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Approximate evaluations based on aggregation functions
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Fuzzy Sets and Systems 220 (2013) 34 – 52 www.elsevier.com/locate/fss
Approximate evaluations based on aggregation functions Slavka Bodjanovaa,∗ , Martin Kalinab a Department of Mathematics, Texas A&M University-Kingsville, MSC 172, Kingsville, TX 78363, USA b Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 81368 Bratislava, Slovakia
Received 29 January 2012; received in revised form 5 June 2012; accepted 25 July 2012 Available online 4 August 2012
Abstract Aggregation of real-valued evaluations of elements x,y from a set U is investigated in order to find an approximate evaluation of the composition of x and y performed by a binary operation defined on U. The notions of a lower bound and an upper bound of an evaluation function with respect to a binary operation are introduced and illustrated with examples. Approximate evaluations of composite concepts described by the union of granules of a finite fuzzy set are studied in detail. © 2012 Elsevier B.V. All rights reserved. Keywords: Evaluation; Aggregation functions; Triangular norms; Granules of a fuzzy set; Fuzzy relations; Approximation; Rough sets
1. Introduction The term “evaluation” is very broad and context dependent. To evaluate an element numerically means to assign a real number to it. This problem has been studied widely by many researchers, especially in the area of soft computing, where elements are described by fuzzy sets [5,8,9,15]. We assume a nonempty set U equipped with a binary operation ∗. The main goal of our study is to investigate how evaluations of elements x and y from U can be used to approximate evaluation of x ∗ y. We consider an evaluation function on U and we propose a lower and an upper bound of (x ∗ y) based on aggregation of (x) and (y) by an appropriate aggregation function [5,6,12]. When evaluation functions are aggregation functions and their boundaries are created by a t-norm T or a t-conorm S, we obtain a T–S-characterization of aggregation functions with respect to ∗. The T–S-characterizations of frequently used aggregation functions (e.g., triangular norms themselves, the arithmetic mean, the geometric mean, etc.) with respect to the union of finite sets will be provided. Knowing the boundaries of (x ∗ y) is beneficial in many areas of applications, especially in approximate reasoning and decision making. It allows researchers to identify elements whose composition will have a particular numerical evaluation within (or out of) given thresholds without performing the actual composition. We apply our theory to approximate evaluations of the union of granules of a finite fuzzy set and to approximate evaluations of the union of granules of a fuzzy relation between two finite universes. It is well known that a vague concept defined on a finite universal set U can be described by a fuzzy set f : U → [0, 1] [8,15]. A set Y ∈ 2U ∗ Corresponding author. Tel.: +1 361 593 3517.
E-mail addresses: [email protected] (S. Bodjanova), [email protected] (M. Kalina). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.07.014
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induces the granule f Y of f. A numerical evaluation of f Y can be interpreted as the degree to which the collection of elements in Y satisfies the vague concept described by f. A fuzzy set R defined on the Cartesian product of two finite universes U × W is referred to as a binary fuzzy relation from U to W [7,10,11,17]. A pair of sets (X, Y ) ∈ 2U × 2W induces the granule R X Y of R. An appropriate numerical evaluation of R X Y provides information about the strength of the relationship of X to Y. This contribution is a continuation and an extension of our results from [3,2]. It is organized as follows. In Section 2 we present the basic definitions, notations and properties of evaluation functions. Section 3 is devoted to evaluation and approximate evaluation of granules of a finite fuzzy set. In Section 4 we focus on evaluation functions based on aggregation functions. Evaluation and approximate evaluation of granules of a fuzzy relation are studied in Section 5. Concluding remarks are in Section 6. 2. Evaluation functions Throughout this paper we will only deal with quantitative evaluations (real numbers) from the unit interval. Definition 1. Let U be a nonempty set. Then a mapping : U → [0, 1] will be called an evaluation function on U. Further properties of an evaluation function depend on the structure of U and on the purpose of evaluation. For example, if U is partially ordered by a relation , we say that is nondecreasing, if for all x, y ∈ U , x y ⇒ (x) ⱕ (y), and is nonincreasing if for all x, y ∈ U , x y ⇒ (x) ⱖ (y). If ⊥ and are the least and the greatest elements of U, respectively, and (⊥) = 0, while () = 1, we say that is bounded. In [3] we introduced the notions of T- and S-evaluators as follows. Definition 2. Consider a lattice (L , , ⊥, ). A function : L → [0, 1] is said to be an evaluator on L if and only if it satisfies the following properties: • (⊥) = 0, () = 1, • for all x, y ∈ L : x y ⇒ (x) ⱕ (y). For a t-norm T and a t-conorm S, an evaluator on L is called a T-evaluator if and only if for all x, y ∈ L T ((x), (y)) ⱕ (x ∧ y),
(1)
and it is called an S-evaluator if and only if S((x), (y)) ⱖ (x ∨ y).
