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Rough SetRough Set
FuzzyFuzzy SetsSets
• Dr. Lotfi Zadeh propose this approach.
• In his approach an element can belong to a
degree k (0 <= k <= 1).
• In classical set theory an element must belong or
not belong to a set.
• Fuzzy membership function can be presented as:
µX(x) є (0,1)
where, X is a set and x is an element.
2
Fuzzy Sets (Continue)Fuzzy Sets (Continue)
• Fuzzy membership function has the following
properties.
• µU -x(x) = 1 - µX(x) for any x є U
• µxUy(x) = max(µX(x), µy(x)) for any x є U
• µx ∩y(x) = min(µX(x), µy(x)) for any x є U
3
Rough SetsRough Sets
• Rough set theory is another approach to handle
vagueness.
• Imprecision in this approach is expressed by a
boundary region of a set, and not by a partial
membership, like in fuzzy set theory.
• Rough set concept can be defined by topological
operation interior and closure called
approximations.
4
Rough Sets (Continue)Rough Sets (Continue)
• Suppose we are given a set of objects U called
the universe and an indiscernibility relation R ⊆ U
× U, representing our lack of knowledge about
elements of U.
• For simplicity we assume that R is an equivalence
relation.
• Let X be a subset of U.
• We want to characterize the set X with respect to
R. To do this we will need the basic concepts of
rough set theory given in next slide. 5
Rough Sets (Continue)Rough Sets (Continue)
• The lower approximation of a set X with respect
to R is the set of all objects, which can be for
certain(sure) classified as X with respect to R (are
certainly X with respect to R).
• The upper approximation of a set X with respect
to R is the set of all objects which can be
possibly(maybe) classified as X with respect to R
(are possibly X in view of R).
• The boundary region of a set X with respect to R is
the set of all objects, which can be classified
neither as X nor as not-X with respect to R. 6
Rough Sets (Continue)Rough Sets (Continue)
• So that,
• Set X is crisp (Exact with respect to R), if the
boundary region of X is empty.
• Set X is rough (Inexact with respect to R), if the
boundary region of X is nonempty.
7
Rough ApproximationRough Approximation
• Formal definitions of approximations and the
boundary region are as follows:
• R-lower approximation of X
R*(x) = U {R(x): R(x) ⊆ X}
• R- upper approximation of X
R*
(x) = U {R(x): R(x) ∩ X ≠ ɸ}
• R-boundary approximation of X
RNR (X) = R*
(X) - R*
(X)
8
Rough ApproximationRough Approximation
• As we can see from the definition, approximations
are expressed in terms of granules of knowledge.
• The lower approximation of a set is union of all
granules which are entirely included in the set.
• The upper approximation − is union of all granules
which have non-empty intersection with the set.
• The boundary region of set is the difference
between the upper and the lower approximation.
9
Rough Membership functionRough Membership function
• Rough sets can be also defined as rough
membership function.
µx
R
: U  (0,1)
Where
µx
R
(x) = |X∩ R(x)| / |R(x)|
And |X| denotes the cardinality of X.
10
Rough MembershipRough Membership
function(Conti)function(Conti)
• The rough membership function can be expresses
conditional probability.
• That x belongs to X given R.
• And can be interpreted as a degree that x belongs
to X in view of information about x expressed by
R.
• The meaning of rough membership function can
be defined as shown in fig 1.
11
Rough MembershipRough Membership
function(Conti)function(Conti)
X
X
X
R(x)
R(x)
R(x)
x
x
x
µx
R
(x) = 0
0 < µx
R
(x) < 1
µx
R
(x) = 1
12
Fig 1
Rough MembershipRough Membership
function(Conti)function(Conti)
• The rough membership function can be used to
define approximations and the boundary region
of a set, as shown below:
R*
(x) = {xєU : µx
R
(x) = 1}
R*
(x) = {xєU : µx
R
(x) > 0}
RNR (X) = {xєU : 0 < µx
R
(x) < 1 }
13
Rough MembershipRough Membership
function(Conti)function(Conti)
• The rough membership function has the
following properties:
• µx
R
(x) = 1 iff x є R*
(x)
• µx
R
(x) = 0 iff x є U - R*
(x)
• 0 < µx
R
(x) < 1 iff x є RNR (X)
• µR
U-x (x) = 1 - µx
R
(x) for any x є U
• µxUy(x) => max(µR
X(x), µR
y(x)) for any x є U
• µx ∩y(x) <= min (µR
X(x), µR
y(x)) for any x є U
14
IndiscernibilityIndiscernibility
• A decision system (i.e. a decision table) express
all the model.
