Mining Hierarchical Decision Rules from Hybrid Data with

Categorical and Continuous Valued Attributes Miao Duoqian, Qian Jin, Li Wen, Zhang Zehua

Outline

Conclusion

Mining Hierarchical decision rules

Attribute reduction

Similarity-based Rough Set Model

Introduction

Introduction

Rough set theory, proposed by Pawlak, is a useful mathematical framework to deal with imprecise, uncertain information.

Classical attribute reduction methods mainly deal with categorical data.

In practice, there exist continuous-valued (numerical) attributes in real application systems.

Discretization methods These methods are too categorical and may bring

information loss in some cases because the degrees of membership of numerical values to discretized values are not considered.

Existing Methods

Extended rough set model Fuzzy rough set model Tolerance rough set model Neighborhood rough set model Similarity rough set model ……

Similarity rough set model

Decision rule

Attribute reduction

Similarity class

Similarity relation

SimilarityThe similarity class of x, denoted by R(x), is the set

of objects which are similar to x.

( ) { : }R x y U yRx 1( ) { : }R x y U xRy

Notice that the statements yRx, which means “y is similar to x”, is directional. It has a subject y and a referent x.

Symmetry and Transitivity?Symmetry?The most controversial property is symmetry.Although yRx is directional, most authors dealing

with similarity relation do impose this property.

Transitivity? Imposing transitivity to R is even more

questionable.The reason for this is that, sometimes, a series of

negligible differences cannot be propagated.

Similarity Measure

max min

| ( ) ( ) |( , ) 1

| |a

a x a ySIM x y

a a

1,( , ) .

0,a

if x ySIM x y

if x y

For numerical attributes

For categorical attributes

Similarity

( , ) , ( , )P a ax y R a P SIM x y

( , )( , )

| |a P a

P

SIM x yx y R

P

Local similarity

Global similarity

( , ) ( , )P aa P

x y R SIM x y

If a global similarity measure threshold equals 1, the similarity-based rough set model degenerates into classical rough set model.

Researchers pointed out empirically that in some contexts, similarity does not necessarily have features like symmetry or subadditivity implied by distance measures.

New Similarity Distance Measure

( , ) min{ ( , ) | }P ax y R SIM x y a P

Similarity distance measure?This inherent weakness of the distance-based

similarity measure comes from a lack of consideration of the contribution of the similarity direction when comparing the similarity of two objects.

Similarity direction measure

( , )ia y x�������������� ( ) ( )

max( ) min( )i i

i i

a y a x

a a

=

……na1a 2a 1na

……1a 2a 1na na

Fig1 Same direction

……na

1a2a

1na

…… na

1a2a

1na

Fig2 Different direction

Similarity direction measure

Definition 9. Given two objects x and y, the similarity direction measure of both objects is defined as

( , )D y x1

1( , )

m

ii

a y xm

��������������=

If D (y, x) >=0, the object y is similar to x; otherwise y is dissimilar to x.

However, if we employ such similarity direction measure, similarity relation is not symmetric in most cases, even if the similarity direction differences between two objects are very small.

Furthermore, each similarity direction measure may not possess subadditivity.

Definition 10. Given two objects x and y, the similarity direction measure of both objects is defined as

( , )D y x max{ ( , ) | }i ia y x a P��������������

min{ ( , ) | }i ia y x a P��������������

= .

If D (y, x)>= , the object y is similar to x; otherwise y is dissimilar to x.

In general, the same similarity direction is good. Here we give a constraint parameter to extend similarity.

Similarity relation

Construction of a rational, reliable and practical similarity measure is a fundamental and substantial research topic in the field of decision making, otherwise the accuracy and validity of a similarity measure could be challenged.

Attribute reductionx U

( , , )DT P ( , , )IDT P All consistent objects set and inconsistent objects set are denoted by and

Definition 11. Let DT be a decision table, and , we will say that x is a consistent object under similarity measure parameters and if for all y; otherwise x is an inconsistent object.

,( , ) ( , ) ( ) ( )Px y R D x y d x d y

P A

Attribute reduction

x U

Definition 12. Let DT be a decision table, and , we will say that x and y are dissimilar under similarity measure parameters and if .

x U

x U P A

,( , ) ( , )Px y R D x y

Definition 13 Let DT be a decision table, and , the discernibility matrix = is defined as

P A, ,PM , ,{ ( , )}pm x y

,

, , ,

{ | ( , ) } { | ( , ) )} , ( , , ) ( ) ( )

( , ) { | ( , ) } { | ( , ) )} ( , , ) ( , , )a

P a

a x y R P D x y x y DT P d x d y

m x y a x y R P D x y x DT P y IDT P

Mining Hierarchical decision rules

Example

Company Asset profit type of product credit

1 105 67 computer software bad

2 54 75 automobile good

3 80 93 automobile bad

4 64 80 automobile good

5 92 92 computer hardware good

6 96 102 computer hardware good

7 111 65 computer software bad

8 58 70 automobile good

9 74 77 automobile bad

10 105 105 computer hardware good

11 85 82 automobile bad

Decision rules

2

11

4 9

6

7

8

5

1

10

3

Fig 3. A similarity relation graph with =0.75 and =-0.01

2

11

4 9

6

7

8

5

1

10

3

?

Without considering similarity direction parameter, we can not discern object 4 and object 9 under =0.75. In such case, we will generate some inconsistent decision rules.

Fig 4. A similarity relation graph with =0.75

Choosing a level in concept hierarchy, we can mine hierarchical decision rules.

Conclusion

This paper mainly discusses similarity distance measure and similarity direction measure, and proposes an algorithm for mining hierarchical decision rules .

Future work Both theoretical and experimental comparison of

mining hierarchical decision rules.