(2)
Recall that the lattice operations x ∧ y = inf{x, y} and x ∨ y = sup{x, y} are known also as the meet and the join, respectively. Further in this paper we will consider the four basic t-norms, i.e., TM (x, y) = min{x, y} (the minimum), TP (x, y) = x · y (the product), TL (x, y) = max{x + y − 1, 0} (the Łukasiewicz t-norm) and the drastic product 0 if max{x, y} < 1, TD (x, y) = min{x, y} if max{x, y} = 1. The basic t-conorms (dual of the four basic t-conorms) are: S M (x, y) = max{x, y} (the maximum), S P (x, y) = x + y − x · y (the probabilistic sum), SL (x, y) = min{x + y, 1} (the Łukasiewicz t-conorm) and the drastic sum 1 if min{x, y} > 0, S D (x, y) = max{x, y} if min{x, y} = 0.
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t-Norms and t-conorms [14,13] are special types of aggregation functions. Let us recall that an aggregation function [5,6,12] is a function A : n∈N [0, 1]n → [0, 1], where N denotes the set of all positive integers, such that A(x1 , . . . , xn ) ⱕ A(y1 , . . . , yn ) whenever xi ⱕ yi for all i = 1, . . . , n, A(x) = x for all x ∈ [0, 1], A(0, . . . , 0) = 0, and A(1, . . . , 1) = 1. Now we will generalize the statements (1) and (2) from Definition 2. Definition 3. Consider an aggregation function A and an evaluation function on a nonempty set U. Let ∗ be a binary operation on U. We say that has a lower A-bound with respect to ∗ if and only if for all x, y ∈ U A((x), (y)) ⱕ (x ∗ y),
(3)
and it has an upper A-bound with respect to ∗ if and only if A((x), (y)) ⱖ (x ∗ y).
(4)
Based on the definition above, a T-evaluator on L is a nondecreasing bounded evaluation function which has a lower T-bound with respect to the meet of elements from L. On the other hand, an S-evaluator on L is a nondecreasing bounded evaluation function which has an upper S-bound with respect to the join of elements from L. We will focus on boundaries created by t-norms and t-conorms. For the best approximation of (x ∗ y) we want to identify the greatest lower bound and the smallest upper bound. We say that an aggregation function A is stronger than and aggregation function B with the notation B ⱕ A, if B(x, y) ⱕ A(x, y) for all (x, y) ∈ [0, 1]2 . It is well known that TD ⱕ TL ⱕ TP ⱕ TM and TD ⱕ T ⱕ TM for any t-norm T. Analogously, S D ⱖ SL ⱖ S P ⱖ S M and S D ⱖ S ⱖ S M for any t-conorm S. Then • • • •
if has a lower TM -bound then it has a lower T-bound for any t-norm T, if has an upper S M -bound then it has an upper S-bound for any t-conorm S, if does not have a lower TD -bound then a lower T-bound of does not exist, if does not have an upper S D -bound then an upper S-bound of does not exist.
Lemma 1. Let U be a partially ordered set and for all x, y ∈ U, x ∨ y = sup{x, y} and x ∧ y = inf{x, y}. Then each nondecreasing evaluation function on U has a lower S M -bound with respect to ∨ and an upper TM -bound with respect to ∧. Each nonincreasing evaluation function on U has a lower S M -bound with respect to ∧ and an upper TM -bound with respect to ∨. Corollary 1. If is a T-evaluator on a lattice L then for all x, y ∈ L T ((x), (y)) ⱕ (x ∧ y) ⱕ TM ((x), (y)).
