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_at_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.

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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?

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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 gt 1.7, 1.8

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Key Concepts

• Information systems/tables and decision tables.
• Indiscernibility.
• Set approximation.
• Reducts.

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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

Object Date High Close
1 1-Jul-91 1434.98 1421.54
2 2-Jul-91 1473.99 1473.99
3 3-Jul-91 1473.99 1467.78
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Decision Table An Example

• Decision Table
• T (U, A d)

Object Date High Close Decision
1 1-Jul-91 1434.98 1421.54 1
2 2-Jul-91 1473.99 1473.99 0
3 3-Jul-91 1473.99 1467.78 -1
U non-empty finite set of objects. A non-empty finite set of conditional attributes.
d one or more decision attributes.
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Indiscernibility

• For any in I ( ), there
  exists an equivalence relation

where is the B-indiscernibility
relation.

• An equivalence relation partitions U into
  equivalence classes

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Set Approximation

• Data may not precisely define distinct, crisp
  sets.
• A rough set has a lower and upper approximation.

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Visualize Rough Sets
Let T (U, A), ,

• Lower Approximation

• Upper Approximation

• Boundary Region

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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

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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.

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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

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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).

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Algebraic Decisions

• Decisions made from algebraic rough set analysis.
• Immediate
• If P(Ax) 1, then x is in POS(A).
• If P(Ax) 0, then x is in NEG(A).
• Delayed
• If 0 lt P(Ax) lt 1, then x is in BND(A).

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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.

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Variable-Precision Decisions

• Pure Immediate
• If P(Ax) 1, then x is in POS1 (A).
• If P(Ax) 0, then x is in NEG0 (A).
• User-Accepted Immediate
• If u P(Ax) lt 1, then x is in POSu (A).
• User-Rejected Immediate
• If 0 lt P(Ax) l, then x is in NEGl (A).
• Delayed
• If l lt P(Ax) lt u, then x is in BNDl,u (A).

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Decision-Theoretic Decisions

• Decisions made from decision-theoretic rough set
  analysis.
• Calculated cost (risk) using Bayesian decision
  procedure provides minimum a, ß values for region
  division.
• Pure Immediate decisions.
• Accepted Loss Immediate decisions.
• Rejected Loss Immediate decisions.
• Delayed decisions.

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Decision-Theoretic Decisions

• Pure Immediate
• If P(Ax) 1, then x is in POS1 (A).
• If P(Ax) 0, then x is in NEG0 (A).
• Accepted Loss Immediate
• If a P(Ax) lt 1, then x is in POSa (A).
• User-Rejected Immediate
• If 0 lt P(Ax) ß, then x is in NEGß (A).
• Delayed
• If ß lt P(Ax) lt a, then x is in BNDa, ß (A).

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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

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Costs of Parking Your Car
(meter) (lot)
(lt 2) 2.00 7.00
(gt 2) 12.00 7.00
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Making Decision Based on Probabilities

• Assume that
• Cost of each action
• Take action park the car on meter

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Determine the Probability Threshold for One Action

• The condition of taking action park the car
  on meter

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Relationships Amongst Rough Set Models
Decision-theoretic model
Variable-precision model
Probabilistic rough set approximations
Loss function
Threshold values
Baysian decision theory
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Summary of Decisions
Region Decision Type
POS1 (A) Pure Immediate
POSu (A) User-accepted Immediate
BNDl,u (A) Delayed
NEGl (A) User-rejected Immediate
NEG0 (A) Pure Immediate
Region Decision Type
POS(A) Immediate
BND(A) Delayed
NEG(A) Immediate
Pawlak Method
Region Decision Type
POS1 (A) Pure Immediate
POSa (A) Accepted Loss Immediate
BNDa, ß (A) Delayed
NEGß (A) Rejected Loss Immediate
NEG0 (A) Pure Immediate
Variable-Precision Method
Decision-Theoretic Method
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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.

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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.

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Rough Set Model Selection forPractical Decision
Making

• Jeseph P. Herbert JingTao Yao
• Department of Computer Science
• University of Regina
• jtyao_at_cs.uregina.ca

29
Where is Regina?