Basic Level in Formal Concept Analysis: Interesting Concepts and Psychological Ramifications Radim Belohlavek, Martin Trnecka Palacky University, Olomouc, Czech Republic ! ! ! Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 1 / 32
Basic Idea Belohlavek R., Klir G.: Concepts and Fuzzy Logic. MIT Press, 2011. Two research directions: A) Formal Concept analysis (data minig in general) benefits from Psychology of Concepts (utilizing phenomena studied by PoC). B) Psychology of Concepts benefits from data mining (FCA). Psychology of Concepts: big area in cognitive psychology, empirical study of human concepts. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 2 / 32
Motivation Number of patterns extracted from data may become too large to be reasonably comprehended by a user. This is in particular true of formal concept analysis. Users finds some extracted concepts important (relevant, interesting), some less important, some even “artificial” and not interesting. Select only important concepts. Several approaches have been proposed, e.g.: – Indices enabling us to sort concepts according to their relevance. Kuznetsov’s stability index – Taking into account additional user’s knowledge (background knowledge) to filter relevant concepts Belohlavek et al.: attribute dependency formulas, constrained concept lattices Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 3 / 32
Previous Work Another possibility is that the user judgment on concepts’ importance is due to certain psychological processes and phenomena. Our approach: important are concepts from the basic level . Belohlavek R., Trnecka M.: Basic Level of Concepts in Formal Concept Analysis. In: F. Domenach, D.I. Ignatov, and J. Poelmans (Eds.): ICFCA 2012, LNAI 7278, Springer, Heidelberg, 2012, pp. 28-44. In our previous work we demonstrated that the basic level phenomenon may be utilized to select important formal concepts. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 4 / 32
Q: What is this? Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 5 / 32
A: Dog . . . Why dog? There is a number of other possibilities: Animal Mammal Canine beast Retriever Golden Retriever Marley . . . So why dog?: Because “dog” is a basic level concept. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 6 / 32
Basic Level Phenomenon Extensively studied phenomenon in psychology of concepts. When people categorize (or name) objects, they prefer to use certain kind of concepts. Such concepts are called the concepts of the basic level. Definition of basic level concepts?: Are cognitively economic to use; “carve the world well”. One feature: Basic level concepts are a compromise between the most general and most specific ones. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 7 / 32
Basic Level Definitions The psychological literature does not contain a single, robust and uniquely intepretable definition of the notion of basic level. Several informal definitions. There exists (semi)formalized metrics from psychologists. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 8 / 32
Our Paper (First Step in B) We formalize within FCA five selected approaches to basic level. Goal: Explore their relationship. Several challenges: – Every formal model of basic level will be simplistic (potentially a point of criticism from the psychological standpoint). – Various issues which are problematic or not yet fully understood from a psychological viewpoint (less knowledge and extensive domain knowledge). Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 9 / 32
Introduction to Formal Concept Analysis (FCA) Method for analysis of tabular data (Rudolf Wille). Input: binary objects (rows) × attributes (columns) data table called formal context (denoted by � X, Y, I � ); the binary relation I ⊆ X × Y tells us whether attribute y ∈ Y applies to object x ∈ X , i.e. � x, y � ∈ I , or not, i.e. � x, y � �∈ I . Output: Hierarchically ordered collection of clusters: 1 – called concept lattice, – clusters are called formal concepts, – hierarchy = subconcept-superconcept. Data dependencies: 2 – called attribute implications, – not all (would be redundant), only representative set. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 10 / 32
