Recent Researches in Applied Computers and Computational Science
Logical Implications SYLVIA ENCHEVA Stord/Haugesund University College Faculty of Technology, Business and Maritime Sciences Bjørnsonsg. 45, 5528 Haugesund NORWAY
[email protected] Abstract: In this article we apply a method for presenting numeric predicates as a closed set of ’closed sets’ associated with the objects instead of a closed set of attributes associated with a set of objects. In order to derive additional knowledge from a dataset we introduce a new measure called ’distance’ that emphasises changes in attribute values. Key–Words: Decision support services, uncertainty management, closed sets
1
Introduction
In a lattice illustrating partial ordering of knowledge values, the logical conjunction is identified with the meet operation and the logical disjunction with the join operation.
Closed set data mining has been applied for discovering deterministic, causal dependencies [6]. Closed set data mining turns to be especially handy when frequent set mining experiences difficulties related to large data sets or too many associations, [1], [4], [11]. One interesting question is how to derive logical implications where atoms representation is not restricted to boolean symbolics.An approach to that is presented in [6] where the extension to ordinal values is based on the fact that orderings are antimatroid closure spaces. Concepts lattices are a very useful visualization and analytical tool. However they can be quite difficult to read when the number of objects and attributes increases. In such cases logical implications can be deduced from adjacent closed concepts in the lattice, rather than to seek clusters of such concepts, [6]. In this work we compare concept lattices and lattices with nonbinary values based on the same dataset. In the case of closed sets of tuples with nonbinary entrances we introduce distance between attributes of objects in order to derive additional knowledge. The rest of the paper is organised as follows. Section 2 contains definitions of terms used later on. Section 3 contains the conclusion of this work.
2 2.1
Definition 2 [10] A context is a triple (G, M, I) where G and M are sets and I ⊂ G × M . The elements of G and M are called objects and attributes respectively, [2] and [10]. For A ⊆ G and B ⊆ M , define A′ = {m ∈ M | (∀g ∈ A) gIm}, B ′ = {g ∈ G | (∀m ∈ B) gIm} where A′ is the set of attributes common to all the objects in A and B ′ is the set of objects possessing the attributes in B. Definition 3 [10] A concept of the context (G, M, I) is defined to be a pair (A, B) where A ⊆ G, B ⊆ M , A′ = B and B ′ = A. The extent of the concept (A, B) is A while its intent is B. A subset A of G is the extent of some concept if and only if A′′ = A in which case the unique concept of the which A is an extent is (A, A′ ). The corresponding statement applies to those subsets B ∈ M which is the intent of some concepts. The set of all concepts of the context (G, M, I) is denoted by B(G, M, I).
Methodology Concept of a Context
2.2
Definition 1 [2] Let P be a non-empty ordered set. If sup{x, y} and inf {x, y} exist for all x, y ∈ P , then P is called a lattice.
ISBN: 978-1-61804-084-8
Closure Concepts
Notations in this subsection are as in [8]. By a ’closure system’ over a ’universe’ U , we mean a collection C
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Recent Researches in Applied Computers and Computational Science
of sets X, Y, ..., Z ⊆ U , including U , satisfying the property that if X, Y ∈ C then X ∪ Y ∈ C. The sets of C are said to be the closed sets of U . As an alternative to this ’intersection’ characterization, one can define a closure operator ϕ : 2U → 2U satisfying the following 3 axioms for all X, Y, Z: X ⊆ X.ϕ, X ⊆ Y implies X.ϕ ⊆ Y.ϕ X.ϕ.ϕ = X.ϕ. (For technical reasons we prefer to use suffix operator notation, so read X.ϕ as ’X closure’.) Readily, a set X is closed in C, if X.ϕ = X. The equivalence of these two alternative definitions is well known, [5] Closure systems can satisfy many other axioms, and those that do give rise to different varieties of mathematical systems. If (X ∪ Y ).ϕ = X.ϕ ∪ Y.ϕ we say is a topological closure. If the system satisfies the exchange axiom, that is if p, q ∈ X.ϕ but q ∈ (X ∪ {p}).ϕ then necessarily p ∈ (X ∪ {q}).ϕ, the system can be viewed as a kind of linear algebra, or more generally a ’matroid’. The Galois closure we will be using in this section satisfies neither of these additional axioms. But later in Section 4, we will be using ’antimatroid’ closure operators, that is those which satisfy the ’anti-exchange axiom’ if p, q ∈ X.ϕ and q ∈ (X ∪ {p}).ϕ then p ∈ (X ∪ {q}).ϕ. Every boolean algebra, or lattice L, is a closure system because x, y ∈ L implies x ∧ y ∈ L. In particular, any predicate p = 0 or 1 is a trivial boolean algebra, or closure system Cp . Each element, 0 or 1, is its own generator. In order to make it a closure lattice, whenever two tuples are covered by a common tuple, their ’intersection tuple’ has been entered into the order, as required in any well-formed closure system. These intersection tuples have been underlined for emphasis.
2.3
Figure 1: Lattice of closed sets implicit in Table 1 it. Intersection and cover tuples are of special interest because show correlations between the closest available groups. A distance between two tuples t1 = (x1 , x2 , ..., xm ) and t2 = (y1 , y2 , ..., ym ) is defined as |x1 − y1 | + |x2 − y2 | + ... + |xm − ym | = Dt1 t2 . The distance between corresponding elements in two tuples d = |xi − yi |, i = 1, .., m is considered to be significant if d ≥ m 2.
