Sequential Three-way Decision with Probabilistic Rough Sets Supervisor: Dr. Yiyu Yao Speaker: Xiaofei Deng 18th Aug, 2011
Dec 31, 2015
Sequential Three-way Decision with Probabilistic Rough Sets
Supervisor: Dr. Yiyu YaoSpeaker: Xiaofei Deng 18th Aug, 2011
Outline
Motivation The main idea Basic concepts and notations Multiple representations of objects in an
information table Three-way decision with a set of attributes Computation of thresholds Sequential three-way decision-making
with a sequence of attributes
Motivation
The three-way decision One single step decision (current) Minimal cost of correct, incorrect
classifications (accuracy, misclassification
errors) Considering the cost of obtaining an
evidence Decision making: supporting evidence An observation -> a piece of evidence
The main idea of sequential three-way decision making
Sequential model should consider the trade-off: Cost Vs. misclassification error
The main idea of the sequential decision making Selecting a sequence of evidence Constructing a multi-level granular structure For sufficient evidence,
Make an acceptance, rejection rules Insufficient evidence: the deferment rules
For deferment rules, Refining with further observation
The main idea (cont.): An example
A task: selecting a set of relevant papers from a set of papers
A granular structure (with increasing evidence)
Title
(Sub)Section headings
Intro, conclusion, paragraphs
Quick decision,Less cost of time
A little bit more cost of reading time
More cost of reading time
Moresupportingevidence
More Info.
More Info.
High level
Low level
Accept, reject
Accept, reject
Accept, reject
Basic concepts
An information table:
An equivalence relation
The equivalence class:
A partition,
:AE U U
( ( ) ( )).A a axE y a A I x I y
[ ] [ ] { | }.AE A Ax x y xE y
/ {[ ] | }.A AU E x x U
( , ,{ | },{ | }).a aS U At V a At I a At
let [ ] [ ] :AE Ax x
Basic concepts (cont.)
A refinement-coarsening relation :
Suppose , we have the monotonic properties:
2 12 1( / ) ( / ).A AU E U E
2 1 1 2 ( ).iff b a a b
1 2A A At
2 1A AE E
2 1[ ] [ ]A Ax x
A short summary
Based on the Information table
For two subsets of attributes: With more details (supporting evidence)
The coarsening-refinement relation Partial ordering between two partitions Construct a granular structure
An information table A set of attributes
An equivalence relation
A partition of the set of objectsThe description of an object
1 2A A At 2A
Multiple representation of objectsConstructing a granular structure
The description of an object (atomic formulas)
A sequence of sets of attributes: (More evidence) (Granules) (Granulations)
A sequence of different descriptions of an object: (Increasing
details) Construct a multi-level granular structure
With above elements For sequential three-way decision
x( ) ( ( )).A a A aDes x a I x
1 2( ), ( ),..., ( ).
kA A ADes x Des x Des x
1 2 ... kA A A At 2 1
[ ] ... [ ] [ ]kA A Ax x x
2 1/ ... / /
kA A AU E U E U E
Three-way decision making with a set of attributesOne single step three-way decision making
is an unknown concept The Conditional probability:
The three probabilistic regions of
| [ ] |Pr( | [ ] ) .
| [ ] |A
AA
C xC x
x
C
C
( , )
( , )
( , )
POS ( ) { | Pr( | [ ] ) },
BND ( ) { | Pr( | [ ] ) },
NEG ( ) { | Pr( | [ ] ) }.
A
A
A
C x U C x
C x U C x
C x U C x
Three-way decision making (Cont.)
Three types of quantitative probabilistic decision rules:
Infer the membership in , based on the description of .
( , )
( , )
( , )
rule of acceptance: [ ] POS ( ),
( ) accept ;
rule of deferment: [ ] BND ( ),
( ) neither accept nor reject ;
rule of rejection: [ ] NEG ( ),
( ) reject ;
A
A
A
A
A
A
x C
Des x x C
x C
Des x x C
x C
Des x x C
xC
Computation of the two thresholds
Computing based on the Bayesian decision theory A decision with the minimal risk
The cost of actions in different states
( , )
States
Action( )C P ( )cC N
Pa
Ba
Na
PP
BP
NP
PN
BN
NN
Computing thresholds (cont.)
The lost function, for
A particular decision with the minimal risk Considering the three regions
An example: the positive rule
( | [ ] ) Pr( | [ ] ) Pr( | [ ] )ci A iP A iN AR a x C x C x
, ,or :i P B N
( , )
If ( | [ ] ) ( | [ ] ) and
( | [ ] ) ( | [ ] )
then decide POS ( );
P A N A
P A B A
R a x R a x
R a x R a x
x C
Computing thresholds (cont.)
The pair of thresholds For
We have:
( ),
( ) ( )
( ).
( ) ( )
PN BN
PN BN BP PP
BN NN
BN NN NP BP
( , ) : 0 1 and
,PP BP NP
NN BN PN
Sequential three-way decision
A sequence of attributes Non-Monotonicity
The new evidence The conditional probability:
Support, is neutral, refutes
2 1A A
2 1
2 1
2 1
Pr( | [ ] ) Pr( | [ ] )
Pr( | [ ] ) Pr( | [ ] )
Pr( | [ ] ) Pr( | [ ] )
A A
A A
A A
C x C x
C x C x
C x C x
C
1 2 ... kA A A At
Sequential three-way decision (cont.)
Trade-off between Revisions and the tolerance of classification errors Refine the deferment rules in the next
lower level Bias: making deferment rules
Higher , lower for a higher level
Conditions of thresholds:
1 2 2 1
0 1,1 , (in the same level)
... ... . (between levels)i i
k k
i k
An sequential algorithm
Step1: One single step three-way
Step i: refines the deferment rules in step (i-1)
1 1
( , ) ( , ) ( , )
, ;
Construct POS ( ), BND ( ), NEG ( );
Rules of acceptance, Rejection, deferment;
U U C C
C C C
1 1
1 1
( , ) 1
( , ) 1
( , ) ( , ) ( , )
(1 ) : Let
BND ( );
BND ( );
Compute POS ( ), BND ( ), NEG ( );
Rules of acceptance, Rejection, deferment;
i i
i i
i i i i i i
i i
i i
i i i
i k
U C
C C C
C C C
(New universe)
(New concept)
Conclusion
Advantages Consider cost of misclassification and
the cost of obtaining an evidence The tolerance of misclassification errors Avoid test or observation to obtain new
evidence at current level Multi-representation of an object: an
important direction in granular computing
Reports the preliminary results
Future work
Future work How to obtaining a sequence of
attributes? How to precisely measure the cost of
obtaining the evidence for a decision? A formal analysis of cost-accuracy
trade-off to further justify the sequential three-way decision making.
Reference
Yao, Y.Y., X.F. Deng, Sequential Three-way Decisions with Probabilistic Rough Sets, 10th IEEE International Conference on Cognitive Informatics and Cognitive Computing, 2011