Embedded Interval Valued Type-2 Fuzzy Sets Robert I John Centre for Computational Intelligence Department of Computer Science De Montfort University The Gateway Leicester LE1 9BH United Kingdom
[email protected] Abstract—Type-2 fuzzy sets are growing in popularity as, for certain applications they model uncertainty and imprecision better than type1 fuzzy sets. However, type-2 fuzzy sets can be difficult to understand and explain. Recent work has introduced embedded type-2 fuzzy sets and the Representation Theorem which enable us to discuss type-2 fuzzy sets in a different way. In particular they allow for alternative proofs of theoretical results which are often easier to follow. Interval valued type-2 fuzzy sets are found to be useful in many application. This paper presents embedded interval valued type-2 fuzzy sets to aid the discussion of interval valued type-2 fuzzy sets and to prove the join and meet of interval valued type-2 fuzzy sets.
For any given x the µ A˜ (x; u), 8u 2 Jx , is a type-1 membership function. The Jx are known as the primary memberships. It can be seen from this definition that a type-2 membership function is three dimensional. To illustrate this, Figure 1 provides an example of type-2 fuzzy set. We have a three dimen-
µA˜ (x,u) 1
I. I NTRODUCTION Type-2 fuzzy sets (originally introduced by Zadeh [1]) have membership grades that are fuzzy. That is, instead of being in [0,1] the membership grades are themselves (type-1) fuzzy sets. Karnik and Mendel [2][page 2] provide this definition of a type-2 fuzzy set: A type-2 fuzzy set is characterised by a fuzzy membership function, i.e. the membership value (or membership grade) for each element of this set is a fuzzy set in [0,1], unlike a type1 fuzzy set where the membership grade is a crisp number in [0,1]. The characterisation in this definition of type-2 fuzzy sets uses the notion that type-1 fuzzy sets can be thought of as a first order approximation to uncertainty and, therefore, type-2 fuzzy sets provide a second order approximation. They play an important role in modelling uncertainties that exist in fuzzy logic systems [3], [4] and are becoming increasingly important in the goal of ‘Computing with Words’ [5] and the ‘Computational Theory of Perceptions’ [6]. There has been much work carried out on the theoretical properties of type-2 fuzzy sets (e.g. [7], [8]). In particular, previous work has provided a complete set of terms that describe type-2 fuzzy sets, a new representation for type-2 fuzzy sets and new derivations of union, intersection and complement of type-2 fuzzy sets without making use of the extension principle [9]. For the purposes of this paper a type-2 fuzzy set is defined in the following way [10]: ˜ is characterised by a Definition 1: A type-2 fuzzy set, A, type-2 membership function µ A˜ (x; u), where x 2 X and u 2 Jx [0; 1] A˜
=
f((x; u); µA˜ (x; u)) j8x 2 X ; 8u 2 Jx [0; 1]g
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(1)
x 0 0.2 0.4 0.6 0.8 1
u
Fig. 1. A Typical Type-2 Fuzzy Set
sional figure with the axes being x, u and µ A˜ (x; u). The ‘spikes’ are in [0,1] and represent µ A˜ (x; u) for a given (x; u). For a given x we have secondary membership function: Definition 2: At each value x (say x 0 ) then µA˜ (x0 ; u) is a secondary membership function of µ A˜ (x; u). An example secondary membership function of the type-2 fuzzy set in Figure 1 is given in Figure 2. Note that the type2 fuzzy set A˜ is the union of all the secondary membership functions. To enable sensible discussion of type-2 fuzzy sets we define the Domain of Uncertainty in the following manner. Definition 3: The Domain of Uncertainty (DOU), for a ˜ is the union of all the primary membertype-2 fuzzy set, A, ˜ ships of A, i.e. Jx (2) DOU (A˜ ) =
[
x2X
µA˜ (x,u)
