Rough Set Model Selection for Practical Decision Making

1. Rough Set Model Selection forPractical Decision Making Jeseph P. Herbert JingTao Yao Department of Computer Science University of Regina jtyao@cs.uregina.ca

2. Introduction • Rough sets have been applied to many areas in order to aid decision making. • Information (rules) derived from multi-attribute data helps users in making decisions. • Rough set reducts minimize the strain on the user by giving them only the necessary information. J T Yao

3. Motivation • Can we further utilize the strengths provided by rough sets in order to make more informed decisions? • Can we differentiate the types of decisions that can be made from using various rough set methods? • Can we provide some sort of support mechanism to the user to help them choose a suitable rough set method for their analysis? J T Yao

4. Rough Sets • Developed in the early 1980s by Zdzislaw Pawlak. • Sets derived from imperfect, imprecise, and incomplete data may not be able to be precisely defined. • Sets must be approximated • Using describable concepts to approximate known concept • 1.76 cm => 1.7, 1.8 J T Yao

5. Key Concepts • Information systems/tables and decision tables. • Indiscernibility. • Set approximation. • Reducts. J T Yao

6. is the set of value for attribute a. Information Table: An Example • Information table I = (U, A) • U = non-empty finite set of objects • A = non-empty finite set of attributes such that: for all J T Yao

7. Decision Table T = (U, A {d}) Decision Table: An Example J T Yao

8. For any in I = ( ), there exists an equivalence relation: Indiscernibility where is the B-indiscernibility relation. • An equivalence relation partitions U into equivalence classes: J T Yao

9. Data may not precisely define distinct, crisp sets. A rough set has a lower and upper approximation. Set Approximation J T Yao

10. Visualize Rough Sets Let T = (U, A), , • Lower Approximation: • Upper Approximation: • Boundary Region: J T Yao

11. Rough Set Methods for Data Analysis • Two type of models are focused on: • Algebraic Method • Probabilistic • Decision-theoretic Method, • Variable-precision Method • Each method has different strengths that can be used to improve decision making J T Yao

12. Types of Decisions • Broadly, there are two main types of decisions that can be made using rough set analysis. • Immediate decisions (Unambiguous). • Delayed decisions (Ambiguous). • We can further categorize decision types by looking at rough set method strengths. J T Yao

13. Immediate Decisions • These types of decisions are based upon classification with the POS and NEG regions. • The user can interpret findings as: • Classification into POS regions can be considered a “yes” answer. • Classification into NEG regions can be considered a “no” answer J T Yao

14. Delayed Decisions • These types of decisions are based on classification in the BND region. • A “wait-and-see” approach to decision making. • A decision-maker can decrease ambiguity with the following: • Obtain more information (more data). • A decreased tolerance for acceptable loss (decision-theoretic) or user thresholds (variable-precision). J T Yao

15. Algebraic Decisions • Decisions made from algebraic rough set analysis. • Immediate • If P(A|[x]) = 1, then x is in POS(A). • If P(A|[x]) = 0, then x is in NEG(A). • Delayed • If 0 < P(A|[x]) < 1, then x is in BND(A). J T Yao

16. Variable-Precision Decisions • Decisions made from variable-precision rough set analysis. • User-defined thresholds u and l representing lower and upper bounds to define regions. • Pure Immediate decisions. • User-Accepted Immediate decisions. • User-Rejected Immediate decisions. • Delayed decisions. J T Yao

17. Variable-Precision Decisions • Pure Immediate • If P(A|[x]) = 1, then x is in POS1(A). • If P(A|[x]) = 0, then x is in NEG0(A). • User-Accepted Immediate • If u≤ P(A|[x]) < 1, then x is in POSu(A). • User-Rejected Immediate • If 0 < P(A|[x]) ≤l, then x is in NEGl(A). • Delayed • If l < P(A|[x]) < u, then x is in BNDl,u(A). J T Yao

18. Decision-Theoretic Decisions • Decisions made from decision-theoretic rough set analysis. • Calculated cost (risk) using Bayesian decision procedure provides minimum α, β values for region division. • Pure Immediate decisions. • Accepted Loss Immediate decisions. • Rejected Loss Immediate decisions. • Delayed decisions. J T Yao

19. Decision-Theoretic Decisions • Pure Immediate • If P(A|[x]) = 1, then x is in POS1(A). • If P(A|[x]) = 0, then x is in NEG0(A). • Accepted Loss Immediate • If α≤ P(A|[x]) < 1, then x is in POSα(A). • User-Rejected Immediate • If 0 < P(A|[x]) ≤β, then x is in NEGβ(A). • Delayed • If β < P(A|[x]) < α, then x is in BNDα, β(A). J T Yao

20. A Simple Example: Parking a Car • Set of states: • : meeting will be over in less than 2 hours, • : meeting will be over in more than 2 hours. • Set of actions: • : park the car on meter • : park the car on parking lot J T Yao

21. Costs of Parking Your Car J T Yao

22. Making Decision Based on Probabilities • Assume that: • Cost of each action: • Take action : park the car on meter J T Yao

23. Determine the Probability Threshold for One Action • The condition of taking action : park the car on meter: J T Yao

24. Relationships Amongst Rough Set Models Decision-theoretic model Variable-precision model Probabilistic rough set approximations Loss function Threshold values Baysian decision theory J T Yao

25. Summary of Decisions Pawlak Method Variable-Precision Method Decision-Theoretic Method J T Yao

26. Choosing a Method • If the user is informed enough to provide thresholds, variable-precision rough sets can be used for data analysis. • If cost or risk information is beneficial to the types of decisions being made, decision-theoretic rough sets can be used for data analysis. J T Yao

27. Conclusions • We can utilize the strengths of various rough set methods in order to improve our decision making capability. • The various rough set methods can each make different types of decisions. • By determining what kind of decisions they wish to make, users can choose a suitable rough set method for data analysis to reach their goals. J T Yao

28. Rough Set Model Selection forPractical Decision Making Jeseph P. Herbert JingTao Yao Department of Computer Science University of Regina jtyao@cs.uregina.ca

29. Where is Regina? J T Yao