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Johanna GOLD

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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.
- We want decision rules.

Approaches

- There are two approaches:
- Rough Sets Theory (RST)
- Logical Analysis of Data (LAD)

- Goal : compare them

Contents

Rough Sets Theory

Logical Analysis Of data

Comparison

Inconsistencies

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.

Covered by RST

- RST doesn’t correct or aggregate inconsistencies.
- For each class : determination of lower and upper approximations.

Approximations

- Lower : objects we are sure they belong to the class.
- Upper : objects than can belong to the class.

Impact on rules

- Lower approximation → certain rules
- Upper approximation → possible rules

Pretreatment

- Rules induction on numerical data → poor rules → too many rules.
- Need of pretreatment.

Discretization

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

Algorithms

- First algorithm: LEM2
- Improved algorithms:
- Include the pretreatment
- MLEM2, MODLEM, …

LEM2

- Induction of certain rules from the lower approximation.
- Induction of possible rules from the upper approximation.
- Same procedure

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]

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]

Definitions (3)

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

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’.

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).

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

Algorithm

principle

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

Algorithm

Heuristics details

Among the possible blocks, we choose the one:

- With the highest priority
- With the highest intersection
- With the smallest cardinal

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.

Illustration

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

Data set

Cut points

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

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}

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.

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)

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)

Minimal complex

- The pair (Hair, Black) is a minimal complex because:

New concept

- B = [(Attraction,-)] – [(Hair,Black)]
= {1,4,5,6} - {4,5,6}

= {1}

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)

Choice of a pair (2)

- Problem :
- (Hair, Blond) is non a minimal complex.
- We chose the following pair:
(Height,160..165).

Minimal Complex

- {(Hair, Blond),(Height,160..165)} is a second minimal complex.

End of the concept

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

Rules

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

Contents

Rough Sets Theory

Logical Analysis Of data

Comparison

Inconsistencies

Principle

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

Definitions (1)

- Let S be the set of all observations.
- Each observation is described by n attributes.
- Each observation belongs to a class.

Definitions (2)

- The classification can be considered as a partition into two sets
- An archiveis represented by a boolean function Φ :

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.

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 :

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).

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.

Example

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.

Pattern generation

- symmetry between positive and negative patterns.
- Two approaches :
- Top-down
- Bottom-up

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.

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.

Objectives

- We prefer short pattern → simplicity principle.
- we also want to cover the maximum of examples with only one model → globality principle.
- hybrid approach bottom-up – top-down.

Hybrid approach

- We fix a degree D.
- We start by a bottom-up approach to generate the models of degree lower or equal to D.
- For all the points which are not covered by the 1st phase, we proceed to the top-down approach.

Extension to the non binary case

- Extension from binary case : binerization.
- Two types of data :
- quantitative : age, height, …
- qualitative : color, shape, …

Qualitative data

- For each value v that a qualitative attribute x can be, we associate a boolean variable b(x,v) :
- b(x,v) = 1 if x = v
- b(x,v) = 0 otherwise

Quantitative data

- there are two types of associated variables:
- Level variables
- Interval variables

Level variables

- For each attribute x and each cut point t, we introduce a boolean variable b(x,t) :
- b(x,t) = 1 if x ≥ t
- b(x,t) = 0 if x < t

Intervals variables

- For each attribute x and each pair of cut points t’, t’’ (t’<t’’), we introduce a boolean variable b(x,t’,t’’) :
- b(x,t’,t’’) = 1 if t’ ≤ x < t’’
- b(x,t’,t’’) = 0 otherwise

Example

Example

Example

Example

Example

Example

Example

Example

Supporting set

- A set of binary attributes is called supporting set if the archive obtained by the elimination of all the other attributes will remained "contradiction-free".
- A supporting set is irredundant if there is no subset of it which is a supporting set.

Variables

- We associate to the attributea variable
such as if the attribute belongs to the supporting set.

- Application : elements a and e are different on attributes 1, 2, 4, 6, 9, 11, 12 and 13 :

Linear program

- We do the same for all pairs of true and false observations :
- Exponential number of solutions : we choose the smallest set :

Solution of

our example

- Positive patterns :
- Negative patterns :

Contents

Rough Sets Theory

Logical Analysis Of data

Comparison

Inconsistencies

Basic idea

- LAD more flexible than RST
- Linear program -> modification of parameters

Comparisonblocks / variables

- RST : couples (attribute, value)
- LAD : binary variables
- Correspondence?

Qualitative data

- For an attribute a taking the values:

Quantitative data

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

Quantitative data

- RST : for each cut point, we have two blocks :

Quantitative data

- LAD : for each cut point, we have a level variable :
- ...

Quantitative data

- LAD : for each pair of cut points, we have a interval variable :
- ...

Quantitative data

- Correspondence :
- Level variable :

Quantitative data

- Correspondence :
- Interval variable :

Variation of LP parameters

- Three parameters can change :
- Right hand side of constraints:
- coefficients of the objective function:
- coefficients of the left hand side of the constraints:

Heuristics adaptation

- We try to adapt the three heuristics :
- The highest priority
- The highest intersection with the concept
- The smallest cardinality

The highest priority

- Priority on blocks -> priority on attributes
- Introduction as weights in the objective function
- Minimization : choice of pairs with first priorities

The highest intersection

- Pb : in LAD, no notion of concept ; everything is done symmetrically, the same time.

