RoughSetsTheoryIntro_KunHsiang

8/12/2019 RoughSetsTheoryIntro_KunHsiang

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Rough Sets Theory

SpeakerKun Hsiang


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Outline

Introduction

Basic concepts of the rough sets theory

Decision table Main steps of decision table analysis

An illustrative example of application of the

rough set approach Discussion


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Introduction

Often, information on the surrounding

world is

Imprecise

Incomplete

uncertain.

We should be able to process uncertain

and/or incomplete information.


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Introduction

When dealing with inexact, uncertain, or

vague knowledge, are the fuzzy setand

the rough settheories.


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Introduction

Fuzzy set theory

Introduced by Zadeh in 1965 [1]

Has demonstrated its usefulness in chemistry

and in other disciplines [2-9]

Rough set theory

Introduced by Pawlak in 1985 [10,11]

Popular in many other disciplines [12]


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Introduction

Fuzzy Sets


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Introduction

Rough Sets

In the rough set theory, membership is not

the primary concept.

Rough sets represent a different

mathematical approach to vagueness and

uncertainty.


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Introduction

Rough Sets The rough set methodology is based on the premise

that lowering the degree of precision in the data

makes the data pattern more visible.

Consider a simple example. Two acids with pKs of

respectively pK 4.12 and 4.53will, in many contexts,

be perceived as so equally weak, that they are

indiscerniblewith respect to this attribute. They are part of a rough set weak acidsas

compared to strong or medium or whatever other

category, relevant to the context of this classification.


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Basic concepts of the rough sets

theory

Information system

IS=(U,A)

Uis the universe ( a finite set of objects,

U={x1,x2,.., xm}

A is the set of attributes (features, variables)

Va is the set of values a, called the domain of

attribute a.


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Consider a data set containing the results of three

measurements performed for 10 objects. The results

can be organized in a matrix10x3.


Measurements

Objects


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IS=(U,A)

U={x1,x2,x3,x4,x5,x6.., x10}

A={a1,a2,a3} The domains of attributes are

V1={1,2,3}

V2={1,2}V3={1,2,3,4}


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If Xis crispwith respect toB.

If Xis roughwith respect toB.

,1)( XB

,1)( XB


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If the set of attributes is dependent, one can be

interested in finding all possible minimal subsets

of attributes.

The concepts of coreand reductare twofundamental concepts of the rough sets theory.

Keep only those attributes that preserve the

indiscernibility relation and, consequently, set

approximation. There are usually several such subsets of attributes

and those which are minimal are called reducts.


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To compute reducts and core, the

discernibility matrix is used.

The discernibility matrix has the dimension

nxn.

where n denotes the number of

elementary sets and its elements are

defined as the set of all attributes which

discern elementary sets [x]iand [x]j.


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Discerns set 1 from both set 2 and set 3

Discerns set 1 from sets 2,3,4 and set 5


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The discernibility function has the following form:


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Classification


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Decision Table


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rough set approach


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rough set approach


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rough set approach


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rough set approach


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rough set approach

f f


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rough set approach


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Discussion

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