• This table may be unnecessarily large.
• The same or indiscernible objects may be
represented several times.
15
Indiscernibility (Conti)Indiscernibility (Conti)
Element Age LEMS Walk
X1 16-30 50 Yes
X2 16-30 0 No
X3 31-45 1-25 No
X4 31-45 1-25 Yes
X5 46-60 26-49 No
X6 16-30 26-49 No
X7 46-60 26-49 Yes
LEMS = Lower Extremity(boundary) Motor Score
Table 1
16
IndiscernibilityIndiscernibility
• A binary relation R ⊆ X x X which is reflexive (i.e.
an object is in relation with itself xRx), symmetric
(if xRy then yRx) and transitive (if xRy and yRz
then xRz) is called an equivalence relation.
• The equivalence class of an element x є X
consists of all objects y є X such that xRy.
• Let A = (U, A) be an information system then with
any B ⊆ A there is associated an equivalence
relation INDA (B) :
INDA (B) = {(x,x’) є U2
| a є B a(x) = a(x’)} 17
IndiscernibilityIndiscernibility
• INDA (B) is called the B-indiscernibility relation. If
(x,x') є INDA (B) then x and x' are indiscernible
from each other by attributes from B.
• The equivalence classes of the B-indiscernibility
relation are denoted [x]B.
Example:
• Let us illustrate how a decision table such as
Table 1 defines an indiscernibility relation.
• The non-empty subsets of the conditional
attributes are {Age}, {LEMS} and {Age, LEMS}. 18
IndiscernibilityIndiscernibility
• If we consider, for instance, {LEMS}, objects X3 and
X4 belong to the same equivalence class and are
indiscernible.
• The relation IND defines three partitions of the
universe.
IND ({Age}) = {{x1,x2,x6},{x3,x4},{x5,x7}}
IND ({LEMS}) = {{x1},{x2},{x6},{x3,x4},{x5,x6,x7}}
IND ({Age,LEMS}) = {{x1},{x2},x6},{x3,x4},{x5,x7},{x6}}
19

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Rough Set and Fuzzy Set Theory Comparison

  • 2. FuzzyFuzzy SetsSets • Dr. Lotfi Zadeh propose this approach. • In his approach an element can belong to a degree k (0 <= k <= 1). • In classical set theory an element must belong or not belong to a set. • Fuzzy membership function can be presented as: µX(x) є (0,1) where, X is a set and x is an element. 2
  • 3. Fuzzy Sets (Continue)Fuzzy Sets (Continue) • Fuzzy membership function has the following properties. • µU -x(x) = 1 - µX(x) for any x є U • µxUy(x) = max(µX(x), µy(x)) for any x є U • µx ∩y(x) = min(µX(x), µy(x)) for any x є U 3
  • 4. Rough SetsRough Sets • Rough set theory is another approach to handle vagueness. • Imprecision in this approach is expressed by a boundary region of a set, and not by a partial membership, like in fuzzy set theory. • Rough set concept can be defined by topological operation interior and closure called approximations. 4
  • 5. Rough Sets (Continue)Rough Sets (Continue) • Suppose we are given a set of objects U called the universe and an indiscernibility relation R ⊆ U × U, representing our lack of knowledge about elements of U. • For simplicity we assume that R is an equivalence relation. • Let X be a subset of U. • We want to characterize the set X with respect to R. To do this we will need the basic concepts of rough set theory given in next slide. 5
  • 6. Rough Sets (Continue)Rough Sets (Continue) • The lower approximation of a set X with respect to R is the set of all objects, which can be for certain(sure) classified as X with respect to R (are certainly X with respect to R). • The upper approximation of a set X with respect to R is the set of all objects which can be possibly(maybe) classified as X with respect to R (are possibly X in view of R). • The boundary region of a set X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R. 6
  • 7. Rough Sets (Continue)Rough Sets (Continue) • So that, • Set X is crisp (Exact with respect to R), if the boundary region of X is empty. • Set X is rough (Inexact with respect to R), if the boundary region of X is nonempty. 7
