Data mining using Rough Sets
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1 Data mining using Rough Sets Alber Sánchez 1 alber.ipia@inpe.br 1 Instituto Nacional de Pesquisas Espaciais, São José dos Campos, SP, Brazil Referata Geoinformatica, / 44
2 Table of Contents Rough Sets Reducts Knowledge reduction Applications Software References 2 / 44
3 Rough Set Theory Who? Zdzis law Pawlak When? In the 80 s What? Classificatory analysis of data tables. Why? To synthesize approximations of concepts from data. 3 / 44
4 Cutting to the chase - It allows to go from this... Table 1 : Decision table Diploma Experience French Reference Decision x 1 MCE Low No Good Stand By x 2 MCE Low No Neutral Stand By x 3 MBA Low No Neutral Rejected x 4 MCE Medium No Good Rejected x 5 MCE Medium No Excellent Accept x 6 Msc Medium No Excellent Accept x 7 Msc High Yes Excellent Accept x 8 Msc High Yes Excellent Accept 4 / 44
5 To this... Table 2 : Core values of conditions U Diploma Experience Reference Decision x 1 MCE Low * Stand By x 2 MBA * * Rejected x 3 * Medium Good Rejected x 4 * * Excellent Accept 5 / 44
6 Venn diagram Figure 1 : A Set. Taken from 6 / 44
7 Information systems Also known as tables A = (U, A) U: non-empty finite set of objects A: non-empty finite set of attributes a : U V a, a A V a is the value set of a. Table 3 : An information system Age LEMS x x x x x x x / 44
8 Decision tables Table 4 : An Decision Table Age LEMS Walk x Yes x No x No x Yes x No x Yes x No 8 / 44
9 Equivalence Equivalence relation R X X Binary Reflexive (xrx) Symmetric (xry yrx) Transitive (xry yrz = xrz) Equivalence class The EC of x X consists all of y X xry 9 / 44
10 Indiscernibility relation IND A (B) is called the B-Indiscernibility relation Let A = (U, A) be an Information System Then B A IND A (B) Where IND A (B) = {(x, x ) U 2 a B a(x) = a(x )} 10 / 44
11 Indiscernibility relation example Table 5 : A decision Table Age LEMS Walk x Yes x No x No x Yes x No x Yes x No IND({Age}) = {{x1, x2, x6}, {x3, x4}, {x5, x7}} IND({LEMS}) = {{x1}, {x2}, {x3, x4}, {x5, x6, x7}} IND({Age, LEMS}) = {{x1}, {x2}, {x3, x4}, {x5, x7}, {x6}} The equivalence classes of the B-indiscernibility relation are denoted [x] B 11 / 44
12 Set approximation The concept walk cannot be defined as a crisp set using Age and LEMS because of {x3, x4} However, we can approximate it using 3 sets. Those objects which fulfil Walk = Yes Those objects which fulfil Walk = No The remaining objects 12 / 44
13 Set approximation 2 Let A = (U, A) be a IS Let B A Let X U X can be approximated using only the information contained in B using 3 sets: B-lower approximation of X, BX = {x [x] B X } B-upper approximation of X, BX = {x [x] B X } B-boundary region, BN B = BX BX 13 / 44
14 Set approximation 3 On the basis of knowledge in B: Objects in BX can be with certainly classified as members of X Objects in BX can be only classified as possible members of X Objects we cannot decisively classify into X Besides, there is the set B-outside region of X which is U BX 14 / 44
15 Rough Set definition A set is said to be rough if the boundary region is non-empty. 15 / 44
16 Rough Set example Table 6 : A decision Table Age LEMS Walk x Yes x No x No x Yes x No x Yes x No Let W = {x Walk(x) = yes}, then: AW = {x1, x6} AW = {x1, x3, x4, x6} BNA (W ) = {x3, x4} U AW = {x2, x5, x7} 16 / 44
17 Rough Set graphic example Figure 2 : A rough set. 17 / 44