(5)
If is an S-evaluator on L then for all x, y ∈ L S M ((x), (y)) ⱕ (x ∨ y) ⱕ S((x), (y)).
(6)
Lemma 2. Assume that an evaluation function on a lattice L has the valuation property, i.e., for all x, y ∈ L (x ∨ y) = (x) + (y) − (x ∧ y). Then has an upper SL -bound and a lower TL -bound with respect to ∨ and also with respect to ∧. In applications, only the nontrivial bounds (i.e., the upper bound less than 1 and the lower bound more than 0) provide an useful information. However, some combinations of aggregation functions do not yield both nontrivial boundaries. Lemma 3. Let has an upper SL -bound and a lower TL -bound with respect to a binary operation ∗ defined on U. Then either the upper bound is equal to 1 or the lower bound is equal to 0. Because a lower T-bound (an upper S-bound) of exists only if has a lower TD -bound (an upper S D -bound), checking the existence of TD and S D boundaries is an important starting point in searching for boundaries of .
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Lemma 4. An evaluation function has an upper S D -bound with respect to a binary operation ∗ defined on U if an only if for all x, y ∈ U (x) = 0 ⇒ (x ∗ y) ⱕ (y),
(7)
and has a lower TD -bound with respect to ∗ if and only if for all x, y ∈ U (x) = 1 ⇒ (x ∗ y) ⱖ (y).
(8)
In general, we may have evaluation functions whose both boundaries are based either on t-norms or on t-conorms. Clearly, if an evaluation function has a lower S-bound than its upper bound, if it exists, is an S1 -bound such that S1 ⱖ S. Analogously, if an evaluation function has an upper T-bound, its lower bound, if it exists, is a T1 -bound such that T1 ⱕ T . The diversity of evaluation functions is very broad. In the next section we will investigate boundaries of some evaluation functions of granules of finite fuzzy sets. 3. Evaluation of fuzzy set granules A vague concept defined on a finite universal set U can be described by a fuzzy set f : U → [0, 1] [8,15]. Note that a fuzzy set itself is an evaluation function on U, where f (x) evaluates the degree to which an element x from U satisfies the concept described by f. We will use the notation F(U ) for the family of all fuzzy sets defined on U. A set Y ∈ 2U induces the granule f Y of f, which is a fuzzy set from F(Y ) given for all x ∈ Y by f Y (x) = f (x).
(9)
We will denote the family of all granules of a fuzzy set f ∈ F(U ) by G( f ). Hence G( f ) = { f Y : Y ∈ 2U }.
(10)
Obviously, fU = f and the granule of f induced by the empty set is the empty set. For f X , f Y ∈ G( f ) we say that f X is included in f Y , denoted by f X ⊂ f Y , if and only if X ⊂ Y . Analogously, the union and the intersection of granules are defined by f X ∪ f Y = f X ∪Y
(11)
f x ∩ f Y = f X ∩Y ,
(12)
and
respectively. An appropriate numerical evaluation of f Y can be interpreted as the degree to which the collection of elements in Y satisfies the vague concept described by f. From a practical point of view only evaluations of f Y where Y ∅ are useful. We will denote by 2˜ U the family of all nonempty subsets from U. Various evaluation functions are used in different areas of application. We will explore just a few of them. A natural way of evaluation of a fuzzy set granule is an aggregation of its membership grades. Then, using an aggregation function A, we define evaluation function A as follows: for all f Y ∈ G( f ), Y ∈ 2˜ U A ( f Y ) = A( f (x), x ∈ Y ).
