Hierarchical Multidimensional Modelling Hierarchical Multidimensional Modelling in the Concept- -Oriented Data Model Oriented Data Model in the Concept Alexandr Savinov Fraunhofer Institute for Autonomous Intelligent Systems Knowledge Discovery Team Germany alexandr.savinov@ais.fraunhofer.de 1 Fraunhofer Institute for Autonomous Intelligent Systems
Contents Contents Introduction � Logical and Physical Structure � Dimensions and Inverse Dimensions � Projection and De-projection � Multidimensional Grouping and Aggregation � Conclusion � FCA Data Modeling ? 2 Fraunhofer Institute for Autonomous Intelligent Systems
Introduction Introduction Data models and dimensionality modelling modelling Data models and dimensionality Entities and relationships (ERM) � Logic and predicates (deductive databases) � Relations (RM) � Facts (ORM) � Objects (OODM) � Dimensions (OLAP, multidimensional databases) � Dimension is a named link between subconcept and superconcept � Orders Products order products OrderParts 3 Fraunhofer Institute for Autonomous Intelligent Systems
Introduction Introduction Assumptions and related work Assumptions and related work Global semantics (URM) � Using the structure for navigation (FDM) � Hierarchical structure (FCA) � Top concept Concept Level of details (OLAP) � Dimension Bottom concept C 4 Fraunhofer Institute for Autonomous Intelligent Systems
Introduction Introduction FCA FCA Concept -> Concept � Object -> Data item � Attribute -> Primitive concept � In FCA concepts depend on data while in COM data � depends on concepts, that is concepts define a structure for data (in FCA the structure is derived from Attributes M data semantics) Items belong to one concept while in FCA object may � belong to many concepts COM concept is a (non-primitive) attribute for � subconcepts Objects G 5 Fraunhofer Institute for Autonomous Intelligent Systems
Introduction Introduction Questions Questions Why we have (primitive) attributes defined at structural level while � concepts are derived from data semantics? Why not to have a possibility to define a (non-primitive) attribute as a concept? Attributes M Attributes are primitive concepts Each concept has ist set of objects Objects G 6 Fraunhofer Institute for Autonomous Intelligent Systems
Physical and and Logical Logical Structure Structure Physical Physical structure Physical structure At physical level an element of the model is a collection of other elements � Physical structure is used for representation and access � Physical structure is used to implement reference � Physical structure is hierarchical where each element has only one parent � , , model root R ∈ - membership in C = K { , , } a b physical collection concepts C U V ∈ ∈ K , , a A b A items e a b c d f 7 Fraunhofer Institute for Autonomous Intelligent Systems
Physical and and Logical Logical Structure Structure Physical Logical structure Logical structure Each element is a combination of other elements (by reference) � Logical structure is used to represent data semantics (properties) � Logical collection is a dual combination � Each element has many parents and many children � = 〈 〉 , K < < K < , , , , , , g a b c , a g b g c g customer date a b c , AND > - membership in order g logical collection OR d e f part1 part2 = > > K > K , , , { , , , } g d e f , d g e g f g 8 Fraunhofer Institute for Autonomous Intelligent Systems
Physical and and Logical Logical Structure Structure Physical Two level model Two level model = [Root] One root element R is a physical collection of concepts, � K { , , , } R C C C 1 2 N [Syntax] Each concept is � – (i) a combination of other concepts called superconcepts (while this = 〈 n ∈ 〉 K , , , C C C C R concept is a subconcept ), 1 2 = ∈ K { , , } C i i R – (ii) a physical collection of data items (or concept instances), 1 2 , , [Semantics] Each data item is � – (i) a combination of other data items called superitems (while this item = 〈 n ∈ 〉 K , 2 , , is a subitem ), i i i i C 1 = {} – (ii) empty physical collection, i 9 Fraunhofer Institute for Autonomous Intelligent Systems
Physical and and Logical Logical Structure Structure Physical Two level model Two level model [Special elements] If a concept does not have a superconcept then it is � referred to as primitive and its superconcept is one common top concept ; and if a concept does not have a subconcept then it is assumed to be one common bottom concept , and an absence of superitem is denoted by one special null item . , [Cycles] Cycles in subconcept-superconcept relation and subitem- � superitem relation are not allowed, , [Syntactic constraints] Each data item from a concept may combine only � items from its superconcepts. 10 Fraunhofer Institute for Autonomous Intelligent Systems
Syntax and Semantics Semantics Syntax and Model syntax Model syntax At syntactic level a concept is a combination of ist superconcepts � = 〈 〉 K : , : , , : C x C x C x n C 1 1 2 2 n Each superconcept is identified by dimension name, that is, � dimension is a relative position of superconcept , Top , Users Categories Prices Dates user category user date Products date price product Auctions auction AuctionBids 11 Fraunhofer Institute for Autonomous Intelligent Systems
Syntax and Semantics Semantics Syntax and Model semantics Model semantics C = K { , , } Each concept is a set of items: � i i 1 2 = 〈 〉 K , 2 , , An item is a combination of its superitems: i i i i � 1 n There is no difference between objects and attribute values: an object � , has values in other objects, and it is a value for other objects , a) b) 12 Fraunhofer Institute for Autonomous Intelligent Systems
Model Dimensionality Dimensionality Model Dimensions Dimensions Dimension is a named position of superconcept � Superconcept is referred to as the domain � Dimensions of higher rank consists of many (local) dimensions � Dimension with the domain in a primitive concept is a primitive dimension � The number of primitive dimensions is the model primitive dimensionality � , , Top Prices Users Dates Categories user category user date Products date price product Auctions auction AuctionBids 13 Fraunhofer Institute for Autonomous Intelligent Systems
Model Dimensionality Dimensionality Model Inverse dimensions Inverse dimensions Inverse dimension has an opposite direction � Inverse dimension identifies a subconcept � Inverse dimensions are multi-valued (while dimensions are one-valued) � The number of primtive dimensions is equal to the number of primtive � inverse dimensions , {AuctionBids.auction.product.category} � , Top Users Prices Dates Categories user category user date Products date price product Auctions auction AuctionBids 14 Fraunhofer Institute for Autonomous Intelligent Systems
Model Dimensionality Dimensionality Model Logical collections Logical collections A concept is a logical collection of its subconcepts � An item is logical collection of its subitems � An item is group for its subitems � , Countries Products Customers , Orders 15 Fraunhofer Institute for Autonomous Intelligent Systems
Model Dimensionality Dimensionality Model Hierarchical coordinate system Hierarchical coordinate system A concept can be interpreted as an axis with items as coordinates � A coordinate has its own coordinates and points can be used as � coordinates for other points , , X X Y XY Y 16 Fraunhofer Institute for Autonomous Intelligent Systems
Projection and and De De- -projection projection Projection Projection Projection Projection of a subset of subitems along some dimension path: � → = ∈ = ∈ ⊆ { | . , } I ⊆ I d u U i d u i I C C = 1 2 k L . . . d d d d U = Dom( d ) U , Projection direction Countries Products Customers , C For each subitem we get its superitem Orders along the dimension I used in projection 17 Fraunhofer Institute for Autonomous Intelligent Systems
Projection and and De De- -projection projection Projection De- -projection projection De De-projection of a subset of superitems along some inverse dimension: � → = ∈ = ∈ ⊆ { } { | . , } = 1 2 I d s S s d i i I C k L { } { . . . } d d d d S = Dom({ d }) , I De-projection direction , Top Prices Users Dates Categories For each superitems we find all subitems user category along inverse S dimension that user date Products reference it date price product Auctions auction AuctionBids 18 Fraunhofer Institute for Autonomous Intelligent Systems
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