18 Rough Set properties 1. B(X ) X B(X ) 2. B( ) = B( ), B(U) = B(U) = U 3. B(X Y ) = B(X ) B(Y ) 4. B(X Y ) = B(X ) B(Y ) 5. X Y implies B(X ) B(Y ) and B(X ) B(Y ) 6. B(X Y ) B(X ) B(Y ) 7. B(X Y ) B(X ) B(Y ) 8. B( X ) = B(X ) 9. B( X ) = B(X ) 10. B(B(X )) = B(B(X )) = B(X ) 11. B(B(X )) = B(B(X )) = B(X ) Where X denotes U X 18 / 44
19 Rough Set classification X is roughly B-definable, iff B(X ) and B(X ) U X is internally B-indefinable, iff B(X ) = and B(X ) U X is externally B-indefinable, iff B(X ) and B(X ) = U X is totally B-indefinable, iff B(X ) = and B(X ) = U 19 / 44
20 Accuracy of approximation α B (X ) = B(X ), where X is the cardinality of X B(X ) 0 α B (X ) 1 if α B (X ) = 1, X is crisp with respect to B If α B (X ) < 1, X is rough with respect to B 20 / 44
21 Quality of approximation γ B (X ) = B(X ) U, where X is the cardinality of X It express the percentage of possible correct decisions when classifying objects employing the knowledge B 21 / 44
22 Table of Contents Rough Sets Reducts Knowledge reduction Applications Software References 22 / 44
23 Reducts Let A = (U, A) A reduct of A is a minimal set of attributes B A such that IND A (B) = IND A (A) A reduct is a minimal set of attributes from A that preserves the partitioning of the universe, and hence, the ability to perform classifications as the whole attribute set A does. 23 / 44
24 Reduct example A = (U, {Diploma, Experience, French, Reference}) Table 7 : An unreduced decision table Diploma Experience French Reference Decision x 1 MBA Medium Yes Excellent Accept x 2 MBA Low Yes Neutral Reject x 3 MCE Low Yes Good Reject x 4 Msc High Yes Neutral Accept x 5 Msc Medium Yes Neutral Reject x 6 Msc High Yes Excellent Accept x 7 MBA High No Good Accept x 8 MCE Low No Excellent Reject 24 / 44
25 Discernibility matrix and function DM is a symmetric nxn matrix which entries are: c ij = {a A a(x i ) a(x j )} for i, j = 1,..., n DF f A is a Boolean function of m Boolean variables a 1,..., a m (corresponding to attributes a 1,..., a m ) defined as below, where c ij = {a a c ij } f A (a 1,..., a m) = { c ij 1 j i n, c ij } The set of all prime implicants of f A determines the set of all reducts of A 1 1 An implicant of a Boolean function f is any conjuntion of literals (variables or their negations) such that if the values of that literals are true under an arbitrary valuation v of variables then thge value of the function f under v is also true. A rpime implicant is a minimal implicant. Here we are interested in implicants of monotone Boolean functions only i.e. functions constructed without negation. 25 / 44
26 k-relative discernibility function & reducts Resulting from constructing a Boolean function by restricting the conjuntion to only run over column k in the discernibility matrix (instead of all the columns). The set of all prime implicants of this function determines the set of all k-relative reducts of A. These reducts reveal the minimum amount of information needed to discern x k U (or more pecisely [x k ] U) from all other objects. 26 / 44
27 Table of Contents Rough Sets Reducts Knowledge reduction Applications Software References 27 / 44
28 Example Table 8 : Decision table Diploma Experience French Reference Decision x 1 MCE Low No Good Stand By x 2 MCE Low No Neutral Stand By x 3 MBA Low No Neutral Rejected x 4 MCE Medium No Good Rejected x 5 MCE Medium No Excellent Accept x 6 Msc Medium No Excellent Accept x 7 Msc High Yes Excellent Accept x 8 Msc High Yes Excellent Accept 28 / 44
