1. Chapter 10 Dissimilarity Analysis Presented by: ASMI SHAH (ID : 24)

2. 10.4 Pattern Recognition Taking an Example of digits display unit in a calculator, the table will be assumed to represent a characterization of “hand written” digits, consist of horizontal and vertical strokes.

3. Table describes each digit in terms of elements a, b, c, d, e, f and g. Uabcdefg 01111110 10110000 21101101 31111001 40110011 51011011 61011111 71110000 81111111 91111011 Our task is to find a minimal description of each digit and the corresponding decision algorithms. Compute the core first and next, find all reducts of attributes. Core is the set {a, b, c, d, e, f, g}

4. Finding the column reduct It is sufficient to consider only the attributes {a, b, e, f, g} as the basis for the decision algorithm, which means that there is following dependency between the attribute {a, b, e, f, g}  {c, d} Means, c, d are dependent on the reduct so they are not necessary for digits’ recognition. Uabefg 011110 101000 211101 311001 401011 510011 610111 711000 811111 911011

5. Computing Core for each decision rule, Checking the consistency by removing the attributes. Reduce each decision rule in the table separately, by computing first the core of each rule. Dropping the core value makes the consistent rule inconsistent. So finding the column reduct first. Finding the core for each rule. Computing the Value Reducts for each decision rule.

6. Removing attribute ‘b’ we get Uaefg 01110 10000 21101 31001 40011 51011 61111 71000 81111 91011 in which rules (6, 8) and (5, 9) are indiscernible; thus value b0 and b1 are core values in these rules respectively. Removing attribute ‘a’ we get Ubefg 01110 11000 21101 31001 41011 50011 60111 71000 81111 91011 Thus removing attribute a makes rule 1, 7 and 4, 9 inconsistent, i.e. we are unable to discern digits 1 and 7, and 4 and 9 without the attribute a. Hence a 0 is the core value in rule 1 and 4 whereas a 1 is the core value in rules 7 and 9.

7. Removing attribute ‘e‘ we get Uabfg 01110 10100 21101 31101 40111 51011 61011 71100 81111 91111 we get three pairs of indiscernible rules (2, 3), (5, 6) and (8, 9), which yields core values for corresponding rules (e1, e0), (e0, e1) and (e1, e0). which yields indiscernible pairs (2, 8), and (3, 9) and core values for each pair are (f0, f1). Removing attribute ‘f‘ we get Uabeg 01110 10100 21111 31101 40101 51001 61011 71100 81111 91101

8. Finally the last attribute ‘g’ removed gives, Uabef 01111 10100 21110 31100 40101 51001 61011 71100 81111 91101 where pairs of rules (0, 8), and (3, 7) are indiscernible, and the values (g0, g1) and (g1, g0) are cores of corresponding decision rules.

9. Core values for all decision rules Uabefg 0----0 10---- 2--10- 3--001 40---- 5-00-- 6-01-- 71---0 8-1111 91101-

10. Finding Value Reducts for each Row Uabefg 0----0 10---- 2--10- 3--001 40---- 5-00-- 6-01-- 71---0 8-1111 91101- Uabefg 011110 101000 211101 311001 401011 510011 610111 711000 811111 911011 U abefg 0xx1x0 0’xxx10 10xx0x 1’0xxx0 2xx10x 3xx001 40xx1x 4’0xxx1 5x00xx 6x01xx 71x0x0 7’1xx00 8x1111 91101x

11. Rules 2, 3, 5, 6, 8 and 9 are already reduced, since these core values discern these rules from the remaining ones, i.e. the rules with the core values only are consistent (true). For the remaining rules, core values make them inconsistent (false), so they do not form reducts and reducts must be computed by adding proper additional attributes to make the rules consistent. U abefg 0xx1x0 0’xxx10 10xx0x 1’0xxx0 2xx10x 3xx001 40xx1x 4’0xxx1 5x00xx 6x01xx 71x0x0 7’1xx00 8x1111 91101x Finding Value Reducts for each Row

