1. 1 Chapter 10 DISSIMILIRATY ANALYSIS Presented by: Turkov. Eugene Class id : 113 and Minfang Tao Class id : 112 Professor: Dr. T.Y. Lin

2. 2 Introduction  In Chapter 8 and 9 we focused on decision tables in which condition and decision attributes were distinguished: CONDITIONSDECISIONS Uabcde 111110 201000 311101 411001 501011 610011 710111

3. 3 Indroduction  In this chapter we are going to discuss Knowledge Representation Systems in which neither condition nor decision attributes are distinguished. CONDITION & DECISION ATTRIBUTES Uabcde 111110 201000 311101 411001 501011 610011 710111

4. 4 Pattern Recognition - Original Table Uabcdefg 11111110 20110000 31101101 41111001 50110011 61011011 71011111 81110000 91111111 101111011

5. 5 Pattern Recognition - Task  “Our task is to find a minimal description of each digit and the corresponding decision algorithms.”

6. 6 Pattern Recognition– Steps for The Task  1. Make sure that the table is consistent. If it is not, create a new consistent table from it.  2. Find the attribute(column) reduct, and create a new table that only includes the attributes which are members of the reduct.  3. Compute core and reduct values for each decision rule.

7. 7 Pattern Recognition - Consistent  Each row in the original table is unique, hence the table is consistent. Uabcdefg 11111110 20110000 31101101 41111001 50110011 61011011 71011111 81110000 91111111 101111011

8. 8 Pattern Recognition – Finding Core Attributes Convert the table to seven equivalence relations: U/a={ {1,3,4,6,7,8,9,10 },{2,5 } } U/b={ {1,2,3,4,5,8,9,10 },{6,7 } } U/c={ {1,2,4,5,6,7,8,9,10 },{3 } } U/d={ {1,3,4,6,7,9,10 },{2,5,8 } } U/e={ {1,3,7,9 },{2,4,5,6,8,10 } } U/f={ {1,5,6,7,9,10 },{2,3,4,8 } } U/g ={ {1,2,8 },{3,4,5,6,7,9,10 } }

9. 9 Pattern Recognition – Find Core Attributes The process for computing the Indiscernibility for all equivalnce relations : U/IND(a,b)={ {1,3,4,8,9,10 },{6,7 },{2,5 } } U/IND(a,b,c)={ {1,4,8,9,10 },{3 },{6,7 },{2,5 } } U/IND(a,b,c,d)={ {1,4,9,10 },{8 },{3 },{6,7 },{2,5 } } U/IND(a,b,c,d,e)={ {1,9 },{4,10 },{8 },{3 },{7 },{6 },{2,5 } } U/IND(a,b,c,d,e,f)={ {1,9 },{10 },{4 },{8 },{3 },{7 },{6 },{5 },{2 } } U/IND(a,b,c,d,e,f,g )={ {1 },{9 },{10 },{4 },{8 },{3 },{7 },{6 },{5 },{2 } }

10. 10 Pattern Recognition – Find Core Attributes Finally, we got: U/IND(ALL R) = U/IND(a,b,c,d,e,f,g )= { {1 },{9 },{10 },{4 },{8 },{3 },{7 },{6 },{5 },{2 } } Computing U/IND(R-x) : U/IND(R-a) = { {1 },{9 },{10 },{4 },{5 },{2,8 },{3 },{7 },{6 } } U/IND(R-a) ! = U/IND(R) U/IND(R-b) = { {1 },{7,9 },{6,10 },{4 },{8 },{3 },{5 },{2 } } U/IND(R-b) ! = U/IND(R) U/IND(R-c) = { {1 },{9 },{3 },{10 },{4 },{8 },{7 },{6 },{5 },{2 } } U/IND(R-c) = = U/IND(R)

11. 11 Pattern Recognition – Find Core Attributes U/IND(R-d) = { {1 },{9 },{10 },{8 },{4 },{3 },{7 },{6 },{5 },{2 } } U/IND(R-d) = = U/IND(R) U/IND(R-e) = { {1 },{9,10 },{4 },{8 },{3 },{6,7 },{5 },{2 } } U/IND(R-e) ! = U/IND(R) U/IND(R-f) = { {1 },{9 },{4,10 },{8 },{3 },{7 },{6 },{2 },{5 } } U/IND(R-f) ! = U/IND(R) U/IND(R-g ) = { {1,9 },{10 },{4 },{8 },{3 },{7 },{6 },{5 },{2 } } U/IND(R-g ) ! = U/IND(R) We have determined that c and d are dispensable and our core attributes are: {a, b, e, f, g}

