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Rough Set Model Selection for Practical Decision Making
Rough Set Model Selection forPractical Decision Making
Jeseph P. Herbert JingTao YaoDepartment of Computer Science
University of Regina
jtyao@cs.uregina.ca
J T Yao 2Rough Set Model Selection for Practical Decision Making
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 3Rough Set Model Selection for Practical Decision Making
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 4Rough Set Model Selection for Practical Decision Making
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 5Rough Set Model Selection for Practical Decision Making
Key Concepts
• Information systems/tables and decision tables.
• Indiscernibility.
• Set approximation.
• Reducts.
J T Yao 6Rough Set Model Selection for Practical Decision Making
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
aVUa :
Aa
is the set of value for attribute a.
aV
J T Yao 7Rough Set Model Selection for Practical Decision Making
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.
J T Yao 8Rough Set Model Selection for Practical Decision Making
Indiscernibility
• For any in I = ( ), there exists an equivalence relation:
xaxaBaUxxBINDI ,, 2
AB AU ,
where is the B-indiscernibility relation.
BINDI
• An equivalence relation partitions U into equivalence classes: Rx
J T Yao 9Rough Set Model Selection for Practical Decision Making
Set Approximation
• Data may not precisely define distinct, crisp sets.
• A rough set has a lower and upper approximation.
J T Yao 10Rough Set Model Selection for Practical Decision Making
Visualize Rough Sets
• Lower Approximation:
Let T = (U, A), , UX AB
AxxAapr B )(
• Upper Approximation:
AxxAapr B)(
• Boundary Region:
)()( AaprAaprABNDB
J T Yao 11Rough Set Model Selection for Practical Decision Making
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 12Rough Set Model Selection for Practical Decision Making
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 13Rough Set Model Selection for Practical Decision Making
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 14Rough Set Model Selection for Practical Decision Making
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 15Rough Set Model Selection for Practical Decision Making
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 16Rough Set Model Selection for Practical Decision Making
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 17Rough Set Model Selection for Practical Decision Making
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 18Rough Set Model Selection for Practical Decision Making
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 19Rough Set Model Selection for Practical Decision Making
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 20Rough Set Model Selection for Practical Decision Making
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 21Rough Set Model Selection for Practical Decision Making
Costs of Parking Your Car
(meter) (lot)
(<= 2) $2.00 $7.00
(> 2) $12.00 $7.00
J T Yao 22Rough Set Model Selection for Practical Decision Making
Making Decision Based on Probabilities
• Assume that:
• Cost of each action:
• Take action : park the car on meter
J T Yao 23Rough Set Model Selection for Practical Decision Making
Determine the Probability Threshold for One Action
• The condition of taking action : park the car on meter:
J T Yao 24Rough Set Model Selection for Practical Decision Making
Relationships Amongst Rough Set Models
Loss function
Thresholdvalues
Probabilisticrough setapproximations
Baysian decisiontheory
Decision-theoreticmodel
Variable-precisionmodel
J T Yao 25Rough Set Model Selection for Practical Decision Making
Summary of Decisions
Region Decision Type
POS(A) Immediate
BND(A) Delayed
NEG(A) Immediate
Region Decision Type
POS1 (A) Pure Immediate
POSu (A) User-accepted Immediate
BNDl,u (A) Delayed
NEGl (A) User-rejected Immediate
NEG0 (A) Pure ImmediateRegion Decision Type
POS1 (A) Pure Immediate
POSα (A) Accepted Loss Immediate
BNDα, β (A) Delayed
NEGβ (A) Rejected Loss Immediate
NEG0 (A) Pure Immediate
Pawlak Method
Variable-Precision Method
Decision-Theoretic Method
J T Yao 26Rough Set Model Selection for Practical Decision Making
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.
Rough Set Model Selection for Practical Decision Making
Rough Set Model Selection forPractical Decision Making
Jeseph P. Herbert JingTao YaoDepartment of Computer Science
University of Regina
jtyao@cs.uregina.ca
J T Yao 29Rough Set Model Selection for Practical Decision Making
Where is Regina?