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