(13)
An evaluation function on G( f ) can also be based on evaluation of -level sets of f, i.e., crisp sets f = {x ∈ U : f (x) ⱖ },
(14)
∈ (0, 1], by a fuzzy measure m on 2U . Recall that m is a nondecreasing and bounded evaluation function on 2U . Then, for f Y ∈ G( f ), Y ∈ 2˜ U we define m, ( f Y ) =
m( f ∩ Y ) . m(Y )
(15)
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When | · | denotes the cardinality of a set, formula (15) can be rewritten as |·|, ( f Y ) =
| f ∩ Y | |Y |
(16)
and one may interpret it as a degree of inclusion of Y in f . Many evaluation functions of f Y are based on an evaluation of the individual membership grades of f by nonnegative real numbers. Assume a mapping h : [0, 1] → [0, b], where b is a positive real number and Y ∅. Then ⎧ ⎪ ⎨ x∈Y h(x) if h(x)0, (17) h ( f Y ) = x∈U x∈U h(x) ⎪ ⎩ 0 otherwise. For example, if h(x) = −x ln(x) − (1 − x) ln(1 − x) with the convention h(0) = 0, then h ( f Y ) evaluates the relative fuzziness of f Y . If h(x) = x, then h ( f Y ) evaluates the relative cardinality of f Y . If we want to evaluate the range of membership grades of granule f Y , we can create an evaluation function which compares the extreme values of f (x), x ∈ Y . For example, 1 ( f Y ) = max f (x) − min f (x)
(18)
⎧ ⎨ min x∈Y f (x) if max f (x) > 0, x∈Y 2 ( f Y ) = maxx∈Y f (x) ⎩ 1 otherwise.
(19)
x∈Y
or
x∈Y
Now we will examine boundaries of evaluation functions presented above with respect to the union of granules of a finite fuzzy set. Proposition 1. Let m be a probability measure on 2U . Then for all f ∈ F(U ) and all ∈ (0, 1] evaluation function m, given by (15) has an upper S P -bound and a lower TP -bound with respect to the union of granules from G( f ). Proof. First we show that m, has a lower TP -bound with respect to the union of granules from G( f ). Let us take an arbitrary couple of granules f X and f Y and choose ∈ (0, 1]. Using the notation n 1 = m(X \ f ∩ X ), n 2 = m( f ∩ X ), n 3 = m(Y \ f ∩ Y ), n 4 = m( f ∩ Y ),
(20)
we obtain that m, ( f X ) = n 2 /(n 1 + n 2 ), m, ( f Y ) = n 4 /(n 3 + n 4 ). Let us denote m, ( f X ∪Y ) = M /M. Obviously there exist nonnegative constants 1 , 2 such that 1 + 2 = n 4 and M = n 2 + 1 .
(21)
Clearly, n 1 + M ≤ M ≤ n 1 + n 3 + M .
(22)
Assume that n 4 ≤ n 2 . Since for all a, b, c > 0, a ≤ b we have a a+c ≤ , b b+c
(23)
we get a lower bound and an upper bound of m, ( f X ∪Y ) by (24) and (25), respectively. M n 2 + 1 n2 ≥ ≥ , M n 1 + n 3 + n 2 + 1 n1 + n2 + n3
(24)
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M n 2 + 1 + 2 n 2 + 1 ≤ ≤ M n 1 + n 2 + 1 n 1 + n 2 + 1 + 2 n2 + n4 ≤ . n1 + n2 + n4
39
(25)
Now we will show that m, ( f X ∪Y ) ≥ m, ( f X ) · m, ( f Y ). Assume that n 4 ≤ n 2 , i.e., n 4 = n 2 + c, c ≥ 0. Then n4 + c n4 n2 ≥ = n3 + n4 + c n3 + n2 n3 + n4 and for m, ( f X ) · m, ( f Y ) we have n4 n2 m, ( f X ) · m, ( f Y ) = n1 + n2 n4 + n3 n2 n2 ≤ n1 + n2 n2 + n3 =
n 22 . (n 1 + n 2 )(n 2 + n 3 )
(26)
Suppose that n 1 + n 2 + n 3 = 1, since this is just a matter of normalization. Then, using (24) and (26) it is enough to show that n2 ≥
n 22 , (n 1 + n 2 )(n 2 + n 3 )
or equivalently (n 1 + n 2 )(n 2 + n 3 ) ≥ n 2 .
