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Sequential Three-way Decision with Probabilistic Rough Sets

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Sequential Three-way Decision with Probabilistic Rough Sets

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### Sequential Three-way Decision with Probabilistic Rough Sets

Supervisor: Dr. Yiyu Yao

Speaker: 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)

Basic concepts

- An information table:
- An equivalence relation
- The equivalence class:
- A partition,

Basic concepts (cont.)

- A refinement-coarsening relation :
- Suppose , we have the monotonic properties:

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

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

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

Three-way decision making (Cont.)

- Three types of quantitative probabilistic decision rules:
- Infer the membership in , based on the description of .

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

Computing thresholds (cont.)

- The lost function, for
- A particular decision with the minimal risk
- Considering the three regions
- An example: the positive rule

Computing thresholds (cont.)

- The pair of thresholds
- For
- We have:

Sequential three-way decision

- A sequence of attributes
- Non-Monotonicity
- The new evidence
- The conditional probability:
- Support, is neutral, refutes

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:

An sequential algorithm

- Step1: One single step three-way
- Step i: refines the deferment rules in step (i-1)

(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

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