  • 8. Rough ApproximationRough Approximation • Formal definitions of approximations and the boundary region are as follows: • R-lower approximation of X R*(x) = U {R(x): R(x) ⊆ X} • R- upper approximation of X R* (x) = U {R(x): R(x) ∩ X ≠ ɸ} • R-boundary approximation of X RNR (X) = R* (X) - R* (X) 8
  • 9. Rough ApproximationRough Approximation • As we can see from the definition, approximations are expressed in terms of granules of knowledge. • The lower approximation of a set is union of all granules which are entirely included in the set. • The upper approximation − is union of all granules which have non-empty intersection with the set. • The boundary region of set is the difference between the upper and the lower approximation. 9
  • 10. Rough Membership functionRough Membership function • Rough sets can be also defined as rough membership function. µx R : U  (0,1) Where µx R (x) = |X∩ R(x)| / |R(x)| And |X| denotes the cardinality of X. 10
  • 11. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function can be expresses conditional probability. • That x belongs to X given R. • And can be interpreted as a degree that x belongs to X in view of information about x expressed by R. • The meaning of rough membership function can be defined as shown in fig 1. 11
  • 13. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function can be used to define approximations and the boundary region of a set, as shown below: R* (x) = {xєU : µx R (x) = 1} R* (x) = {xєU : µx R (x) > 0} RNR (X) = {xєU : 0 < µx R (x) < 1 } 13
  • 14. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function has the following properties: • µx R (x) = 1 iff x є R* (x) • µx R (x) = 0 iff x є U - R* (x) • 0 < µx R (x) < 1 iff x є RNR (X) • µR U-x (x) = 1 - µx R (x) for any x є U • µxUy(x) => max(µR X(x), µR y(x)) for any x є U • µx ∩y(x) <= min (µR X(x), µR y(x)) for any x є U 14
  • 15. IndiscernibilityIndiscernibility • A decision system (i.e. a decision table) express all the model. • This table may be unnecessarily large. • The same or indiscernible objects may be represented several times. 15
  • 16. Indiscernibility (Conti)Indiscernibility (Conti) Element Age LEMS Walk X1 16-30 50 Yes X2 16-30 0 No X3 31-45 1-25 No X4 31-45 1-25 Yes X5 46-60 26-49 No X6 16-30 26-49 No X7 46-60 26-49 Yes LEMS = Lower Extremity(boundary) Motor Score Table 1 16
  • 17. IndiscernibilityIndiscernibility • A binary relation R ⊆ X x X which is reflexive (i.e. an object is in relation with itself xRx), symmetric (if xRy then yRx) and transitive (if xRy and yRz then xRz) is called an equivalence relation. • The equivalence class of an element x є X consists of all objects y є X such that xRy. • Let A = (U, A) be an information system then with any B ⊆ A there is associated an equivalence relation INDA (B) : INDA (B) = {(x,x’) є U2 | a є B a(x) = a(x’)} 17
  • 18. IndiscernibilityIndiscernibility • INDA (B) is called the B-indiscernibility relation. If (x,x') є INDA (B) then x and x' are indiscernible from each other by attributes from B. • The equivalence classes of the B-indiscernibility relation are denoted [x]B. Example: • Let us illustrate how a decision table such as Table 1 defines an indiscernibility relation. • The non-empty subsets of the conditional attributes are {Age}, {LEMS} and {Age, LEMS}. 18
  • 19. IndiscernibilityIndiscernibility • If we consider, for instance, {LEMS}, objects X3 and X4 belong to the same equivalence class and are indiscernible. • The relation IND defines three partitions of the universe. IND ({Age}) = {{x1,x2,x6},{x3,x4},{x5,x7}} IND ({LEMS}) = {{x1},{x2},{x6},{x3,x4},{x5,x6,x7}} IND ({Age,LEMS}) = {{x1},{x2},x6},{x3,x4},{x5,x7},{x6}} 19