12. 12 Pattern Recognition – Find Rreduct Attributes  Finding reduct attributes by using core attributes: U/IND(a,b,e,f,g) = { {1 },{9 },{10 },{8 }, {4 },{3 },{7 },{6 },{5 },{2 } } U/IND(a,b,e,f,g) == U/IND(a,b,c,d,e,f,g) (a,b,e,f,g) is one and only one reduct of the original table. (a,b,e,f,g)=>(c,d).

13. 13 Pattern Recognition – Find Core Attributes Uabefg 111110 201000 311101 411001 501011 610011 710111 811000 911111 1011011 Reduct Table – It is consistent

14. 14 Pattern Recognition – Find Core Attributes After removing attribute {a}, the table is inconsistent Ubefg 11110 21000 31101 41001 51011 60011 70111 81000 91111 101011

15. 15 Pattern Recognition – Find Core Attributes After removing attribute {b}, the table is inconsistent Uaefg 11110 20000 31101 41001 50011 61011 71111 81000 91111 101011

16. 16 Pattern Recognition – Find Core Attributes After removing attribute {e}, the table is inconsistent Uabfg 11110 20100 31101 41101 50111 61011 71011 81100 91111 101111

17. 17 Pattern Recognition – Find Core Attributes After removing attribute {f}, the table is inconsistent Uabeg 11110 20100 31111 41101 50101 61001 71011 81100 91111 101101

18. 18 Pattern Recognition – Find Core Attributes After removing attribute {g}, the table is inconsistent Uabef 11111 20100 31110 41100 50101 61001 71011 81100 91111 101101

19. 19 Pattern Recognition – Decision Rules Conditions & Decisions Uabefg 111110 201000 311101 411001 501011 610011 710111 811000 911111 1011011 Attributes {a, b, e, f, g} are not only conditions, but they are also decisions.

20. 20 Pattern Recognition – Decision Rules For the sake of illustration, we extend this table. ConditionsDecisions Uabefg a’a’b’b’e’e’f’f’g’g’ 11111011110 20100001000 31110111101 41100111001 50101101011 61001110011 71011110111 81100011000 91111111111 101101111011

21. 21 Pattern Recognition – Decision Rules  U/IND(a,b,e,f,g) = U/IND(a ’,b ’,e ’,f ’,g ’ )= {{1},{2},{3},{4}, {5},{6},{7},{8},{9}, {10}}.  To simplify the table, we use one attribute t= {1,2,3,4,5,6,7,8,9,10} to replace attribute (a ’,b ’,e ’,f ’,g ’ ).  U/IND(t) == U/IND(a ’,b ’,e ’,f ’,g ’ )  {a, b, e, f, g} are conditions, the {t} is decision.

22. 22 Pattern Recognition – Decision Rules CONDITIONSDECISIONS Uabefgt 1111101 2010002 3111013 4110014 5010115 6100116 7101117 8110008 9111119 1011011 Computing core and reducts values will depend on this regular decision table.

23. 23 Pattern Recognition - Decision Rules  Method 1 for finding reducts for each rule. F={{a}, {b}, {e}, {f}, {g}} All subfamilies G ⊆ to F + G={{a}, {b}, {e}, {f}, {g}, {ab}, {ae}, {af},.. {be}, …..{eg}, {fg} {abe}, {abf}, {abg}, {bef}, {beg}, {ebg}, {abef}, {abeg}, {befg},{abefg} }. The relationship for the elements in G is intersection :{abe}={a}  {b}  {e} Using G to find reducts.

24. 24 Pattern Recognition - Decision Rules  Method 2 for finding reducts for each rule. (Pawlak ’ s method) Find core value for every rule Testing this core value is reduct value? If it is reduct value, we can say this rule has only one reduct value. If it is not reduct value, we will add an uncore value into it, then test are they reduct value? Repeat, until find all reduct values.

