1. On Applications of Rough Sets theory to Knowledge Discovery Frida Coaquira UNIVERSITY OF PUERTO RICO MAYAGÜEZ CAMPUS frida_cn@math.uprm.edu

2. Introduction One goal of the Knowledge Discovery is extract meaningful knowledge. Rough Sets theory was introduced by Z. Pawlak (1982) as a mathematical tool for data analysis. Rough sets have many applications in the field of Knowledge Discovery: feature selection, discretization process, data imputations and create decision Rules. Rough set have been introduced as a tool to deal with, uncertain Knowledge in Artificial Intelligence Application.

3. Equivalence Relation Let X be a set and let x, y, and z be elements of X. An equivalence relation R on X is a Relation on X such that: Reflexive Property: xRx for all x in X. Symmetric Property: if xRy, then yRx. Transitive Property: if xRy and yRz, then xRz.

4. Rough Sets Theory Let, be a Decision system data, Where: U is a non-empty, finite set called the universe, A is a non-empty finite set of attributes, C and D are subsets of A, Conditional and Decision attributes subsets respectively. for is called the value set of a, The elements of U are objects, cases, states, observations. The Attributes are interpreted as features, variables, characteristics conditions, etc.

5. Indiscernibility Relation The Indecernibility relation IND(P) is an equivalence relation. Let,, the indiscernibility relation IND(P), is defined as follows: for all

6. Indiscernibility Relation The indiscernibility relation defines a partition in U. Let, U/IND(P) denotes a family of all equivalence classes of the relation IND(P), called elementary sets. Two other equivalence classes U/IND(C) and U/IND(D), called condition and decision equivalence classes respectively, can also be defined.

7. R-lower approximation Let and, R is a subset of conditional features, then the R-lower approximation set of X, is the set of all elements of U which can be with certainty classified as elements of X. R-lower approximation set of X is a subset of X

8. R-upper approximation the R-upper approximation set of X, is the set of all elements of U such that: X is a subset of R-upper approximation set of X. R-upper approximation contains all data which can possibly be classified as belonging to the set X the R-Boundary set of X is defined as:

9. Representation of the approximation sets If then, X is R-definible (the boundary set is empty) If then X is Rough with respect to R. ACCURACY := Card(Lower)/ Card (Upper)

10. Decision Class The decision d determines the partition of the universe U. Where for will be called the classification of objects in T determined by the decision d. The set X k is called the k-th decision class of T

11. Decision Class This system data information has 3 classes, We represent the partition: lower approximation, upper approximation and boundary set.

12. Rough Sets Theory Lets consider U={x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8 } and the equivalence relation R with the equivalence classes: X 1 ={x 1,x 3,x 5 }, X 2 ={x 2,x 4 }and X 3 ={x 6,x 7,x 8 } is a Partition. Let the classification C={Y 1,Y 2,Y 3 } such that Y 1 ={x 1, x 2, x 4 }, Y 2 ={x 3, x 5, x 8 }, Y 3 ={x 6, x 7 } Only Y 1 has lower approximation, i.e.

13. Positive region and Reduct Positive region POS R (d) is called the positive region of classification CLASS T (d) is equal to the union of all lower approximation of decision classes. Reducts,are defined as minimal subset of condition attributes which preserve positive region defined by the set of all condition attributes, i.e. A subset is a relative reduct iff 1, 2 For every proper subset condition 1 is not true.

14. Dependency coefficient Is a measure of association, Dependency coefficient between condition attributes A and a decision attribute d is defined by the formula: Where, Card represent the cardinality of a set.

15. Discernibility matrix Let U={x 1, x 2, x 3,…, x n } the universe on decision system Data. Discernibility matrix is defined by:, where, is the set of all attributes that classify objects x i and x j into different decision classes in U/D partition. for some i, j }.

16. Dispensable feature Let R a family of equivalence relations and let P R, P is dispensable in R if IND(R) = IND(R-{P}), otherwise P is indispensable in R. CORE The set of all indispensable relation in C will be called the core of C. CORE(C)= ∩RED(C), where RED(C) is the family of all reducts of C.