29 Encode values Diploma 0 MBA 1 MCE 2 Msc Experience 0 Low 1 Medium 2 High French 0 No 2 Yes Reference 0 Neutral 1 Good 2 Excellent Decision 0 Rejected 1 Stand By 2 Accept Table 9 : Encoded decision table U a b c d e x x x x x x x x / 44
30 Compute indiscernibility relation Table 10 : Encoded decision table U a b c d e x x x x x x x IND{a} = {{x 1, x 2, x 4, x 5 }, {x 3 }, {{x 6, {x 7 }} IND{a, b, c} = {{x 1, x 2 }, {x 3 }, {x 4, x 5 }, {x 6 }, {x 7 }} (...) IND{a, b, d} = {{x 1 }, {x 2 }, {x 3 }, {x 4 }, {x 5 }, {x 6 }, {x 7 }} IND{a, b, c, d} = {{x 1 }, {x 2 }, {x 3 }, {x 4 }, {x 5 }, {x 6 }, {x 7 }} Attribute c is superfluous because IND{a, b, d} = IND{a, b, c, d} 30 / 44
31 Compute core values of conditions Table 11 : Reduced decision table U a b d e x x x x x x x For rule x 1 : F = {[x 1 ] a, [x 1 ] b, [x 1 ] d } F = {{x 1, x 2, x 4, x 5 }, {x 1, x 2, x 3 }, {x 1, x 4 }} Consider that [x 1 ] a,b,d = [x 1 ] a [x 1 ] b [x 1 ] d = {x 1 } [x 1 ] e = {x 1, x 2 } Find a smaller relation being a subset of [x 1 ] e [x 1 ] b [x 1 ] d = {x 1 } [x 1 ] e [x 1 ] a [x 1 ] d = {x 1, x 4 } [x 1 ] a [x 1 ] b = {x 1, x 2 } [x 1 ] e So, b(x 1 ) = 0 is a core value because it is present in [x 1 ] b [x 1 ] d and [x 1 ] a [x 1 ] b, both are subsets of [x 1 ] e 31 / 44
32 Result of computing the core values of conditions Table 12 : Core values of conditions U a b d e x x x x x x x / 44
33 Compute value reducts For rule x 1 : F = {[x 1 ] a, [x 1 ] b, [x 1 ] d } = {{x 1, x 2, x 4, x 5 }, {x 1, x 2, x 3 }, {x 1, x 4 }} We need to find all subfamilies G F G [x 1 ] e = {x 1, x 2 } [x 1 ] b [x 1 ] d = {x 1, x 2, x 3 } {x 1, x 4 } = {x 1 } [x 1 ] e [x 1 ] a [x 1 ] d = {x 1, x 2, x 4, x 5 } {x 1, x 4 } = {x 1, x 4 } [x 1 ] a [x 1 ] b = {x 1, x 2, x 4, x 5 } {x 1, x 2, x 3 } = {x 1, x 2 } [x 1 ] e So, only [x 1 ] b [x 1 ] d and [x 1 ] a [x 1 ] b are reducts of the family F 33 / 44
34 Results of computing value reducts Table 13 : Core values of conditions U a b d e x * 1 x 1 * x * 1 x 2 1 * 0 1 x 3 0 * * 0 x 4 * x 5 * * 2 2 x 6 * * 2 2 x 6 2 * * 2 x 7 * * 2 2 x 7 * 2 * 2 x 7 2 * * 2 34 / 44
35 Many possible solutions Table 14 : Core values of conditions U a b d e x * 1 x 2 1 * 0 1 x 3 0 * * 0 x 4 * x 5 * * 2 2 x 6 * * 2 2 x 7 2 * * 2 Table 15 : Core values of conditions U a b d e x * 1 x * 1 x 3 0 * * 0 x 4 * x 5 * * 2 2 x 6 * * 2 2 x 7 * * / 44
36 Minimal solution After removing duplicates and re-numbering Table 16 : Core values of conditions U a b d e x * 1 x 2 0 * * 0 x 3 * x 4 * * / 44
37 Minimal solution decoded Table 17 : Core values of conditions U Diploma Experience Reference Decision x 1 MCE Low * Stand By x 2 MBA * * Rejected x 3 * Medium Good Rejected x 4 * * Excellent Accept 37 / 44
38 Table of Contents Rough Sets Reducts Knowledge reduction Applications Software References 38 / 44
39 Applications Data mining AI 39 / 44
40 Sense, plan, act Figure 3 : Sense, plan, act cycle. 40 / 44
41 Table of Contents Rough Sets Reducts Knowledge reduction Applications Software References 41 / 44
42 Software R packages RoughSetKnowledgeReduction and RoughSets RSES - Rough Set Exploration System Infobright Community Edition 42 / 44
43 Table of Contents Rough Sets Reducts Knowledge reduction Applications Software References 43 / 44
44 References I Pawlak, Zdzis law. Rough Sets: Theoretical Aspects of Reasoning about Data. Springer Netherlands, Komorowski, Jan et al. Rough sets: A tutorial. Rough fuzzy hybridization: A new trend in decision-making, pages 3 98, / 44
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