25. 25 Pattern Recognition - Decision Rules Computing core for every rule -- rule 1 In rule 1 {a={1,3,4,6,7,8,9,10}, b={1,2,3,4,5,8,9,10}, e={1,3,7,9}, f={1,5,6,7,9,10}, g={1,2,8}} and Decision for rule 1 is [1] t ={1} Removing a, Intersection (b,e,f,g) = {1} == [1]t Removing b, Intersection (a,e,f,g) = {1} == [1]t Removing e, Intersection (a,b,f,g) = {1} == [1]t Removing f, Intersection (a,b,e,g) = {1} == [1]t Removing g, Intersection (a,b,e,f) = {1,9} != [1]t g is the core value.

26. 26 Pattern Recognition - Decision Rules Computing core for every rule -- rule 2 In rule 2 {a={2,5}, b={1,2,3,4,5,8,9,10}, e={2,4,5,6,8,10}, f={2,3,4,8}, g={1,2,8}} and Decision for rule 2 is [2]t ={2} Removing a, Intersection (b,e,f,g) = {2,8} != [2]t ={2} Removing b, Intersection (a,e,f,g) = {2} == [2]t ={2} Removing e, Intersection (a,b,f,g) = {2} == [2]t ={2} Removing f, Intersection (a,b,e,g) = {2} == [2]t ={2} Removing g, Intersection (a,b,e,f) = {2} == [2]t ={2} a is the core value.

27. 27 Pattern Recognition - Decision Rules Computing core for every rule -- rule 3 In rule 3 {a={1,3,4,6,7,8,9,10}, b={1,2,3,4,5,8,9,10}, e={1,3,7,9},f={2,3,4,8}, g={3,4,5,6,7,9,10}} and Decision for rule 3 is [3]t ={3} Removing a, Intersection (b,e,f,g) = {3} == [3]t Removing b, Intersection (a,e,f,g) = {3} == [3]t Removing e, Intersection (a,b,f,g) = {3,4} != [3]t Removing f, Intersection (a,b,e,g) = {3,9} != [3]t Removing g, Intersection (a,b,e,f) = {3} == [3]t e and f are the core values.

28. 28 Pattern Recognition - Decision Rules Computing core for every rule -- rule 4 Core values :e, f, g Computing core for every rule -- rule 5 Core values :a Computing core for every rule -- rule 6 Core values :b, e Computing core for every rule -- rule 7 Core values :b, e Computing core for every rule -- rule 8 Core values :a, g Computing core for every rule -- rule 9 Core values :b, e, f, g Computing core for every rule -- rule 10 Core values :a, b, e, f

29. 29 Pattern Recognition - Decision Rules The core values table: Uabefgt 1----01 20----2 3--10-3 4--0014 50----5 6-00--6 7-01--7 81---08 9-11119 101101-

30. 30 Pattern Recognition - Decision Rules Computing reduct values by using core values -- rule 1 In rule 1 {a={1,3,4,6,7,8,9,10}, b={1,2,3,4,5,8,9,10}, e={1,3,7,9}, f={1,5,6,7,9,10}, g={1,2,8}} and Intersection(a,b,e,f,g) = [1]t ={1} and the core value is g. g ! = [1]t, so g is not reduct value. Intersection (a,g)={1,8}!= [1]t = {1} ; Intersection(b,g)={1,2,8} != [1]t = {1}; Intersection(e,g)={1} == [1]t = {1} ; Intersection(f,g)={1} == [1]t = {1} ; Intersection (a,b,g)={1,8}!= [1]t = {1} ; Reducts values are { {e, g}, {f, g} }

31. 31 Pattern Recognition - Decision Rules Computing reduct values by using core values -- rule 2 In rule 2 {a={2,5}, b={1,2,3,4,5,8,9,10}, e={2,4,5,6,8,10}, f={2,3,4,8}, g={1,2,8}}and Intersection(a,b,e,f,g) = [2]t ={2} and the core value is a. a ! = Intersection(a,b,e,f,g), so a is not reduct value. Intersection(a,b)={2,5} != [2]t = {2} ; Intersection(a,e)={2,5} != [2]t = {2}; Intersection(a,f)={2} == [2]t = {2} ; Intersection(a,g)={2} == [2]t = {2} ; Intersection (a,b,e)={2, 5}!= [2]t = {2} ; Reducts values are { {a, f}, {a, g} }

32. 32 Pattern Recognition - Decision Rules Computing reduct values by using core values -- rule 3 In rule 3 {a={1,3,4,6,7,8,9,10},b={1,2,3,4,5,8,9,10},e={1,3,7,9},f={2,3,4,8}, g={3,4,5,6,7,9,10}}and Intersection(a,b,e,f,g) = [3]t ={3} and the core value is e, f. Intersection(e,f) ={3} == [3]t ={3} { {e, f} } are not only core value,but they also are reduct value and they are only on reduct value for rule 3.