(27)
At the left-hand-side we have n 1 n 2 + n 22 + n 1 n 3 + n 2 n 3 = n 2 (n 1 + n 2 + n 3 ) +n 1 n 3 ,
=1
and hence the inequality (27) is valid. Now, we show that m, has an upper S P -bound with respect to the union of granules from G( f ). We take an arbitrary couple of granules f X and f Y and we keep that notation (20). Then the following yields: S P (m, ( f X ), m, ( f Y )) =
n2 n4 n2n4 + − . n1 + n2 n3 + n4 (n 1 + n 2 )(n 3 + n 4 )
Assume that n 1 ≥ n 3 . Then n4 n4 ≤ n1 + n4 n3 + n4 and the monotonicity of S P implies S P (m, ( f X ), m, ( f Y )) ≥
n2 n4 n2n4 . + − n1 + n2 n1 + n4 (n 1 + n 2 )(n 1 + n 4 )
(28)
Using (25) and (28) it is enough to show that n2 + n4 n2 n4 n2n4 . ≤ + − n1 + n2 + n4 n1 + n2 n3 + n4 (n 1 + n 2 )(n 1 + n 4 )
(29)
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As a matter of normalization we will assume that n 1 + n 2 + n 4 = 1. Then inequality (29) can be reformulated into n2 + n4 ≤
n 2 (n 1 + n 4 ) + n 4 (n 1 + n 2 ) − n 2 n 4 , (n 1 + n 2 )(n 1 + n 4 )
or equivalently (n 2 + n 4 )(n 1 + n 2 )(n 1 + n 4 ) ≤ n 1 n 2 + n 2 n 4 + n 1 n 4 .
(30)
At the left-hand-side of (30) we have the following expression: n 1 n 2 (n 1 + n 2 + n 4 ) +n 2 n 4 (n 1 + n 2 + n 4 ) +n 1 n 4 (n 1 + n 4 ),
=1
=1
≤1
which proves inequality (30). The next example shows that m, has neither the lower TM -bound nor the upper S M -bound with respect to the union of granules from G( f ). Example 1. Let X = {x1 , x2 }, Y = {x2 , x3 }. Assume f ∈ F(U ) and ∈ (0, 1] such that f (x1 ) = f (x3 ) = 0 and f (x2 ) = 1. Then by formula (16) we get |·|, ( f X ) = |·|, ( f Y ) =
1 2
and |·|, ( f X ∪Y ) = 13 .
However, |·|, ( f X ∪Y ) < min{|·|, ( f X ), |·|, ( f Y )}, therefore |·|, does not have a lower TM -bound. Now assume g ∈ F(U ) and ∈ (0, 1] such that g (x1 ) = g (x3 ) = 1 and f (x2 ) = 0. Then |·|, (g X ) = |·|, (gY ) =
1 2
and |·|, (g X ∪Y ) = 23 .
However, |·|, (g X ∪Y ) > max{|·|, (g X ), |·|, (gY )}, therefore |·|, does not have an upper S M -bound. When we consider only disjoint granules of a fuzzy set, we get the following result. Proposition 2. Let m be a probability measure on 2U . Then for all f ∈ F(U ) and all ∈ (0, 1] evaluation function m, has an upper S M -bound and a lower TM -bound with respect to the union of disjoint granules from G( f ). Proof. We will use the notations from the proof of Proposition 1. Assume that, e.g., m, ( f X ) ⱕ m, ( f Y ). Then, because X ∩ Y = ∅, n2 + n4 m, ( f X ∪Y ) = n1 + n2 + n3 + n4 =
(n 1 + n 2 )m, ( f X ) + (n 3 + n 4 )m, ( f Y ) n1 + n2 + n3 + n4
≤
(n 1 + n 2 )m, ( f Y ) + (n 3 + n 4 )m, ( f Y ) n1 + n2 + n3 + n4
= m, ( f Y ) = max{m, ( f Y ), m, ( f X )}.
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On the other hand, m, ( f X ∪Y ) =
(n 1 + n 2 )m, ( f X ) + (n 3 + n 4 )m, ( f Y ) n1 + n2 + n3 + n4
ⱖ
(n 1 + n 2 )m, ( f X ) + (n 3 + n 4 )m, ( f X ) n1 + n2 + n3 + n4
= m, ( f X ) = min{m, ( f Y ), m, ( f X )}.