The highest intersection

- Modification of the heuristic : difference between the intersection with a concept and the intersection with the other.
- The highest, the better.

The highest intersection

- Goal of RST : find minimal complexes:
- Find blocks covering the most examples of the concept : highest possible intersection with the concept
- Find blocks covering the less examples of the other concept : difference of intersections

The highest intersection

- For LAD : difference between the number of times a variable takes the value 1 in
and in .

- Introduction as weights in the constraints : we choose first the variable with the highest difference.

The smallest cardinality

- Simple : number of times a variable takes the value 1.
- Introduction as weight in the constraints.

Weight of the constraints

- Two calculations to be introduced :
- The highest difference
- The smallest cardinality

- Difference of the two calculations

Right hand side of the constraints

- Before : everything is 1.
- Pb : modification of the weights of the left hand side has no signification.

Ideas of modification

- Average of compared to the number of attributes.
- Average of in each constraint
- Inconvenient : not a real signification

Ideas of modification

- Not touch the weight in the constraints: introduce everything in the coefficients of the objective function:

Contents

Rough Sets Theory

Logical Analysis Of data

Comparison

Inconsistencies

For RST

- Use of two approximations : lower and upper.
- Rules generation: sure and possible.

For LAD

- Classification mistakes: positive point classified as negative or the other way.
- Two different cases.

Pos. Point

classified as neg.

- All other points are well classify : our point will not be covered.
- If the number of non covered points is high: generation of longer patterns.
- If this number is small : erroneous classification and we forgot the points for the following.

Neg. Point

classified as pos.

- Terms covering a lot of positive points : also some negative points.
- Probably wrongly classified : not taken into account for the evaluation of candidates terms.

Ratio

- We introduce a ratio.
- A term is still candidate if the ratio between negative and positive points is smallest than:

Inconsistenciesand mistakes

- An inconsistence can be considered as a mistake of classification
- Inconsistence : two « identical » objects differently classified.
- One of them is wrongly classified (approximations)

Equivalence?

- Let consider an inconsistence in LAD :
- two points :
- two classes :

- There are two possibilities :
- is not covered by small degree patterns
- is covered by patterns of

1st case

- We have only one inconsistence.
- The covered point is isolated ; it’s not taken into account.
- Patterns of will be generated without the inconsistence point
-> lower approximation

2nd case

- A point covered by the other concept patterns is wrongly classified.
- It’s not taken into account for the candidate terms.
- It’s not taken into account for the pattern generation of
-> lower approximation

2nd case

- Not taken into account for but not a problem for
- For : upper approximation

Equivalence?

- According to a ratio, LAD decide if a point is well classified or not.
- For an inconsistence, it’s the same as consider:
- The upper approximation of a class
- The lower approximation of the other

- On more than 1 inconsistence : we re-classify the points.

Conclusion

- Complete data : we can try to match LAD and RST.
- Inconsistencies : classification mistakes of LAD can correspond to approximations.
- Missing data : different management

Sources (1)

- Jerzy W. Grzymala-Busse, MLEM2 - Discretization During Rule Induction, Proceedings of the IIPWM'2003, International Conference on Intelligent Information Processing and WEB Mining Systems, Zakopane, Poland, June 2-5, 2003, 499-508. Springer-Verlag.
- Jerzy W. Grzymala-Busse, Jerzy Stefanowski, Three Discretization Methods for Rule Induction, International Journal of Intelligent Systems, 2001.
- Endre Boros, Peter L. Hammer, Toshihide Ibaraki, Alexander Kogan, Eddy Mayoraz, Ilya Muchnik, An Implementation of Logical Analysis of Data, Rutcor Research Raport 22-96, 1996.

Sources (2)

- Endre Boros, Peter L. Hammer, Toshihide Ibaraki, Alexander Kogan, Logical Analysis of Numerical Data, Rutcor Research Raport 04-97, 1997.
- Jerzy W. Grzymala-Busse, Rough Set Strategies to Data with Missing Attribute Values,Proceedings of theWorkshop on Foundation and New Directions in Data Mining, Melbourne, FL, USA. 2003.
- Jerzy W. Grzymala-Busse, Sachin Siddhaye, Rough Set Approaches to Rule Induction from Incomplete Data, Proceedings of the IPMU'2004, the 10th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based System[C],Perugia,Italy, July 4, 2004 2 : 923- 930.

Sources (3)

- Jerzy Stefanowski, Daniel Vanderpooten, Induction of Decision Rules in Classi_cation and Discovery-Oriented Perspectives, International Journal of Intelligent Systems, 16 (1), 2001, 13-28.
- Jerzy Stefanowski, The Rough Set based Rule Induction Technique for Classification Problems, Proceedings of 6th European Conference on Intelligent Techniques and Soft Computing EUFIT 98, Aachen 7-10 Sept., (1998) 109.113.
- Roman Slowinski, Jerzy Stefanowski, Salvatore Greco, Benedetto Matarazzo, Rough Sets Processing of Inconsistent Information in Decision Analysis, Control and Cybernetics 29, 379±404, 2000.