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 objects Constructing 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 attributes One 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 States Action

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