Proposition 3. For an arbitrary f ∈ F(U ), with respect to the union of granules from G( f ), • the evaluation function h given by (17) has a lower S M -bound and an upper SL -bound, • the evaluation function 1 given by (54) has a lower S M -bound and it does not have an upper S-bound for any t-conorm S, • the evaluation function 2 given by (19) has an upper TM -bound and it does not have a lower T-bound for any t-norm T. Proof. (i) Evaluation function h is increasing and it has the valuation property. Therefore, according to Lemma 1 it has a lower S M -bound and according to Lemma 2 it has an upper SL -bound. (ii) Evaluation function 1 is increasing and therefore it has a lower S M -bound. In Example 2 we will show that 1 does not have an upper S D -bound, and therefore it does not have an upper S-bound for any t-conorm S. (iii) Evaluation function 2 is nonincreasing and therefore it has an upper TM -bound. In Example 2 we will show that 2 does not have a lower TD -bound, and therefore it does not have a lower T-bound for any t-norm T. Example 2. Let X = {x1 , x2 } and Y = {x3 , x4 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 ) = f (x2 ) = 0.6 and f (x3 ) = f (x4 ) = 0.4. Then 1 ( f X ) = 1 ( f Y ) = 0 and 1 ( f X ∪Y ) = 0.6 − 0.4 = 0.2. Clearly, 0.2 > max{0, 0}, and therefore 1 does not have an upper S D -bound. 2 When we use the evaluation function 2 , we obtain that 2 ( f X ) = 2 ( f Y ) = 1 and 2 (FX ∪Y ) = 0.4 0.6 = 3 . However, 2 3 < min{1, 1} and hence 2 does not have a lower TD -bound. Boundaries of the evaluation function A given by (13) will be studied in the next section. 4. Evaluation functions based on aggregation functions First of all we will examine evaluation functions based on t-norms and t-conorms. The next proposition is obvious. Proposition 4. When A is an arbitrary t-norm T, the evaluation function A given by (13) has an upper TM -bound and a lower T-bound with respect to the union of granules from G( f ). When A is an arbitrary t-conorm S, then A has an upper S-bound and a lower S M -bound with respect to the union of granules from G( f ). Now we will analyze evaluation of f X ∈ G( f ) by the arithmetic mean M. Hence f (x) M ( f X ) = M( f (x), x ∈ X ) = x∈X . |X |
(31)
Proposition 5. Evaluation function M has a lower TL -bound and an upper SL -bound with respect to the union of granules from G( f ), f ∈ F(U ). Proof. For X, Y ∈ 2U we have that
f (x) ⱕ f (x) + f (x) x∈X ∪Y
x∈X
x∈Y
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and |X ∪ Y | ⱖ max{|X |, |Y |}. Therefore f (x) f (x) x∈X ∪Y f (x) M( f X ∪Y ) = ⱕ x∈X + x∈Y |X ∪ Y | |X ∪ Y | |X ∪ Y | f (x) x∈X f (x) ⱕ + x∈Y |X | |Y | ⱕ min{M( f X ) + M( f Y ), 1} = SL (M( f X ), M( f Y ). Now denote n 1 = |X \ X ∩ Y |, n 2 = |X ∩ Y | and n 3 = |Y \ X ∩ Y |. Obviously, n 1 ⱖ n 1 M( f Y ), n 3 ⱖ n 3 M( f X ) and n 2 ⱖ X ∩Y f (x). Therefore
n 1 + n 2 + n 3 ⱖ n 1 M( f Y ) + f (x) + n 3 M( f X ). (32) X ∩Y
We will add to both sides of (32) the expression
f (x) = f (x) + f (x) + f (x). x∈X ∪Y
x∈X \X ∩Y
x∈Y \X ∩Y
x∈X ∩Y
Then on the left side of (32) we obtain
f (x) = (n 1 + n 2 + n 3 )(1 + M( f X ∪Y )) n1 + n2 + n3 +
(33)
x∈X ∪Y
and on the right side of (32) we have
n 1 M( f Y ) + n 3 M( f X ) + f (x) + X ∩Y
= n 1 M( f Y ) + n 3 M( f X ) +
x∈X
x∈X \X ∩Y
f (x) +
f (x) +
f (x) +
x∈Y \X ∩Y
f (x)
x∈X ∩Y
f (x)
x∈Y
= n 1 M( f Y ) + n 3 M( f X ) + (n 1 + n 2 )M( f X ) + (n 3 + n 2 )M( f Y ) = (n 1 + n 2 + n 3 )M( f X ) + (n 1 + n 2 + n 3 )M( f Y ).