33. 33 Pattern Recognition - Decision Rules For rule 4, the reduct value is:{ {e, f, g} } For rule 5, the reduct values are:{ {a, f}, {a, g} } For rule 6, the reduct value is:{ {b, e} } For rule 7, the reduct value is:{ {b, e} } For rule 8, the reduct values are:{{a, e, g}, {a, f, g}} For rule 9, the reduct value is:{ {b, e, f, g} } For rule 10, the reduct value is:{ {a, b, e, f} }

34. 34 Pattern Recognition - Decision Rules Uabefgt 1(1)xx1x01 1(2)xxx101 2(1)0xx0x2 2(2)0xxx02 3(1)xx10x3 4(1)xx0014 5(1)0xx1x5 5(2)0xxx15 6(1)x00xx6 7(1)x01xx7 8(1)1x0x08 8(2)1xx008 9(1)x11119 10(1)1101x10 For rule 8, there are two reduct values:{ {a, e, g} and {a, f, g} }

35. 35 Pattern Recognition - Decision Rules  The rule {3, 4, 6, 7, 9,10} have only one reduct value, so they are already reducted.  Because the four decision rules {1, 2, 5,8 } have two reduced forms, we have altogether 16 (2*2*2*2) minimal decision algorithms.

36. 36 Pattern Recognition - Decision Rules Uabefgt 1(1)xx1x01 1(2)xxx101 2(1)0xx0x2 2(2)0xxx02 3(1)xx10x3 4(1)xx0014 5(1)0xx1x5 5(2)0xxx15 6(1)x00xx6 7(1)x01xx7 8(1)1x0x08 8(2)1xx008 9(1)x11119 10(1)1101x10 Getting 16 minimal decision algorithms

37. 37 Pattern Recognition - Decision Rules Uabefg 1(1)xx1x0 2(1)0xx0x 3(1)xx10x 4(1)xx001 5(1)0xx1x 6(1)x00xx 7(1)x01xx 8(1)1x0x0 9(1)x1111 10(1)1101x {1, 2, 4,8 } have two reduced forms, we have altogether 16 minimal decision algorithms -- Table 1

38. 38 Pattern Recognition - Decision Rules Uabefg 1(2)xxx10 2(1)0xx0x 3(1)xx10x 4(1)xx001 5(1)0xx1x 6(1)x00xx 7(1)x01xx 8(1)1x0x0 9(1)x1111 10(1)1101x {1, 2, 4,8 } have two reduced forms, we have altogether 16 minimal decision algorithms -- Table 2

39. 39 Pattern Recognition - Decision Rules Uabefg 1(2)xxx10 2(2)0xxx0 3(1)xx10x 4(1)xx001 5(2)0xxx1 6(1)x00xx 7(1)x01xx 8(2)1xx00 9(1)x1111 10(1)1101x {1, 2, 4,8 } have two reduced forms, we have altogether 16 minimal decision algorithms -- Table 16

40. 40 Pattern Recognition - Decision Rules Uabefg 1(2)xxx10f1g0-->1 2(2)0xxx0a0g0-->2 3(1)xx10xe1f0-->3 4(1)xx001e0f0g1-->4 5(2)0xxx1a0g1-->5 6(1)x00xxb0e0-->6 7(1)x01xxb0e1-->7 8(2)1xx00a1f0g0-->8 9(1)x1111b1e1f1g1-->9 10(1)1101xa1b1e0f1-->10 Other format to represent this algorithm

41. 41 The End for Pattern Recognition

42. 42 Appendix  A: Split decision table to consistent and totally inconsisten tables  B: Using G to find reduct values

43. 43 Appendix-A  Split decision table to consistent and totally inconsisten tables

44. 44 Decision table abcde 10220 01112 20011 11022 10201 22011 21112 01101  a b c are conditions, d e are decisions