Formal Concept The notion of a concept is inspired by Port-Royal logic (traditional logic): concept := extent + intent extent = objects covered by concept intent = attributes covered by concept. A formal concept of � X, Y, I � is any pair � A, B � consisting of A ⊆ X and B ⊆ Y satisfying A ↑ = B and B ↓ = A where A ↑ = { y ∈ Y | for each x ∈ X : � x, y � ∈ I } , and B ↓ = { x ∈ X | for each y ∈ Y : � x, y � ∈ I } . That is, A ↑ is the set of all attributes common to all objects from A and B ↓ is the set of all objects having all the attributes from B , respectively. Example (Concept dog) extent = collection of all dogs (foxhound, poodle, . . . ), intent = { barks, has four limbs, has tail, . . . } Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 11 / 32
Concept lattice The set B ( X, Y, I ) of all formal concepts of � X, Y, I � equipped with the subconcept-superconcept ordering ≤ is called the concept lattice of � X, Y, I � . � A 1 , B 1 � ≤ � A 2 , B 2 � means that � A 1 , B 1 � is more specic than � A 2 , B 2 � . ≤ captures the intuition behind DOG ≤ MAMMAL (the concept of a dog is more specific than the concept of a mammal). Example 1,2,3,4 c a b c d a, c b, c c, d 1 × × 1,3 2,4 3,4 2 × × 3 × × × a, b, c b, c, d 4 × × × 3 4 a, b, c, d Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 12 / 32
Basic Level Metrics We created a metrics (functions assigning numbers to concepts). Large number is stronger indicates that the concept is basic level concept. For formal context � X, Y, I � which describes all the available information regarding the objects and attributes. For a given approach M to basic level, we define a function BL M mapping every concept � A, B � in the concept lattice B ( X, Y, I ) to [0 , ∞ ) or to [0 , 1] BL M ( A, B ) is interpreted as the degree to which � A, B � belongs to the basic level. A basic level is thus naturally seen as a graded (fuzzy) set rather than a clear-cut set of concepts. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 13 / 32
Similarity Approach (S) Based on informal (one of the first) definition from Eleanor Rosch (1970s). Basic level concept satisfies three conditions: 1. The objects of this concept are similar to each other. 2. The objects of the superordinate concepts are significantly less similar. 3. The objects of the subordinate concepts are only slightly more similar. Formalized in our previous paper (Belohlavek, Trnecka, ICFCA 2012). BL S ( A, B ) = α 1 ⊗ α 2 ⊗ α 3 , Where α i represents the degree of validity of the above condition i = 1 , 2 , 3 and ⊗ represents a truth function of many-valued conjunction. The definition of degrees α i involves definitions of appropriate similarity measures and utilizes basic principles of fuzzy logic. Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 14 / 32
Cue Validity Approach (CV) Proposed by Eleanor Rosch (1976). Based on the notion of a cue validity of attribute y for concept c , i.e. the conditional probability p ( c | y ) that an object belongs to c given that it has y . The total cue validity for c is defined the sum of cue validities of each of the attributes of c . Basic level concepts are those with a high total cue validity. We consider the following probability space: X (objects) are the elementary events, 2 X 1 (sets of objects) are the events, the probability distribution is given by P ( { x } ) = | X | for every object x ∈ X . For an event A ⊆ X then, P ( A ) = | A | / | X | . The event corresponding to a set { y, . . . } ⊆ Y of attributes is { y, . . . } ↓ . | A ∩ { y } ↓ | � P ( A |{ y } ↓ ) = � BL CV ( A, B ) = . |{ y } ↓ | y ∈ B y ∈ B Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 15 / 32
Category Feature Collocation Approach (CFC) Proposed by Jones (1983). Defined as product p ( c | y ) · p ( y | c ) of the cue validity p ( c | y ) and the so-called category validity p ( y | c ) . The total CFC for c may then be defined as the sum of collocations of c and each attribute. Basic level concepts may then again be understood as concepts with a high total CFC. � | A ∩ { y } ↓ | · | A ∩ { y } ↓ | � � � BL CFC ( A, B ) = ( p ( c | y ) · p ( y | c )) = . |{ y } ↓ | | A | y ∈ Y y ∈ Y Belohlavek R., Trnecka M. (DAMOL) Basic Level in Formal Concept Analysis August 8, 2013 16 / 32
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