not received their due attention. The main objective of this thesis is to present a further exploration of structured and adaptive approximations. We propose a generalized definition of structured rough set approximations from the view of semantics. It can be verified that the proposed structured approximations cover the same sets of objects as Pawlak, Bryniarski and adaptive approximations. In this sense, they are consistent and mathematically equiv- alent. However, the constituents of these approximations are quite different. The new formulation highlights the semantics of approximations and displays a well-defined in- ternal structure, which will benefit the rule learning process in concept analysis with rough set theory. The comparisons between the proposed structured rough set ap- proximations and Pawlak, Bryniarski, and adaptive approximations are investigated. We also analyze the relationships between the new framework and Grzyma la-Busse's LERS systems which are complementary to the new framework."]},{"key":"dc:title","label":"Title","values":["Structured Rough Set Approximations"]}]}],"canonical_facts":{"dc:contributor.advisor":["Yao, Yiyu"],"dc:contributor.committeemember":["Yao, JingTao","Butz, Cortney J."],"dc:creator":["Hu, Mengjun"],"dc:date.accessioned":["2015-07-22T19:18:19Z"],"dc:date.available":["2015-07-22T19:18:19Z"],"dc:date.issued":["2014-10"],"dc:description":["A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Science, University of Regina. x, 103 p."],"dc:description.abstract":["Rough set theory is widely used in many areas, such as artificial intelligence, machine learning and data mining. Lower and upper approximations are two fundamental notions for concept analysis with rough set theory. In rough set theory, one can obtain two kinds of sets in an information table, namely, definable and undefinable sets. Intuitively, a definable set represents some- thing we can describe precisely. On the other hand, for an undefinable set, one cannot describe it precisely due to limited available information. One of the main issues in rough set theory is to approximate an undefinable set by a pair of definable sets, called the lower and upper approximations. By introducing these two approxima- tions, approximate inferences can be made about an undefinable set. There are several formulations of rough set approximations. Pawlak proposed to construct the two approximations as unions of equivalence classes, which is now used in main stream research in rough set theory. By explicitly expressing the individual equivalence classes in Pawlak approximations, Bryniarski used a pair of families of equivalence classes as rough set approximations, which are also known as structured Pawlak approximations. Moreover, Deng et al. proposed the adaptive approximations by using a sequence of equivalence relations with different granularities. Although the latter two formulations take the structure and semantics of the approximations into consideration, they have not received their due attention. The main objective of this thesis is to present a further exploration of structured and adaptive approximations. We propose a generalized definition of structured rough set approximations from the view of semantics. It can be verified that the proposed structured approximations cover the same sets of objects as Pawlak, Bryniarski and adaptive approximations. In this sense, they are consistent and mathematically equiv- alent. However, the constituents of these approximations are quite different. The new formulation highlights the semantics of approximations and displays a well-defined in- ternal structure, which will benefit the rule learning process in concept analysis with rough set theory. The comparisons between the proposed structured rough set ap- proximations and Pawlak, Bryniarski, and adaptive approximations are investigated. We also analyze the relationships between the new framework and Grzyma la-Busse's LERS systems which are complementary to the new framework."],"dc:identifier.doi":["https://doi.org/10.82465/4557"],"dc:identifier.uri":["https://hdl.handle.net/10294/5848"],"dc:language.iso":["en"],"dc:publisher":["Faculty of Graduate Studies and Research, University of Regina"],"dc:title":["Structured Rough Set Approximations"],"dc:type":["master thesis"],"thesis:degree_discipline":["Computer Science"],"thesis:degree_level":["Master's"],"thesis:degree_name":["Master of Science (MSc)"],"thesis:institution_name":["Faculty of Graduate Studies and Research, University of Regina"]},"updated_at":"2026-07-24T04:03:39Z"}