Imprecise Categories, Approximation, and Rough Sets

Review

X U X is a concept / category in U

A family of concepts in U will be referred to as abstract knowledge about U

U/ R The family of all equivalence classes of R( or classification of U) referred to as categories of R

Equivalence relations

Classification

[x]R A category in R containing an element x U

Contents

IntroductionRough SetsLower and Upper ApproximationsProperties of ApproximationsApproximations and Membership RelationNumerical Characterization of ImprecisionTopological Characterization of ImprecisionApproximation of ClassificationsRough Equality of Sets

Introduction

Definitions

Sets

a.a. Classical setsClassical sets – either an element belongs to the set or it does not. For example, for the set of integers, either an integer is even or it is not (it is odd).

Examples

Classical sets are also called crisp (sets). Lists: A = {apples, oranges, cherries, mangoes}

Formulas: A = {x | x is an even natural number}

Membership or characteristic function

Ax

Axx

A if 0

if 1)(

b.b. Fuzzy setsFuzzy sets – admits gradation such as all tones between black and white.

Described by a membership function.Example

µA: U −→ [0, 1].

C. C. Rough Sets:Rough Sets: A rough set, R(A), is a given representation of a classical (crisp) set A by two subsets of X/R, and

that approach A as closely as possible from the inside and outside (respectively) and

where and are called the lower and upper approximation of A.

)(AR

)(AR

)( ),()( ARARAR

)(AR

)(AR

Theories of fuzzy sets and rough sets are generalizationsgeneralizations of classical set theory for modeling vagueness and uncertainty

As We had seen before that:

CategoriesCategories are features (i.e. subsets) of objects which can be worded using knowledge available in a given knowledge base.

Some categories are definable in one knowledge base but undefinable in another one.

If category is not definable in a given knowledge base, the question arises whether it can be defined “approximately” in the knowledge base (the vague categories).

The rough set is a useful notion for the classification of objects when the available information is not adequate to represent classes using precise sets.

We will use rough set notion here forhandling the vagueness of knowledge

1.Rough Sets

R-definable setsR-definable sets are those subsets of the universe which can be exactly defined in the knowledge base K, whereas the R-R-undefinable setsundefinable sets are subsets which can not be defined in the knowledge base K.

R-definable sets also called R-exact sets.

R-undefinable sets also called R-inexact or R-rough.

2. Lower and Upper Approximations

• Because the available knowledge is not enough for us to specify categories for some objects, we will use two exact sets for approximation of one set.

• The two approximations are:– Upper Approximation:

– Lower Approximation:

}:/{ XYRUYXR

}:/{ XYRUYXR

Take a closer look!

U

The universe of discourse is the finite set of all objects under consideration.

The attribute (equivalence relation) R1 divides the universe of discourse into a set of equivalence classes (elementary categories) as shown.

Classification using R1

Classification using R2

The attribute (equivalence relation) R2 divides the universe of discourse into a set of equivalence classes (elementary categories) as shown.

Applying the family of attributes (equivalence relations) R simultaneously divides the universe of discourse into a set of basic categories as shown.

Classification using

R={R1, R2}

Take a closer look!

X

A set that can not be precisely determined using the available knowledge is called a Rough Set.

Our goal is to use the concepts of Rough Set theory to approximately determine the set using available knowledge.

XR

The set RX is the set of all elements of U which can be certainty classified as elements of X in the Knowledge R

Lower approximation of X:x R X if and only if [x]R X

Lower Approximation

of X

Take a closer look!

XR

Upper approximation of X:x iff [x]R X

The set is the set of elements of U which can be possibly classified as elements of X, in employing knowledge R

XR

Upper Approximation

of X

XR

Take a closer look!

XRXBNXR R )(XRXRXBNR )(

Take a closer look!

XRXNEGR )(

The Negative Region of

X

Still U can’t understand?!

U

U/R R : subset of attributes

set X

XRXR ∴ X is R-definable

U/R

U

set X

∴ X is R-rough (undefinable)XR XR

X is R-definable (or crisp) if and only if ( i.e X is the union of some R-basic categories, called

R-definable set, R-exact set)X is R-undefinable (rough) with respect to R if and only if

( called R-inexact, R-rough)

XRXR

XRXR XR

XR is the maximal R-definable set contained in X

is the minimal R-definable set containing X

Example

• I = <U, Ω>, let R={a, c} , X={x | d(x) = yes}={1, 4, 6}

► approximate set X using only the information contained in R

the family of all equivalence classes of IND(R)

U/IND(R) = U/R = {{1}{ 2}{6} {3,4}{5,7}

R-lower approximations of X

R-upper approximations of X

• ※ The set X is R-rough since the boundary region is not empty

}4,3,6,1{}][|{R XxxX B

}6,1{}][|{R XxxX B

}4,3{)(R XBN }7,5,2{)(R XNEG}6,1{)(R XPOS

UU a c d1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

yes

yes/no

no

{x1, x6}

{x3, x4}

{x2, x5,x7}

XR

XR

4.Properties of Approximations

4.Properties of Approximations Cont’

5.Approximations and Membership Relation

Imprecise Knowledge need two membership relations to properly classify elements of U.

