Formal Concept Analysis III Knowledge Discovery Robert J aschke - PowerPoint PPT Presentation

Formal Concept Analysis III Knowledge Discovery Robert J¨ aschke Asmelash Teka Hadgu FG Wissensbasierte Systeme/L3S Research Center Leibniz Universit¨ at Hannover Robert J¨ aschke (FG KBS) Formal Concept Analysis 1 / 33

Agenda Triadic Formal Concept Analysis 7 Motivation Folksonomies Motivation Triadic Formal Concept Concept-Tri-Lattice Visualization of Tri-Lattices Iceberg Tri-Lattices Computing Tri-Concepts Qualitative Evaluation Neighborhoods Robert J¨ aschke (FG KBS) Formal Concept Analysis 2 / 33

Motivation: Collaborative Tagging Systems Robert J¨ aschke (FG KBS) Formal Concept Analysis 3 / 33

Motivation: Collaborative Tagging Systems Robert J¨ aschke (FG KBS) Formal Concept Analysis 3 / 33

Motivation: Collaborative Tagging Systems Robert J¨ aschke (FG KBS) Formal Concept Analysis 3 / 33

Motivation: Collaborative Tagging Systems manage your web bookmarks and publication references open for the public since beginning of 2006, → 5 000 active users developed and operated at L3S Research Center Robert J¨ aschke (FG KBS) Formal Concept Analysis 4 / 33

Agenda Triadic Formal Concept Analysis 7 Motivation Folksonomies Motivation Triadic Formal Concept Concept-Tri-Lattice Visualization of Tri-Lattices Iceberg Tri-Lattices Computing Tri-Concepts Qualitative Evaluation Neighborhoods Robert J¨ aschke (FG KBS) Formal Concept Analysis 5 / 33

Folksonomies data structure of collaborative tagging systems connects users, tags, and resources conceptual structure created by the people Forschungszentrum L3S Wissenschaft & Forschung in den Schlüsselbereichen Wissen, Information und Lernen to science l3s center hannover research by jaeschke and 1 other person on 2006-01-27 10:39:07 edit | delete Robert J¨ aschke (FG KBS) Formal Concept Analysis 6 / 33

Folksonomies: Hypergraph, Tensor U u 4 u 3 u 2 u 4 u 1 u 3 t 1 u 2 r 1 t 2 u 1 r 2 t 3 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 7 / 33

Folksonomies: Hypergraph, Tensor U u 4 u 3 u 2 u 4 u 1 u 3 t 1 u 2 r 1 t 2 u 1 r 2 t 3 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 7 / 33

Folksonomies: Hypergraph, Tensor U u 4 u 3 u 2 u 4 u 1 u 3 t 1 u 2 r 1 t 2 u 1 r 2 t 3 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 7 / 33

Folksonomies U Definition (Folksonomy) u 4 F : ✏ ♣ U, T, R, Y q with U , T , R finite sets of users, tags, and resources, resp. u 3 Y ❸ U ✂ T ✂ R ternary relation tripartite hypergraph u 2 boolean 3-dimensional tensor triadic formal context u 1 u 3 u 2 u 4 u 1 t 1 r 1 t 1 t 2 r 2 r 1 t 2 t 3 T r 2 t 3 R Robert J¨ aschke (FG KBS) Formal Concept Analysis 8 / 33

♣ q ❸ ✂ ✂ Motivation U u 4 conceptual clustering of folksonomies find interessting concepts/clusters support browsing, community u 3 detection, recommendations get an overview into the structure of a folksonomy u 2 u 1 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 9 / 33

Motivation U u 4 conceptual clustering of folksonomies find interessting concepts/clusters support browsing, community u 3 detection, recommendations get an overview into the structure of a folksonomy u 2 tri-concept ♣ A, B, C q ❸ U ✂ T ✂ R : maximal cuboid in which every user from A has tagged every ressource from u 1 C with all tags from B ➞ shared conceptualization t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 9 / 33

Motivation Frequent Tri-Concepts Iceberg Concept Lattices / Closed Triadic Concept Itemset Mining Analysis (Lehmann, (Lakhal/Stumme/ Wille 1995) Zaki 1999) Formal Concept Association Analysis Rules (Agrawal, (Wille 1982) Srikant 1993) Robert J¨ aschke (FG KBS) Formal Concept Analysis 10 / 33

Motivation U u 4 u 3 We regard F ✏ ♣ U, T, R, Y q as triadic formal context . In general, the elements of U , T and R are u 2 then called objects , attributes and conditions and ♣ u, t, r q P Y is read as “ object u has the attribute t under condition r ”. u 1 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 11 / 33

Triadic Formal Concept U u 4 Definition (tri-concept) u 3 triple ♣ A, B, C q with A ❸ U , B ❸ T , C ❸ R and A ✂ B ✂ C ❸ Y , such that none of the three components can be enlarged without u 2 violating the condition A ✂ B ✂ C ❸ Y . We call A the extent , B the intent and C the modus of the formal tri-concept. u 1 ➞ natural extension of formal concepts t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 12 / 33

Concept-Tri-Lattice U u 4 three quasi orders ➚ 1 , ➚ 2 , ➚ 3 : ♣ A 1 , A 2 , A 3 q ➚ i ♣ B 1 , B 2 , B 3 q : ô A i ❸ B i , for i ✏ 1 , 2 , 3 . u 3 not antisymmetric , i. e. from ♣ A 1 , A 2 , A 3 q ➚ i ♣ B 1 , B 2 , B 3 q and u 2 ♣ B 1 , B 2 , B 3 q ➚ i ♣ A 1 , A 2 , A 3 q does not follow ♣ A 1 , A 2 , A 3 q ✏ ♣ B 1 , B 2 , B 3 q concept tri-lattice B ♣ K q of the triadic u 1 context K not a real (mathematical) lattice! t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 13 / 33