That is, the secondary membership grade is unity 8u 2 J x . More formally the secondary membership function of an IVFS can be defined in the following way:
1
Definition 5: At each value x (say x 0 ) then the secondary membership function of A˜ iv is given by µ A˜ iv (x0 ; u) = 1 8u 2 Jx . Figure 3 provides a three dimensional view of an IVFS. As
x 0 0.2
0
0.4 0.6 0.8 1
µA˜ (x,u)
u 1
Fig. 2. An Example Secondary Membership Function
The DOU is, essentially, the domain of a type-2 fuzzy set. It is a particularly useful concept since we often wish to consider the whole domain - not just for a particular member of the type-2 fuzzy set. Type-2 fuzzy logic is growing in importance as an approach that models uncertainties in membership functions and the various components of rules in rule based type-1 fuzzy systems. It is not the purpose of this paper to provide background on type-2 fuzzy logic. A more detailed discussion of type-2 fuzzy sets, type-2 fuzzy logic and the applications can be found elsewhere( e.g. [11], [12], [3], [13], [8], [2], [14], [4]). This paper is interested in interval valued type-2 fuzzy sets as they are, arguably, the most useful type-2 representation in real applications [15]. An interval valued type-2 fuzzy sets is a special case of a type-2 fuzzy set where the secondary membership functions all have the value unity. To enable the development of type-2 fuzzy systems using if-then rules and interval valued type-2 fuzzy sets we need the join and meet of interval valued type-2 fuzzy sets. The join and meet of interval valued type-2 fuzzy sets have been proven elsewhere, by the use of mathematical induction [10, Pages 224-227]. This paper extends earlier work [9] to provide alternative proofs by use of embedded type-2 fuzzy sets, from a different perspective. In Section II we define embedded interval valued type-2 fuzzy sets. These are used in Section III to develop the proof of the join and meet for interval valued type-2 fuzzy sets. Section IV provides a conclusion.
x 0 0.2 0.4 0.6 0.8 1
u
Fig. 3. An Interval Valued Type-2 Fuzzy Set
can be seen, the spikes in this case are all the same size, having the value unity. Since IVFS are easier to compute with than general type-2 fuzzy sets they are increasing in popularity. The properties of IVFS and the growing number of useful applications have been reported elsewhere (e.g.[10], [16], [17], [18], [19], [20], [21], [22]).
1
r2 r1
J2 J1
l2
rN JN
l1
lN
0
x1
x2
xN
Fig. 4. An IVFS in Two Dimensions
II. E MBEDDED I NTERVAL VALUED T YPE -2 F UZZY S ETS This Section provides a definition of interval valued type2 fuzzy sets and, for the first time, embedded interval valued type-2 fuzzy sets. Definition 4: An interval valued type-2 fuzzy set (IVFS), A˜ iv , is characterised by a type-2 membership function µA˜ iv (x; u), where x 2 X and u 2 Jx [0; 1] A˜ iv
=
f((x; u); 1)j8x 2 X ; 8u 2 Jx [0; 1]g
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(3)
Using the concept of the Domain of Uncertainty it is useful to represent an IVFS in two dimensions 1 as in Figure 4. In this Figure for each x i (i = 1; : : : ; N ) the Ji has an upper bound of r i and a lower bound of l i . In this paper embedded type-2 fuzzy sets [9], [23] are used to provide a proof for the join and meet of IVFS. A definition of an embedded type-2 fuzzy set is now given. 1 Note that throughout this paper the discrete case only is considered. The results also hold for the continuous case. These results will be included in a longer journal paper
µ A˜ ( x, u)
1
Definition 7: For discrete universes of discourse X and U, ˜ iv an embedded IVFS A˜ iv e has N elements, where Ae contains exactly one element from J x1 ; Jx2 ; : : : ; JxN , namely u1 ; u2 ; : : : ; uN , each with its associated secondary grade of unity, i.e. 2
3
4
5
N
x
A˜ iv e = ∑ [1=ui ]=xi
0.2
ui 2 Jxi U
= [0; 1]
(9)
i=1
Figure 6 provides a three dimensional view of an embedded IVFS.