12. Reduced Decision Algorithm Because the four decision rules 0, 1, 4 and 7 have two reduced forms, we have altogether 16 minimal decision algorithms. e 1 g 0 (f 1 g 0 ) 0 a 0 f 0 (a 0 g 0 )1 e 1 f 0 2 e 0 f 0 g 1 3 a 0 f 1 (a 0 g 1 ) 4 b 0 e 0 5 b 0 e 1 6 a 1 e 0 g 0 (a 1 f 0 g 0 ) 7 b 1 e 1 f 1 g 1 8 a 1 b 1 e 0 f 1 9 eg’(f’g) 0 a’f’(a’g’)1 ef’ 2 e’f’g 3 a’f(a’g) 4 b’e’ 5 b’e 6 ae’g’(af’g’) 7 befg 8 abe’f 9 where x and x’ denote variable and its negation. In parenthesis alternative reducts are given.

13. A better Way for Digits Recognition: To make the example more realistic assume that instead of seven elements we are given a grid of sensitive pixels, say 9 * 12, in which seven areas marked a, b, c, d, e, f and g as shown in Figure are distinguished. After slight modification the algorithm will recognize much more realistic handwritten digits. --------- XXXXXXXXX --------X ----X- ------X-- -----X--- ----X---- -X- ---X----- ---X----- --X------ --X--- -X------- -X------- X-------- a e d c b f g

14. Eg: Buying A Car CarPriceMileageSizeMax-SpeedAcceleration 1LowMediumFullLowGood 2Medium CompacthighPoor 3HighMediumFullLowGood 4Medium FullHighExcellent 5LowHighFullLowGood need minimal description of each car in terms of available features, i.e. we can have to find a minimal decision algorithm as discussed previously. Uabcde 1-0+-0 200-+- 3+0+-0 400+++ 5-++-0

15. Where, a – Price b – Mileage c – Size d – Max Speed e – Acceleration and values of attributes are coded as follows: V Price = { low (-), medium (0), high (+) } V Mileage = { low (-), medium (0), high (+) } V Size = { compact (-), medium (0), full (+) } V Max-Speed = { low (-), medium (0), high (+) } V Acceleration = { poor (-), good (0), excellent (+) } The attribute values are coded by symbols in parenthesis. Uabcde 1-0+-0 200-+- 3+0+-0 400+++ 5-++-0

16. Compute first the core of attributes.. Removing the attribute a we get Ubcde 10+-0 20-+- 30+-0 40+++ 5++-0 which is inconsistent, because of two identical rows 1 and 3 Dropping the attribute b we get inconsistent Table Uacde 1-+-0 20-+- 3++-0 40+++ 5-+-0 rows 1 and 5 are identical

17. Removing attributes c, d or e Uabde 1-0-0 200+- 3+0-0 400++ 5-+-0 Uabce 1-0+0 200-- 3+0+0 400++ 5-++0 UaBcd 1-0+- 200-+ 3+0+- 400++ 5-++- which are consistent.

18. Thus the core of attributes is the set {a, b}. There are two reducts {a, b, c} and {a, b, e} of the set of attributes, i.e. there are exactly two consistent and independent tables. Uabc 1-0+ 200- 3+0+ 400+ 5-++ Uabe 1-00 200- 3+00 400+ 5-+0 Thus we have the following dependencies {a,b,c} {d,e} and {a,b,c} {d,c}

19. Uabc 1-0 2- 3++ 40+ 5+ Finding the core values for the reduct {a,b,c} It turns out the core values are reducts of the decision rules. So we have the following decision algorithm a _ b 0 1 c _ 2 a + 3 a 0 c + 4 b + 5 or (price, low) ^ (mileage, medium) 1 (Acceleration, poor) 2 (Price, high) 3 (Acceleration, excellent) 4 (Mileage, high) 5

20. Thus… Each car is uniquely characterized by a proper decision rule and this characterization can serve as a basis for car evaluation. Knowing differences between various options is often the departure point in deciosion making. The rough set approach seems to be a useful tool to trace the dissimilarities between objects, states, opinions, processes, etc.