Visualization of Tri-Lattices Since it is not really a lattice, we can not draw a lattice diagram ❍ Alternative: t 1 ✉ 3 every quasi-order is written along the edge of a virtual triangle O 3 the tri-concepts are drawn into the t 1 ✉ O 2 triangle 1 example to the right: smallest ❍ non-trivial tri-lattice O 1 2 B 3 ✏ B ♣t 1 ✉ , t 1 ✉ , t 1 ✉ , ❍q ❍ visualization not always possible t 1 ✉ satisfied tetrahedron condition violated Thomson condition Robert J¨ aschke (FG KBS) Formal Concept Analysis 14 / 33

Visualization of Tri-Lattices Robert J¨ aschke (FG KBS) Formal Concept Analysis 15 / 33

Iceberg Tri-Lattices U u 4 u 3 Given support constraints τ u , τ t , τ r : tri-concept ♣ A, B, C q frequent : ô ⑤ A ⑤ ➙ τ u , ⑤ B ⑤ ➙ τ t , and ⑤ C ⑤ ➙ τ r u 2 ➞ iceberg tri-lattice u 1 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 16 / 33

Computing Tri-Concepts Given sets U , T , R ternary relation Y ❸ U ✂ T ✂ R support constraints τ u , τ t , τ r Find ♣ A, B, C q with A ❸ U , B ❸ T , C ❸ R ⑤ A ⑤ ➙ τ u , ⑤ B ⑤ ➙ τ t , ⑤ C ⑤ ➙ τ r A ✂ B ✂ C ❸ Y such that none of the sets A, B or C can be enlarged without violating the former condition Robert J¨ aschke (FG KBS) Formal Concept Analysis 17 / 33

✏ ♣ ✂ q ♣ q ♣ q ♣ q ✏ t♣ ♣ qq ⑤ ♣ q P ✉ ♣ q ♣ ✂ q Computing Tri-Concepts U computes the iceberg tri-lattice of a triadic formal context u 4 u 3 u 2 u 1 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

✏ ♣ ✂ q ♣ q ♣ q ♣ q ✏ t♣ ♣ qq ⑤ ♣ q P ✉ ♣ q ♣ ✂ q Computing Tri-Concepts U computes the iceberg tri-lattice of a triadic formal context u 4 Algorithm u 3 u 2 u 1 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

✏ ♣ ✂ q ♣ q ♣ q ♣ q ♣ q ♣ ✂ q Computing Tri-Concepts U computes the iceberg tri-lattice of a triadic formal context u 4 Algorithm u 3 Let ˜ Y : ✏ t♣ u, ♣ t, r qq ⑤ ♣ u, t, r q P Y ✉ u 2 u 1 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

✏ ♣ ✂ q ♣ q ♣ q ♣ q Computing Tri-Concepts U computes the iceberg tri-lattice of a triadic formal context u 4 Algorithm u 3 Let ˜ Y : ✏ t♣ u, ♣ t, r qq ⑤ ♣ u, t, r q P Y ✉ Loop: Find (frequent) concepts ♣ A , I q in ♣ U, T ✂ R, ˜ Y q u 2 u 1 t 1 r 1 t 2 r 2 t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

✏ ♣ ✂ q ♣ q ♣ q ♣ q Computing Tri-Concepts U computes the iceberg tri-lattice of a triadic formal context u 4 Algorithm u 3 Let ˜ Y : ✏ t♣ u, ♣ t, r qq ⑤ ♣ u, t, r q P Y ✉ Loop: Find (frequent) concepts ♣ A , I q in ♣ U, T ✂ R, ˜ Y q u 2 u 1 In the example: t 1 r 1 � ✟ t 2 r 2 ♣ A, I q ✏ t u 2 , u 3 ✉ , t♣ t 1 , r 1 q , ♣ t 1 , r 2 q , ♣ t 2 , r 1 q✉ t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

✏ ♣ ✂ q ♣ q Computing Tri-Concepts U computes the iceberg tri-lattice of a triadic formal context u 4 Algorithm u 3 Let ˜ Y : ✏ t♣ u, ♣ t, r qq ⑤ ♣ u, t, r q P Y ✉ Loop: Find (frequent) concepts ♣ A , I q in ♣ U, T ✂ R, ˜ Y q u 2 Loop: Find (frequent) concepts ♣ B , C q in ♣ T, R, I q u 1 In the example: t 1 r 1 t 2 r 2 ♣ T, R, I q ✏ ♣ T, R, t♣ t 1 , r 1 q , ♣ t 1 , r 2 q , ♣ t 2 , r 1 q✉q t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

✏ ♣ ✂ q ♣ q Computing Tri-Concepts U computes the iceberg tri-lattice of a triadic formal context u 4 Algorithm u 3 Let ˜ Y : ✏ t♣ u, ♣ t, r qq ⑤ ♣ u, t, r q P Y ✉ Loop: Find (frequent) concepts ♣ A , I q in ♣ U, T ✂ R, ˜ Y q u 2 Loop: Find (frequent) concepts ♣ B , C q in ♣ T, R, I q u 1 In the example: t 1 r 1 t 2 r 2 ♣ B, C q ✏ ♣t t 1 ✉ , t r 1 , r 2 ✉q t 3 T R Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33