Chapter 2
Upcoming SlideShare
Loading in...5
×

Like this? Share it with your network

Share
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
198
On Slideshare
198
From Embeds
0
Number of Embeds
0

Actions

Shares
Downloads
1
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 1. Imprecise Categories, Approximation, and Rough Sets Chapter 2
  • 2. Review Classification Equivalence relations A category in R containing an element x  U [x] R The family of all equivalence classes of R( or classification of U) referred to as categories of R U/ R X is a concept / category in U A family of concepts in U will be referred to as abstract knowledge about U X  U
  • 3. Contents
    • Introduction
    • Rough Sets
    • Lower and Upper Approximations
    • Properties of Approximations
    • Approximations and Membership Relation
    • Numerical Characterization of Imprecision
    • Topological Characterization of Imprecision
    • Approximation of Classifications
    • Rough Equality of Sets
  • 4. Introduction
    • Definitions
    • Sets
    • a. Classical 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
  • 5.
    • b. Fuzzy sets – admits gradation such as all tones between black and white.
    • Described by a membership function .
    • Example
      • µA: U −-> [0, 1].
    • C. 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.
    Theories of fuzzy sets and rough sets are generalizations of classical set theory for modeling vagueness and uncertainty
  • 6.
    • As We had seen before that:
    • Categories 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 for handling the vagueness of knowledge
  • 7. 1.Rough Sets
      • R-definable sets are those subsets of the universe which can be exactly defined in the knowledge base K, whereas the R-undefinable 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.
  • 8. 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:
  • 9. Take a closer look! The universe of discourse is the finite set of all objects under consideration. The attribute (equivalence relation) R 1 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) R 2 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}
  • 10. Take a closer look! 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. The set R X 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
  • 11. Take a closer look!
    • 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 Upper Approximation of X
  • 12. Take a closer look!
  • 13. Take a closer look! The Negative Region of X
  • 14. Still U can’t understand?! ∴ X is R -rough (undefinable) U U/R R : subset of attributes set X ∴ X is R -definable U/R U set X 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 ) is the maximal R-definable set contained in X is the minimal R-definable set containing X
  • 15. 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
    yes 3 1 6 d c a U 7 5 4 3 2 1 no 3 3 no 3 3 Yes 2 2 no 2 2 no 1 1 yes 4 1
  • 16. yes yes/no no {x1, x6} {x3, x4 } {x2, x5,x7}
  • 17. 4.Properties of Approximations
  • 18. 4.Properties of Approximations Cont’
  • 19. 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.
  • 20.
    • 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.
  • 21. Propositions
  • 22. 6.Numerical Characterization of Imprecision
    • Inexactness of a set is due to the existence of a borderline region.
    • accuracy measure α R ( 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 ).
  • 23. Example
    • let R={a, c} , X ={x | d(x) = yes}= {1, 4, 6}
    yes 3 1 6 d c a U 7 5 4 3 2 1 no 3 3 no 3 3 Yes 2 2 no 2 2 no 1 1 yes 4 1
  • 24.
    • 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}}
    yes 3 1 6 d c a U 7 5 4 3 2 1 no 3 3 no 3 3 Yes 2 2 no 2 2 no 1 1 yes 4 1
  • 25. 7. Topological Characterization of Imprecision
    • There are four important and different kinds of rough sets defined as shown below:
  • 26. Take a closer look! Universe with Classification 1 U|R Universe with Classification 2 U|R
  • 27. Take a closer look! R X X Roughly R-definable
  • 28. Take a closer look! X Internally R-undefinable
  • 29. Take a closer look! R X X Externally R-undefinable
  • 30. Take a closer look! X Totally R-undefinable
  • 31. 8.Approximation of Classifications
    • This is a simple extension of the definition of approximations of sets.
    • F ={ X 1 , X 2 , ..., X n } : 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}}
    yes 3 1 6 d c a U 7 5 4 3 2 1 no 3 3 no 3 3 Yes 2 2 no 2 2 no 1 1 yes 4 1
  • 32.
    • 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
  • 33. Example
    • R={a, c}
    • F={ X , Y }={{1,4,6}{2,3,6,7}} , X ={x | d(x) = yes}, Y ={x | d(x) = no}
  • 34. 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.
  • 35.  
  • 36. Propositions
  • 37. Proposition
  • 38.