Membership relation is important when speaking about sets.

In Set theory: Absolute knowledge is required to classify x as x U or x U Precise categories don’t require two membership relations, One

“classical” membership relation suffices

In Rough theory The membership relation is not a primitive notion but one based on

knowledge we have about the objects to be classified Tow membership relations are required.

• The two membership relations are defined as follows:

• R x surely belongs to X with respect to R and called upper

membership relation

• R x possibly belongs to X with respect to R and called lower

membership relation.

Propositions

6.Numerical Characterization of Imprecision

• Inexactness of a set is due to the existence of a borderline region.

• accuracy measure αR(X)

: the degree of completeness of our knowledge R about the set X

• If , the R-borderline region of X is empty

and the set X is R-definable (i.e X is crisp with respect to R).

• If , the set X has some non-empty R-borderline region

and X is R-undefinable (i.e X is rough with respect to R).

Rcard

RcardXR )( .X.10 R

1)( XR

1)( XR

Example

let R={a, c} , X={x | d(x) = yes}= {1, 4, 6}

}4,3,6,1{}][|{R XxxX B

}6,1{}][|{R XxxX B

5.04

2)(R

Rcard

RcardX

UU a c d1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2 Yes

5 3 3 no

6 1 3 yes

7 3 3 no

• R-roughness of X : the degree of incompleteness of knowledge R

about the set X

Example: let R={a, c} , X={x | d(x) = yes}={1, 4, 6} Y={x | d(x) = no}={2, 3, 6, 7}

U/IND(R) = U/R = {{1}{ 2}{6} {3,4}{5,7}}

)(1)( XX RR

5.05.01)(1)(R XX R

}4,3,6,1{}][|{R R XxxX

}6,1{}][|{R R XxxX

5.04/2)(R X

}6,2{}][|{R R YxxY

}7,5,6,4,3,2{}][|{R R YxxY

67.06/4)(1)( RR YY

33.06/2)(R Y

UU a c d1 1 4 yes

2 1 1 no

3 2 2 no

4 2 2Yes

5 3 3 no

6 1 3 yes

7 3 3 no

7. Topological Characterization of Imprecision

• There are four important and different kinds of rough sets defined as shown below:

Take a closer look!

Universe withClassification 1Classification 1

U|R

Universe withClassification 2Classification 2

U|R

Take a closer look!

XR

RXXX

Roughly R-

definable

Take a closer look!

XR

XX

Internally R-undefinable

Take a closer look!

XR

RXXX

Externally R-undefinable

Take a closer look!

XX

Totally R-undefinabl

e

8.Approximation of Classifications

• This is a simple extension of the definition of approximations of sets.

• F={X1, X2, ..., Xn} : a family of non-empty sets and• R-lower approximation of the family F : • R-upper approximation of the family F :

• Example R={a, c}

F={X, Y}={{1,4,6}{2,3,5,7}} , X={x | d(x) = yes},

Y={x | d(x) = no}

U/IND(R) = U/R = {{1}{ 2}{6} {3,4}{5,7}}

},,{ 21 nXRXRXRFR

},,,{ 11 nXRXRXRFR

UF

UU a c d1 1 4 yes2 1 1 no3 2 2 no4 2 2 Yes5 3 3 no6 1 3 yes7 3 3 no

}7,6,5,2,1{}}7,5,2}{6,1{{},{ YRXRFR

}7,6,5,4,3,2,1{}}7,6,5,4,3,2}{6.4.3.1{{},{ YRXRFR

• the accuracy of approximation of F: the percentage of possible correct decisions when classifying objects employing the knowledge R

• the quality of approximation of F : the percentage of objects which can be correctly

classified to classes of F employing the knowledge R

i

iR

XRcard

XRcardF )(

Ucard

XRcardF i

R)(

Example

• R={a, c}

F={X, Y}={{1,4,6}{2,3,6,7}} , X={x | d(x) = yes},Y={x | d(x) = no}

7/5)( FR2/1)64/()32()( FR

}7,6,5,2,1{}}7,5,2}{6,1{{},{ YRXRFR

}7,6,5,4,3,2,1{}}7,6,5,4,3,2}{6.4.3.1{{},{ YRXRFR

9.Rough Equality of Sets

• Effect of rough on equality.• In set theory,

• two sets are equal if they have exactly the same elements • two sets can be unequal in set theory,

• In rough theory,• we need another concept of equality of sets, namely

(approximate (rough) equality. • two sets can be approximately equal from our point of view.

• There are three kinds of approximate equality of sets.

Propositions

Proposition