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### Chapter 2

1. 1. Imprecise Categories, Approximation, and Rough Sets Chapter 2
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. 3. Contents <ul><li>Introduction </li></ul><ul><li>Rough Sets </li></ul><ul><li>Lower and Upper Approximations </li></ul><ul><li>Properties of Approximations </li></ul><ul><li>Approximations and Membership Relation </li></ul><ul><li>Numerical Characterization of Imprecision </li></ul><ul><li>Topological Characterization of Imprecision </li></ul><ul><li>Approximation of Classifications </li></ul><ul><li>Rough Equality of Sets </li></ul>
4. 4. Introduction <ul><li>Definitions </li></ul><ul><li>Sets </li></ul><ul><li> 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). </li></ul><ul><li>Examples </li></ul><ul><ul><li>Classical sets are also called crisp (sets). </li></ul></ul><ul><ul><ul><ul><li>Lists: A = {apples, oranges, cherries, mangoes} </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Formulas: A = {x | x is an even natural number} </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Membership or characteristic function </li></ul></ul></ul></ul>
5. 5. <ul><li>b. Fuzzy sets – admits gradation such as all tones between black and white. </li></ul><ul><li>Described by a membership function . </li></ul><ul><li>Example </li></ul><ul><ul><li>µA: U −-> [0, 1]. </li></ul></ul><ul><li>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 </li></ul><ul><li>that approach A as closely as possible from the inside and outside (respectively) and </li></ul><ul><li>where and are called the lower and upper approximation of A. </li></ul>Theories of fuzzy sets and rough sets are generalizations of classical set theory for modeling vagueness and uncertainty
6. 6. <ul><li>As We had seen before that: </li></ul><ul><li>Categories are features (i.e. subsets) of objects which can be worded using knowledge available in a given knowledge base. </li></ul><ul><li>Some categories are definable in one knowledge base but undefinable in another one. </li></ul><ul><li>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). </li></ul><ul><li>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. </li></ul>We will use rough set notion here for handling the vagueness of knowledge
7. 7. 1.Rough Sets <ul><ul><li>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. </li></ul></ul><ul><ul><li>R-definable sets also called R-exact sets. </li></ul></ul><ul><ul><li>R-undefinable sets also called R-inexact or R-rough. </li></ul></ul>
8. 8. 2. Lower and Upper Approximations <ul><li>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. </li></ul><ul><li>The two approximations are: </li></ul><ul><ul><li>Upper Approximation: </li></ul></ul><ul><ul><li>Lower Approximation: </li></ul></ul>
9. 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. 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. 11. Take a closer look! <ul><li>Upper approximation of X: </li></ul><ul><ul><li>x  iff [x] R  X  </li></ul></ul>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. 12. Take a closer look!
13. 13. Take a closer look! The Negative Region of X
14. 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. 15. Example <ul><li>I = < U, Ω >, let R={a, c} , X ={x | d(x) = yes}={1, 4, 6} </li></ul><ul><li>► approximate set X using only the information contained in R </li></ul><ul><li>the family of all equivalence classes of IND (R) </li></ul><ul><li> U/ IND (R) = U/R = {{1}{ 2}{6} {3,4}{5,7} </li></ul><ul><li> R-lower approximations of X </li></ul><ul><li>R-upper approximations of X </li></ul><ul><li>※ The set X is R- rough since the boundary region is not empty </li></ul>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. 16. yes yes/no no {x1, x6} {x3, x4 } {x2, x5,x7}
17. 17. 4.Properties of Approximations
18. 18. 4.Properties of Approximations Cont’