0.4 0.6 0.8
1
J1
u
J2
J3
J4
J5
µA˜ (x,u)
Fig. 5. An Embedded Type-2 Fuzzy Set
1
Definition 6: For discrete universes of discourse X and U, an embedded type-2 fuzzy set A˜ e has N elements, where A˜ e contains exactly one element from J x1 ; Jx2 ; : : : ; JxN , namely u1 ; u2 ; : : : ; uN , each with its associated secondary grade f xi (ui ) (i = 1; : : : ; N), i.e.
x 0 0.2 0.4 0.6
N
A˜ e = ∑ [ fxi (ui )=ui ]=xi
ui 2 Jxi U
= [0; 1]
0.8
(4)
1
i=1
u
Figure 5 gives an example of an embedded type-2 fuzzy set. As can be seen we now have what we might call a ‘wavy slice’ where we have one element (only) from each vertical slice contained in the embedded type-2 fuzzy set. The definitions so far in this paper provide enough detail to understand the Representation Theorem (see [9] for a full proof). Theorem j Let A˜ e denote the jth type-2 embedded fuzzy set for type-2 ˜ fuzzy set A. j j A˜ ej f(ui ; fxi (ui )); i = 1; : : : ; N g
where
j
ui 2 fuik ; k = 1; : : : ; Mi g
(5) (6)
Here, each uik is a member of the secondary membership grade for xk . A˜ can be represented as the union of all its type-2 embedded fuzzy sets. That is: A˜ =
We now have the necessary building blocks to prove the join and meet for IVFS. III. T HE J OIN AND M EET The proof of the join and meet for IVFS relies on results published elsewhere [9] for general embedded type-2 fuzzy sets. The join (union) and meet (intersection) of embedded type2 fuzzy sets are as follows [9]. Suppose we have two embedj ded type-2 fuzzy sets A˜ e and B˜ ie . The secondary grades at x l j j are denoted as f xl (ul ) and gxl (wl ) respectively then A˜ ej [ B˜ ie
j j i i [Fx1 (u1 ; w1 )=u1 _ w1 ]=x1 + : : :
j j i i +[FxN (uN ; wN )=uN _ wN ]=xN
n
∑ A˜ ej
(7)
j =1
where
Fig. 6. An Embedded IVFS
where, for each l = 1; : : : ; N, j
(8)
i=1
We are able to show, then, that a type-2 fuzzy set A˜ is the union of all its type-2 embedded fuzzy sets. This Theorem has allowed for the development of union, intersection and complement of type-2 fuzzy sets without use of the extension principle [9]. Since an IVFS is a special case of a type-2 fuzzy set we can easily define an embedded IVFS.
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j
Fxl (ul ; wil ) = h[ fxl (ul ); gxl (wil )]
N
n ∏ Mi
(10)
and h is a t-norm. The intersection is given by A˜ ej \ B˜ ie
j j i i [Fx1 (u1 ; w1 )=u1 ^ w1 ]=x1 + : : : j j i i +[FxN (uN ; wN )=uN ^ wN ]=xN
(11)
These results which hold for all type-2 fuzzy sets are next used to develop the proof of the join and meet for IVFS.
A. The Join for IVFS
B. The Meet for IVFS
Suppose we have two IVFS, A˜ iv ; B˜ iv . Remember, for an IVFS the secondary membership grades are unity. Equation 10 has a function applied to secondary membership grades which is a t-norm. The properties of a t-norm are such that iv( j ) iv(i) and B˜ e , h(1; 1) = 1. So, for two embedded IVFS, A˜ e Equation 10 can be re-written as µ ˜ iv( j) [ µ ˜ iv(i) Be
Ae
j i [1=u1 _ w1 ]=x1 + : : :
(12)
Notice that this is a type-2 representation of a type-1 fuzzy set [10, Page 102]. Since IVFS are special cases of type-2 fuzzy sets all results for type-2 fuzzy sets hold. Now, for two IVFS A˜ iv ; B˜ iv the Representation Theorem can be used to show the following [9]: A˜ iv [ B˜ iv =