45. 45 Condition Table: abc 102 011 200 110 102 220 211 011

46. 46 Decision Table: de 20 12 11 22 01 11 12 01

47. 47 Ind(Condition) and Ind(Decision)  U/IND(ALL R) =U/IND(a,b,c)= { {1,5 },{4 },{2,8 },{3 },{7 }, {6 } }  U/IND(ALL R) =U/IND(d,e)= { {1 },{4 },{2,7 },{3,6 },{5,8 } }

48. 48 Computing POS C (D): Computing POSc(D), Checking U/IND(C) to U/IND(D) : { {1,5 },{4 },{2,8 },{3 },{7 },{6 } } is belong to { {1 },{4 },{2,7 },{3,6 },{5,8 } }:

49. 49 Computing…..  Check Set:{3} is belong to Set:{5,8}, it is false, {3} is throwed.  Check Set:{3} is belong to Set:{3,6}, it is true.The set {3} is selected, {3}  Check Set:{3} is belong to Set:{1}, it is false, {3} is throwed.  Check Set:{3} is belong to Set:{2,7}, it is false, {3} is throwed.  Check Set:{3} is belong to Set:{4}, it is false, {3} is throwed.  Check Set:{2,8} is belong to Set:{5,8}, it is false, {2,8} is throwed.  Check Set:{2,8} is belong to Set:{3,6}, it is false, {2,8} is throwed.  Check Set:{2,8} is belong to Set:{1}, it is false, {2,8} is throwed.  Check Set:{2,8} is belong to Set:{2,7}, it is false, {2,8} is throwed.  Check Set:{2,8} is belong to Set:{4}, it is false, {2,8} is throwed.  Check Set:{4} is belong to Set:{5,8}, it is false, {4} is throwed.  Check Set:{4} is belong to Set:{3,6}, it is false, {4} is throwed.  Check Set:{4} is belong to Set:{1}, it is false, {4} is throwed.  Check Set:{4} is belong to Set:{2,7}, it is false, {4} is throwed.  Check Set:{4} is belong to Set:{4}, it is true.The set {4} is selected, {3} U {4}  Check Set:{1,5} is belong to Set:{5,8}, it is false, {1,5} is throwed.

50. 50 Computing…..  Check Set:{1,5} is belong to Set:{3,6}, it is false, {1,5} is throwed.  Check Set:{1,5} is belong to Set:{1}, it is false, {1,5} is throwed.  Check Set:{1,5} is belong to Set:{2,7}, it is false, {1,5} is throwed.  Check Set:{1,5} is belong to Set:{4}, it is false, {1,5} is throwed.  Check Set:{6} is belong to Set:{5,8}, it is false, {6} is throwed.  Check Set:{6} is belong to Set:{3,6}, it is true.The set {6} is selected, {3} U {4} U {6}  Check Set:{6} is belong to Set:{1}, it is false, {6} is throwed.  Check Set:{6} is belong to Set:{2,7}, it is false, {6} is throwed.  Check Set:{6} is belong to Set:{4}, it is false, {6} is throwed.  Check Set:{7} is belong to Set:{5,8}, it is false, {7} is throwed.  Check Set:{7} is belong to Set:{3,6}, it is false, {7} is throwed.  Check Set:{7} is belong to Set:{1}, it is false, {7} is throwed.  Check Set:{7} is belong to Set:{2,7}, it is true.The set {7} is selected, {3} U {4} U {6} U {7}  Check Set:{7} is belong to Set:{4}, it is false, {7} is throwed. POSc(D):{3,4,6,7}

51. 51 Split Table  Using POSc(D):{3,4,6,7} to split the table.  Using 3, 4, 6,7 rows to make the consistent table.  Using 1,2,5,8 rows to make the totally inconsistent table.