19. 19. 5.Approximations and Membership Relation <ul><li>Imprecise Knowledge need two membership relations to properly classify elements of U. </li></ul><ul><li>Membership relation is important when speaking about sets. </li></ul><ul><li>In Set theory: </li></ul><ul><ul><ul><li>Absolute knowledge is required to classify x as x U or x U </li></ul></ul></ul><ul><ul><ul><li>Precise categories don’t require two membership relations, One “classical” membership relation suffices </li></ul></ul></ul><ul><li>In Rough theory </li></ul><ul><ul><ul><li>The membership relation is not a primitive notion but one based on knowledge we have about the objects to be classified </li></ul></ul></ul><ul><ul><ul><li>Tow membership relations are required. </li></ul></ul></ul>
20. 20. <ul><li>The two membership relations are defined as follows: </li></ul><ul><ul><ul><li>R x surely belongs to X with respect to R and called upper membership relation </li></ul></ul></ul><ul><ul><ul><li>R x possibly belongs to X with respect to R and called lower membership relation. </li></ul></ul></ul>
21. 21. Propositions
22. 22. 6.Numerical Characterization of Imprecision <ul><li>Inexactness of a set is due to the existence of a borderline region. </li></ul><ul><li>accuracy measure α R ( X ) </li></ul><ul><li>: the degree of completeness of our knowledge R about the set X </li></ul><ul><li>If , the R-borderline region of X is empty </li></ul><ul><li> and the set X is R -definable (i.e X is crisp with respect to R ) . </li></ul><ul><li>If , the set X has some non-empty R-borderline region </li></ul><ul><li>and X is R- undefinable (i.e X is rough with respect to R ). </li></ul>
23. 23. Example <ul><li>let R={a, c} , X ={x | d(x) = yes}= {1, 4, 6} </li></ul>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. 24. <ul><li>R -roughness of X </li></ul><ul><li>: the degree of incompleteness of knowledge R about the set X </li></ul><ul><li>Example: </li></ul><ul><li>let R={a, c} , X ={x | d(x) = yes}={1, 4, 6} </li></ul><ul><li>Y ={x | d(x) = no}={2, 3, 6, 7} </li></ul><ul><li>U/ IND (R) = U/R = {{1}{ 2}{6} {3,4}{5,7}} </li></ul>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. 25. 7. Topological Characterization of Imprecision <ul><li>There are four important and different kinds of rough sets defined as shown below: </li></ul>
26. 26. Take a closer look! Universe with Classification 1 U|R Universe with Classification 2 U|R
27. 27. Take a closer look! R X X Roughly R-definable
28. 28. Take a closer look! X Internally R-undefinable
29. 29. Take a closer look! R X X Externally R-undefinable
30. 30. Take a closer look! X Totally R-undefinable
31. 31. 8.Approximation of Classifications <ul><li>This is a simple extension of the definition of approximations of sets. </li></ul><ul><li>F ={ X 1 , X 2 , ..., X n } : a family of non-empty sets and </li></ul><ul><ul><ul><li>R-lower approximation of the family F : </li></ul></ul></ul><ul><ul><ul><li>R-upper approximation of the family F : </li></ul></ul></ul><ul><li>Example </li></ul><ul><li>R={a, c} </li></ul><ul><li> F={ X , Y } ={{1,4,6}{2,3,5,7}} , X ={x | d(x) = yes}, </li></ul><ul><li> Y ={x | d(x) = no} </li></ul><ul><li>U/ IND (R) = U/R = {{1}{ 2}{6} {3,4}{5,7}} </li></ul>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. 32. <ul><li>the accuracy of approximation of F </li></ul><ul><li>: the percentage of possible correct decisions when classifying objects employing the knowledge R </li></ul><ul><li>the quality of approximation of F </li></ul><ul><li>: the percentage of objects which can be correctly classified to classes of F employing the knowledge R </li></ul>
33. 33. Example <ul><li>R={a, c} </li></ul><ul><li>F={ X , Y }={{1,4,6}{2,3,6,7}} , X ={x | d(x) = yes}, Y ={x | d(x) = no} </li></ul>
34. 34. 9.Rough Equality of Sets <ul><li>Effect of rough on equality. </li></ul><ul><li>In set theory , </li></ul><ul><ul><ul><li>two sets are equal if they have exactly the same elements </li></ul></ul></ul><ul><ul><ul><li>two sets can be unequal in set theory, </li></ul></ul></ul><ul><li>In rough theory , </li></ul><ul><ul><ul><li>we need another concept of equality of sets, namely (approximate (rough) equality. </li></ul></ul></ul><ul><ul><ul><li>two sets can be approximately equal from our point of view. </li></ul></ul></ul><ul><li>There are three kinds of approximate equality of sets. </li></ul>
35. 36. Propositions
36. 37. Proposition