∑ ∑ A˜ e
iv( j )
µ ˜ iv( j) \ µ ˜ iv(i)
iv(i)
[ B˜ e
j i +[1=uN ^ wN ]=xN
A˜ iv [ B˜ iv
nA˜ iv nB˜iv
=
∑∑
nA˜ iv nB˜iv
∑ ∑ A˜ e
iv( j )
iv(i) \ B˜ e
(17)
j =1 i=1
where A˜ iv and B˜ iv have nA˜ iv and nB˜ iv embedded type-2 fuzzy sets respectively. Equation 17 can be re-written, using Equation 16, as: A˜ iv \ B˜ iv
nA˜ iv nB˜ iv
=
∑ ∑ [(1
j
u1 ^ wi1 )=x1 + : : :
=
j =1 i=1
(13)
=
j i +(1=uN ^ wN )=xN ] nA˜ iv nB˜ iv j i [(1=u1 ^ w1 )=x1 + : : : j =1 i=1 nA˜ iv nB˜iv j i + [(1=uN ^ wN )=xN j =1 i=1
∑∑
∑∑
(18)
Consider a particular x k in Equation 18 nA˜ iv nB˜iv
j i [(1=u1 _ w1 )=x1 + : : :
∑ ∑ (1
j
uk ^ wik )=xk
=
j =1 i=1
=
(16)
For two IVFS A˜ iv ; B˜ iv the Representation Theorem can be used [9]:
j =1 i=1
where A˜ iv and B˜ iv have nA˜ iv and nB˜ iv embedded type-2 fuzzy sets respectively. Equation 13 can be re-written, using Equation 12, as:
j i [1=u1 ^ w1 ]=x1 + : : :
Be
Ae
A˜ iv \ B˜ iv =
j i +[1=uN _ wN ]=xN
nA˜ iv nB˜iv
For the meet of two IVFS, Equation 11 can be re-written as
j =1 i=1
j i +(1=uN _ wN )=xN ] nA˜ iv nB˜iv j i (1=u1 _ w1 )=x1 + : : : j =1 i=1 nA˜ iv nB˜iv j i + (1=uN _ wN )=xN j =1 i =1
j
∑∑
∑∑
(14)
Using minimum as the t-norm then the maximum of u k ^ wik ˜ iv ˜ iv ˜ iv j j ˜ iv is rkA ^ rkB since all uk rkA and wk rkB . The mini˜ j ˜ mum of u k ^ wik has to be lkA ^ lkB . Therefore the secondary membership function for x k is an interval set with domain ˜ iv ˜ iv ˜ iv ˜ iv [(lkA ^ lkB ),(rkA ^ rkB )]. Since Equation 18 is essentially the union of all the x k terms then we have the meet of the IVFS as N
Consider a particular x k in Equation 14
∑ [(lkA
˜ iv
˜ iv
^ lkB
A˜ iv ); (rk
˜ iv
^ rkB
)]=xk
(19)
k =1
nA˜ iv nB˜iv
∑ ∑ (1
j
uk _ wik )=xk
=
This concurs with the result obtained by Mendel [10, Page 226] using the minimum t-norm. The result is also, trivially, true for the product t-norm.
j =1 i=1
j
Using maximum as the t-conorm then the maximum of u k _ wik ˜ iv ˜ iv ˜ iv j ˜ iv is rkA _ rkB . The minimum of u k _ wik has to be lkA _ lkB ˜ iv j j ˜ iv since all uk lkA and wk lkB . Therefore the secondary membership function for x k is an interval set with domain ˜ iv ˜ iv ˜ iv ˜iv [(lkA _ lkB ),(rkA _ rkB )]. Since Equation 14 is essentially the union of all the x k terms then we have the join of the IVFS as N
∑ [(lkA
˜ iv
˜ iv
_ lkB
A˜ iv ); (rk
˜ iv
_ rkB
)]=xk
(15)
k=1
This concurs with the result obtained by Mendel [10, Page 225].
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IV. C ONCLUSION Interval valued type-2 fuzzy sets are computationally less expensive than other type-2 fuzzy sets and have been proven to be useful in a number of applications. This work considers IVFS from a different perspective. In particular, this paper has presented the notion of embedded interval valued type-2 fuzzy sets. They have been used to prove the join and meet of interval valued type-2 fuzzy sets under certain t-norms and t-conorms. This paper has, again, shown the usefulness of the Representation Theorem for embedded type-2 fuzzy sets. It is anticipated that this theorem will be used in future work to help our understanding of type-2 fuzzy logic theory.
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