52. 52 abcde 20011 11022 22011 21112 abcde 10220 01112 10201 01101 Consistent Table Inconsistent Table:

53. 53 Appendix-B  Using G to find reduct values

54. 54 Pattern Recognition - Decision Rules Computing reducts for every rule -- rule 1 Find reducts in {a={1,3,4,6,7,8,9,10}, b={1,2,3,4,5,8,9,10}, e={1,3,7,9}, f={1,5,6,7,9,10}, g={1,2,8}} for decision rule 1 {1} Checking that if [1]Set(a):{1,3,4,6,7,8,9,10} is belong to decision {1} Checking that if [1]Set(b):{1,2,3,4,5,8,9,10} is belong to decision {1} Checking that if [1]Set(e):{1,3,7,9} is belong to decision {1} Checking that if [1]Set(f):{1,5,6,7,9,10} is belong to decision {1} Checking that if [1]Set(g):{1,2,8} is belong to decision {1} Checking that if [1]Intersection(a,b):{1,3,4,8,9,10} is belong to decision {1} Checking that if [1]Intersection(a,e):{1,3,7,9} is belong to decision {1} Checking that if [1]Intersection(a,f):{1,6,7,9,10} is belong to decision {1} Checking that if [1]Intersection(a,g):{1,8} is belong to decision {1} Checking that if [1]Intersection(b,e):{1,3,9} is belong to decision {1} Checking that if [1]Intersection(b,f):{1,5,9,10} is belong to decision {1} Checking that if [1]Intersection(b,g):{1,2,8} is belong to decision {1}

55. 55 Pattern Recognition - Decision Rules Computing reducts for every rule -- rule 1 Checking that if [1]Intersection(e,f):{1,7,9} is belong to decision {1} Checking that if [1]Intersection(e,g):{1} is belong to decision {1} [1]Intersection(e,g):{1} is one reduct Checking that if [1]Intersection(f,g):{1} is belong to decision {1} [1]Intersection(f,g):{1} is one reduct Checking that if [1]Intersection(a,b,e):{1,3,9} is belong to decision {1} Checking that if [1]Intersection(a,b,f):{1,9,10} is belong to decision {1} Checking that if [1]Intersection(a,b,g):{1,8} is belong to decision {1} Checking that if [1]Intersection(a,e,f):{1,7,9} is belong to decision {1} Checking that if [1]Intersection(a,e,g):{1} is belong to decision {1} Because Intersection(a,e,g): contains Intersection(e,g):, so skip it. Checking that if [1]Intersection(a,f,g):{1} is belong to decision {1} Because Intersection(a,f,g): contains Intersection(f,g):, so skip it. Checking that if [1]Intersection(b,e,f):{1,9} is belong to decision {1}

56. 56 Pattern Recognition - Decision Rules Computing reducts for every rule -- rule 1 Checking that if [1]Intersection(b,e,g):{1} is belong to decision {1} Because Intersection(b,e,g): contains Intersection(e,g):, so skip it. Checking that if [1]Intersection(b,f,g):{1} is belong to decision {1} Because Intersection(b,f,g): contains Intersection(f,g):, so skip it. Checking that if [1]Intersection(e,f,g):{1} is belong to decision {1} Because Intersection(e,f,g): contains Intersection(e,g):, so skip it. Checking that if [1]Intersection(a,b,e,f):{1,9} is belong to decision {1} Checking that if [1]Intersection(a,b,e,g):{1} is belong to decision {1} Because Intersection(a,b,e,g): contains Intersection(e,g):, so skip it. Checking that if [1]Intersection(a,b,f,g):{1} is belong to decision {1} Because Intersection(a,b,f,g): contains Intersection(f,g):, so skip it. Checking that if [1]Intersection(a,e,f,g):{1} is belong to decision {1} Because Intersection(a,e,f,g): contains Intersection(e,g):, so skip it.

57. 57 Pattern Recognition - Decision Rules Computing reducts for every rule -- rule 1 Checking that if [1]Intersection(b,e,f,g):{1} is belong to decision {1} Because Intersection(b,e,f,g): contains Intersection(e,g):, so skip it. Checking that if [1]Intersection(a,b,e,f,g):{1} is belong to decision {1} Because Intersection(a,b,e,f,g): contains Intersection(e,g):, so skip it. Finally we found reducts:{ {e, g}, {f, g} }

