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| |aa x a ySIM x ya 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
PSIM x y
x y RP
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 xa a
=
……na1a 2a 1na
……1a 2a 1na na
Fig1 Same direction
……na
1a2a
1na
…… na1a
2a1na
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 good3 80 93 automobile bad4 64 80 automobile good5 92 92 computer hardware good6 96 102 computer hardware good7 111 65 computer software bad8 58 70 automobile good9 74 77 automobile bad10 105 105 computer hardware good11 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.