2.4
Consider students enrolled in a subject where preliminary knowledge is a prerequisite for understanding new content. Students are first suggested to take a test detecting lack of necessary preliminary knowledge or skills. Suppose the test outcome implies insufficient knowledge or skills. The system then provides personalized help to each student based on his/her individual needs. If the test outcome does not imply lack of preliminary knowledge or skills the student is then directed to Chapter 1. Students are divided in sixteen groups g1, g2, ..., g16, according to gender and results from a test providing information about their preliminary knowledge in calculus. Meaning of entrances in Table 1: 0 - indicates amount of correct results being below 40% 1 - indicates amount of correct results being at least 40% and below 70%
Intersection and Cover Tuples
Suppose tuple t1 is an intersection tuple for tuples t2 and t3. This means that topics mastered by the group represented by t1 are exactly the topics mastered by the groups represented by t2 and t3 with the same amount of correct results. Suppose tuple u1 is a cover tuple for tuples u2 and u3. This means that topics mastered by the group represented by u1 are mastered at least as well as the groups represented by t2 and t3 with amount of correct results being better or equal to the corresponding results indicated by t2 and t3. Information obtained via intersection and cover tuples can be used to find out what is causing a lesser learning outcome and what should be done to improve
ISBN: 978-1-61804-084-8
Application
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Recent Researches in Applied Computers and Computational Science
Groups g1 g2 g3 g4 g5 g6 g7 g8 g9 g 10 g 11 g 12 g 13 g 14 g 15 g 16
Table 1: Test outcomes for groups g1, g2, ..., g16 Polynomials Fractions Logarithms Trigonom. functions 2 2 2 1 2 2 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 2 2 2 1 2 2 1 1 1 1 1 0 2 1 2 1 0 1 0 1 2 0 0 1 1 1 1 0 0 0 0 0
Differentiation 2 1 1 0 0 0 0 0 1 2 1 1 0 0 0 1
2 - indicates amount of correct results being at least 70% The lattice in Fig. 2 shows distances between related tuples. Thus there are 3 couples of tuples at distance 4, one couple of tuples at distance 3, 5 couples of tuples at distance 2, and 16 couples of tuples at distance 1. The two lattices in Fig. 3 and Fig. 4 illustrate correlations between different tuples and for male and female students respectively. These lattices facilitate clearer visualisation of dependences in the two groups of students. The lattice in Fig. 3 has two cover tuples 101000 and 11110 and no intersection tuples. The lattice in Fig. 4 has three cover tuples 101000 and 11110 and one intersection tuple 20010.The intersection tuples in Fig. 1 and Fig. 4 are underlined.
2.5
Comparison of a Concept Lattice and a Lattice of Closed Sets
The concept lattice in Fig. 5 is based on the context in Table 2 which is a binary version of Table 1. The object g1 in Table 2 represents all groups from Table 1 with nonzero entrances. The number of objects is thus reduced from 15 to 11. The concept lattice in Fig. 5 does not show differences with respect to degrees to which an object possess an attribute. The lattice in Fig. 1 shows two degrees to which an object possess an attribute. On the other hand lattices relating objects and attributes taking values larger than 3 can be too
ISBN: 978-1-61804-084-8
Figure 2: Lattice of closed sets implicit in Table 1 with distances
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Recent Researches in Applied Computers and Computational Science
Groups g1 g4 g5 g6 g7 g8 g 11 g 13 g 14 g 15 g 16
Table 2: Test outcomes for some of the groups g1, ..., g16 Polynomials Fractions Logarithms Trigonom. functions Differentiation P F L T D × × × × × × × × × × × × × × × × × × × × × × × × × ×
Figure 4: Lattice of closed sets implicit in Table 1 considering groups g9, ..., g16
Figure 3: Lattice of closed sets implicit in Table 1 considering groups g1, ..., g8
ISBN: 978-1-61804-084-8
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[5] Pfaltz, J. L., Closure Lattices, Discrete Mathematics, 154, 1996, 217–236 [6] Pfaltz, J. L., Representing Numeric Values in Concept Lattices, International Conference on Concept Lattices and Their Applications, CLA, 2007. [7] Pfaltz, J. L., A Category of Discrete Partially Ordered Sets, Mid-Atlantic Algebra Conference George Mason University, Fairfax VA, Nov. 2004. [8] Pfaltz, J. L., Establishing Logical Rules from Empirical Data Intern. Journal on Artificial Intelligence Tools , Vol. 17, no. 5 (2008) 985–1001 [9] Pfaltz, J. L. and Taylor, C. M., Closed Set Mining of Biological Data, Workshop on Data Mining in Bioinformatics, SIGKDD 02, 43–58, 2002 [10] Wille, R., Concept lattices and conceptual knowledge systems, Computers Math. Applications, 23(6-9), 1992, 493–515 [11] Zaki M. J., Generating Non-Redundant Association Rules. In 6th ACM SIGKDD Intern’l Conf. on Knowledge Discovery and Data Mining, 34– 43, Boston, MA, 2000.
Figure 5: Concept lattice of the context in Table 2 laborious to construct and difficult to read.
3
Conclusion
Employing nonbinary values for attributes and applying lattices of closed sets instead of concept lattices provides additional useful information. This turns out to be very useful in cases containing attributes that benefit from measuring distances among the attributes. References: [1] Brossette S. E. and Sprague A. P., Medical surveillance, frequent sets and closure operations, Journal of Combinatorial Optimization, 5, 81–94, 2001. [2] Davey, B. A., and Priestley, H.A.: Introduction to lattices and order. Cambridge University Press, Cambridge, (2005) [3] Garriga G. C., Khardon R., and Raedt L., On mining closed sets in multi-relational data, Proceedings of the 20th international joint conference on Artificial intelligence, IJCAI’07, Morgan Kaufmann Publishers Inc. San Francisco, CA, USA, 2007 [4] Godin R. and Missaoui R., An incremental concept formation approach for learning from databases. Theoretical Computer Science, 133, 387–419, 1994. ISBN: 978-1-61804-084-8
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