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Johanna GOLD

Rough Sets Theory Logical Analysis of Data. Monday , November 26, 2007. Johanna GOLD. Introduction. Comparison of two theories for rules induction. Different methodologies Same results?. Generalities. Set of objects described by attributes. Each object belongs to a class.

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Johanna GOLD

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  1. Rough Sets Theory Logical Analysis of Data. Monday, November 26, 2007 Johanna GOLD

  2. Introduction • Comparison of two theories for rules induction. • Different methodologies • Same results?

  3. Generalities • Set of objects described by attributes. • Each object belongs to a class. • We want decision rules.

  4. Approaches • There are two approaches: • Rough Sets Theory (RST) • Logical Analysis of Data (LAD) • Goal : compare them

  5. Contents Rough Sets Theory Logical Analysis Of data Comparison Inconsistencies

  6. Inconsistencies • Two examples having the exact same values in all attributes, but belonging to two different classes. • Example: two sick people have the same symptomas but different disease.

  7. Covered by RST • RST doesn’t correct or aggregate inconsistencies. • For each class : determination of lower and upper approximations.

  8. Approximations • Lower : objects we are sure they belong to the class. • Upper : objects than can belong to the class.

  9. Impact on rules • Lower approximation → certain rules • Upper approximation → possible rules

  10. Pretreatment • Rules induction on numerical data → poor rules → too many rules. • Need of pretreatment.

  11. Discretization • Goal : convert numerical data into discrete data. • Principle : determination of cut points in order to divide domains into successive intervals.

  12. Algorithms • First algorithm: LEM2 • Improved algorithms: • Include the pretreatment • MLEM2, MODLEM, …

  13. LEM2 • Induction of certain rules from the lower approximation. • Induction of possible rules from the upper approximation. • Same procedure

  14. Definitions (1) • For an attribute x and its value v, a block [(x,v)] of attribute-value pair (x,v) is all the cases where the attribute x has the value v. • Ex : [(Age,21)]=[Martha] [(Age,22)]=[David ; Audrey]

  15. Definitions (2) • Let B be a non-empty lower or upper approximation of a concept represented by a decision-value pair (d,w). • Ex : (level,middle)→B=[obj1 ; obj5 ; obj7]

  16. Definitions (3) • Let T be a set of pairs attribute-value (a,v). • Set B depends on set T if and only if:

  17. Definitions (4) • A set T is minimal complex of B if and only if B depends on T and there is no subset T’ of T such as B depends on T’.

  18. Definitions (5) • Let T be a non-empty collection of non-empty set of attribute-value pairs. • T is a set of T. • T is a set of (a,v).

  19. Definitions (6) • T is a local cover of B if and only if: • Each member T of T is a minimal complex of B. • T is minimal

  20. Algorithm principle • LEM2’s output is a local cover for each approximation of the decision table concept. • It then convert them into decision rules.

  21. Algorithm

  22. Heuristics details Among the possible blocks, we choose the one: • With the highest priority • With the highest intersection • With the smallest cardinal

  23. Heuristics details • As long as it is not a minimal complex, pairs are added. • As long as there is not a local cover, minimal complexes are added.

  24. Illustration • Illustration through an example. • We consider that the pretreatment has already been done.

  25. Data set

  26. Cut points • For the attribute Height, we have the values 160, 170 and 180. • The pretreatment gives us two cut points: 165 and 175.

  27. Blocks [(a,v)] • [(Height, 160..165)]={1,3,5} • [(Height, 165..180)]={2,4} • [(Height, 160..175)]={1,2,3,5} • [(Height, 175..180)]={4} • [(Hair, Blond)]={1,2} • [(Hair, Red)]={3} • [(Hair, Black)]={4,5,6}

  28. First concept • G = B = [(Attraction,-)] = {1,4,5,6} • Here there is no inconsistencies. If there were some, it’s at this point that we have to chose between the lower and the upper approximation.

  29. Eligible pairs • Pair (a,v) such as [(a,v)]∩[(Attraction,-)]≠Ø • (Height,160..165) • (Height,165..180) • (Height,160..175) • (Height,175..180) • (Hair,Blond) • (Hair,Black)

  30. Choice of a pair • We chose the most appropriate, which is to say (a,v) for which | [(a,v)] ∩ [(Attraction,-)] | is the highest. • Here : (Hair, Black)

  31. Minimal complex • The pair (Hair, Black) is a minimal complex because:

  32. New concept • B = [(Attraction,-)] – [(Hair,Black)] = {1,4,5,6} - {4,5,6} = {1}

  33. Choice of a pair (1) • Through the pairs (Height,160..165), (Height,160..175) and (Hair, Blond). • Intersections having the same cardinality, we chose the pair having the smallest cardinal: (Hair, Blond)

  34. Choice of a pair (2) • Problem : • (Hair, Blond) is non a minimal complex. • We chose the following pair: (Height,160..165).

  35. Minimal Complex • {(Hair, Blond),(Height,160..165)} is a second minimal complex.

  36. End of the concept • {{(Hair, Black)}, {(Hair, Blond), (Height, 160..165)}} is a local cover of [(Attraction,-)].

  37. Rules • (Hair, Red) → (Attraction,+) • (Hair, Blond) & (Height,165..180 ) → (Attraction,+) • (Hair, Black) → (Attraction,-) • (Hair, Blond) & (Height,160..165 ) → (Attraction,-)

  38. Contents Rough Sets Theory Logical Analysis Of data Comparison Inconsistencies

  39. Principle • Work on binary data. • Extension of boolean approach on non-binary case.

  40. Definitions (1) • Let S be the set of all observations. • Each observation is described by n attributes. • Each observation belongs to a class.

  41. Definitions (2) • The classification can be considered as a partition into two sets • An archive is represented by a boolean function Φ :

  42. Definitions (3) • A literal is a boolean variable or its negation: • A term is a conjunction of literals : • The degree of a term is the number of literals.

  43. Definitions (4) • A term Tcovers a point if T(p)=1. • A characteristic term of a point p is the unique term of degree n covering p. • Ex :

  44. Definitions (5) • A term T is an implicant of a boolean function f if T(p) ≤ f(p) for all • An implicant is called prime if it is minimal (its degree).

  45. Definitions (6) • A positive prime patternis a term covering at least one positive example and no negative example. • A negative prime patternis a term covering at least one negative example and no positive example.

  46. Example

  47. Example • is a positive pattern : • There is no negative example such as • There is one positive example : the 3rd line. • It's a positive prime pattern : • covers one negative example : 4th line. • covers one negative example : 5th line.

  48. Pattern generation • symmetry between positive and negative patterns. • Two approaches : • Top-down • Bottom-up

  49. Top-down • we associate each positive example to its characteristic term→ it’s a pattern. • we take out the literals one by one until having a prime pattern.

  50. Bottom-up • we begin with terms of degree one: • if it does not cover a negative example, it is a pattern • If not, we add literals until having a pattern.

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