17. Small Example Let, the universe set., the conditional features set., Decision features set. d 10211 10201 12002 12210 21002 21102 21211 {,,{,{{,{,{,,,{,,{,,,{,,,,,{,,,,,,,

18. Discernibility Matrix - - -- --

19. Example Then, the Core(C) = {a 2 } The partition produces by Core is U/{a 2 } = {{ x 1,x 2 },{x 5, x 6,x 7 },{x 3,x 4 }}, and the partition produces by the decision feature d is U/{d}={{ x 4 },{ x 1,x 2,x 7 },{x 3,x 5,x 6 }}

20. Similarity relation A similarity relation on the set of objects is, It contain all objects similar to x. Lower approximation, is the set of all element of U which can be with certainty classified as elements of X. Upper approximation SIM-Possitive region of partition Let

21. Similarity measures a b are parameters, this measure is not symmetric. Similarity for nominal attribute

22. Quality of approximation of classification Is the ratio of all correctly classified objects to all objects. Relative Reduct is s relative reduct for SIM A {d} iff 1) 2) for every proper subset condition 1) is not true.

23. Attribute Reduction The purpose is select a subset of attributes from an Original set of attributes to use in the rest of the process. Selection criteria: Reduct concept description. Reduct is the essential part of the knowledge, which define all basic concepts. Other methods are: Discernibility matrix (n×n) Generate all combination of attributes and then evaluate the classification power or dependency coefficient (complete search).

24. Discretization Methods The purpose is development an algorithm that find a consistent set of cuts point which minimizes the number of Regions that are consistent. Discretization methods based on Rough set theory try to find These cutpoints A set of S points P1, …, Pn in the plane R2, partitioned into two disjoint categories S1, S2 and a natural number T. Is there a consistent set of lines such that the partition of the plane into region defined by them consist of at most T regions?

25. Consistent Def. A set of cuts P is consistent with A (or A-consistent) iff, where and are general decisions of A and A P respectively. Def. A set P irr of cuts is A-irreducible iff P irr is A-consistent and any its proper subfamily P’ ( P’ PP irr ) is not A-inconsistent.

26. Level of Inconsistency Let B a subset of A and Where X i is a classification of U and, i = 1,2,…,n L c represents the percentage of instances which can be Correctly classified into class X i with respect to subset B.

27. Imputation Data The rules of the system should have Maximum in terms of consistency. The relevant attributes for x is defined by. is defined } And the relation for all x and y are consistent if. Example Let x=(1,3,?,4), y=(2,?,5,4) and z=(1,?,5,4) x and z are consistent x and y are not consistent

28. Decision rules F1F2F3F4DRules O30001LR1 O50013LR1 O10102LR2 O40110MR3 O21102HR4 Rule1 if (F2=0) then (D=L) Rule2 if (F1=0) then (D=L) Rule3 if (F4=0) then (D=M) Rule4 if (F1=0) then (D=H) The algorithm should minimize the number of features included in decision rules.

29. References [1] Gediga, G. And Duntsch, I. (2002) Maximum Consistency of Incomplete Data Via Non-invasive Imputation. Artificial Intelligence. [2] Grzymala, J. and Siddhave, S. (2004) Rough set Approach to Rule Induction from Incomplete Data. Proceeding of the IPMU’2004, the10th International Conference on information Processing and Management of Uncertainty in Knowledge-Based System. [3] Pawlak, Z. (1995) Rough sets. Proccedings of the 1995 ACM 23rd annual conference on computer science. [4]Tay, F. and Shen, L. (2002) A modified Chi2 Algorithm for Discretization. In IEEE Transaction on Knowledge and Data engineering, Vol 14, No. 3 may/june. [5] Zhong, N. (2001) Using Rough Sets with Heuristics for Feature Selection. Journal of Intelligent Information Systems, 16, 199-214, Kluwer Academic Publishers.

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