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).

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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

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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

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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

Loss function

Thresholdvalues

Probabilisticrough setapproximations

Baysian decisiontheory

Decision-theoreticmodel

Variable-precisionmodel

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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

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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.

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?