58. 58 Pattern Recognition - Decision Rules Computing reducts for every rule -- rule 2 Find reducts in {a={2,5}, b={1,2,3,4,5,8,9,10}, e={2,4,5,6,8,10}, f={2,3,4,8}, g={1,2,8}} for decision rule 2 {2} Checking that if [2]Set(a):{2,5} is belong to decision {2} Checking that if [2]Set(b):{1,2,3,4,5,8,9,10} is belong to decision {2} Checking that if [2]Set(e):{2,4,5,6,8,10} is belong to decision {2} Checking that if [2]Set(f):{2,3,4,8} is belong to decision {2} Checking that if [2]Set(g):{1,2,8} is belong to decision {2} Checking that if [2]Intersection(a,b):{2,5} is belong to decision {2} Checking that if [2]Intersection(a,e):{2,5} is belong to decision {2} Checking that if [2]Intersection(a,f):{2} is belong to decision {2} [2]Intersection(a,f):{2} is one reduct Checking that if [2]Intersection(a,g):{2} is belong to decision {2} [2]Intersection(a,g):{2} is one reduct

59. 59 Pattern Recognition - Decision Rules Computing reducts for every rule -- rule 2 Checking that if [2]Intersection(b,e):{2,4,5,8,10} is belong to decision {2} Checking that if [2]Intersection(b,f):{2,3,4,8} is belong to decision {2} Checking that if [2]Intersection(b,g):{1,2,8} is belong to decision {2} Checking that if [2]Intersection(e,f):{2,4,8} is belong to decision {2} Checking that if [2]Intersection(e,g):{2,8} is belong to decision {2} Checking that if [2]Intersection(f,g):{2,8} is belong to decision {2} Checking that if [2]Intersection(a,b,e):{2,5} is belong to decision {2} Checking that if [2]Intersection(a,b,f):{2} is belong to decision {2} Because Intersection(a,b,f): contains Intersection(a,f):, so skip it. Checking that if [2]Intersection(a,b,g):{2} is belong to decision {2} Because Intersection(a,b,g): contains Intersection(a,g):, so skip it. Checking that if [2]Intersection(a,e,f):{2} is belong to decision {2} Because Intersection(a,e,f): contains Intersection(a,f):, so skip it.

60. 60 Pattern Recognition - Decision Rules Computing reducts for every rule -- rule 2 Checking that if [2]Intersection(a,e,g):{2} is belong to decision {2} Because Intersection(a,e,g): contains Intersection(a,g):, so skip it. Checking that if [2]Intersection(b,e,f):{2,4,8} is belong to decision {2} Checking that if [2]Intersection(b,e,g):{2,8} is belong to decision {2} Checking that if [2]Intersection(b,f,g):{2,8} is belong to decision {2} Checking that if [2]Intersection(e,f,g):{2,8} is belong to decision {2} Checking that if [2]Intersection(a,b,e,f):{2} is belong to decision {2} Because Intersection(a,b,e,f): contains Intersection(a,f):, so skip it. Checking that if [2]Intersection(a,b,e,g):{2} is belong to decision {2} Because Intersection(a,b,e,g): contains Intersection(a,g):, so skip it. Checking that if [2]Intersection(b,e,f,g):{2,8} is belong to decision {2} Finally we found reducts:{ {a, f}, {a, g} }

61. 61 Pattern Recognition - Decision Rules For rule 3, the reducts are:{ {e, f} } For rule 4, the reducts are:{ {e, f, g} } For rule 5, the reducts are:{ {a, f}, {a, g} } For rule 6, the reducts are:{ {b, e} } For rule 7, the reducts are:{ {b, e} } For rule 8, the reducts are:{{a, e, g}, {a, f, g}} For rule 9, the reducts are:{ {b, e, f, g} } For rule 10, the reducts are:{ {a, b, e, f} }

62. 62 Pattern Recognition - Decision Rules Uabefgw 1(1)xx1x01 1(2)xxx101 2(1)0xx0x2 2(2)0xxx02 3(1)xx10x3 4(1)xx0014 5(1)0xx1x5 5(2)0xxx15 6(1)x00xx6 7(1)x01xx7 8(1)1x0x08 8(2)1xx008 9(1)x11119 10(1)1101x10 For rule 8, the reducts are:{ {a, e, g}, {a, f, g} }

63. 63 Pattern Recognition - Decision Rules Uabefgw 1(1)xx1x01 1(2)xxx101 Core value:1----01 2(1)0xx0x2 2(2)0xxx02 Core value :20----2 3(1)xx10x3 Core value : 3--10-3 Intersection to get core values for all decision rules.

64. 64 Pattern Recognition - Decision Rules The core values table: Uabefgw 1----01 20----2 3--10-3 4--0014 50----5 6-00--6 7-01--7 81---08 9-11119 101101-