(34)
After dividing (33) and (34) by (n 1 + n 2 + n 3 ) we obtain that M( f X ∪Y ) + 1 ⱖ M( f X ) + M( f Y ), and therefore M( f X ∪Y ) ⱖ max{M( f X ) + M( f Y ) − 1, 0} = TL (M( f X ), M( f Y )).
The next example shows that M has neither an upper S M -bound nor a lower TM -bound with respect to the union of fuzzy set granules. Example 3. Let X = {x1 , x2 } and Y = {x2 , x3 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 = f (x3 ) = 1 and f (x2 ) = 0.9. Then M ( f X ∪Y ) = (1 + 0.9 + 1)/3 = 0.967, M ( f X ) = M ( f Y ) = (1 + 0.9)/2 = 0.95. Because 0.967 > max{0.95, 0.95}, evaluation function M does not have an upper S M -bound. Now let g(x1 ) = g(x3 ) = 0.1 and g(x2 ) = 0.2. Then M ( f X ∪Y ) = (0.1 + 0.2 + 0.1)/3 = 0.13, M ( f X ) = M ( f Y ) = (0.1 + 0.2)/2 = 0.15. Because 0.13 < min{0.15, 0.15}, evaluation function M does not have a lower TM -bound. However, if we consider only disjoint granules, we have the following result. Proposition 6. Evaluation function M has a lower TM -bound and an upper S M -bound with respect to the union of disjoint granules from G( f ), f ∈ F(U ).
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Proof. Consider X, Y ∈ 2˜ U such that X ∩ Y = ∅. Let n 1 = |X | and n 2 = |Y |. Assume that M( f X ) ⱕ M( f Y ). Then n 1 M( f X ) + n 2 M( f Y ) M( f X ∪Y ) = n1 + n2 ≤
n 1 M( f Y ) + n 2 M( f Y ) n1 + n2
= M( f Y ) = SM (M( f X ), M( f Y )). On the other hand, n 1 M( f X ) + n 2 M( f Y ) M( f X ∪Y ) = n1 + n2 ≥
n 1 M( f X ) + n 2 M( f X ) n1 + n2
= M( f X ) = TM (M( f X ), M( f Y )).
The arithmetic mean can be used to aggregate outputs of other aggregation functions. We will analyze evaluation function min x∈X f (x) + maxx∈X f (x) 3 ( f X ) = 2 = M(TM ( f (x), x ∈ X ), S M ( f (x), x ∈ X )). (35) Proposition 7. Evaluation function 3 given by (35) has an upper S M -bound and a lower TM -bound with respect to the union of granules from G( f ), f ∈ F(U ). Proof. For X, Y ∈ 2˜ U and a mapping f : U → [0, 1], let min x∈X f (x) = a, min x∈Y f (x) = b, maxx∈X f (x) = c and maxx∈Y f (x) = d. Then a ⱕ c, b ⱕ d and we have that K( f X ) =
a+c b+d and K( f Y ) = . 2 2
We will show that max{K( f X ), K( f Y )} ⱖ K( f X ∪Y )
(36)
min{K( f X ), K( f Y )} ⱕ K( f X ∪Y ).
(37)
and
The following four cases may occur: Case 1: If a ⱕ b and d ⱕ c then a ⱕ b ⱕ d ⱕ c and we obtain a+c b+d a+c max , ⱖ = K( f X ∪Y ) 2 2 2 a+c b+d , . ⱖ min 2 2 Case 2: If a ⱕ b and c ⱕ d then a+c b+d b+d a+d max , = > 2 2 2 2
a+c b+d , = K( f X ∪Y ) ⱖ min 2 2
=
a+c . 2
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Case 3: If b ⱕ a and c ⱕ d then b ⱕ a ⱕ c ⱕ d and b+d a+d a+c b+d , = > = K( f X ∪Y ) max 2 2 2 2 a+c b+d , . ⱖ min 2 2 Case 4: If b ⱕ a and d ⱕ c then a+c b+d a+c b+c max , = ⱖ 2 2 2 2
a+c b+d , = K( f X ∪Y ) ⱖ min 2 2
=
b+d . 2
Boundaries of the evaluation function G based on the geometric mean defined by G ( f X ) = G( f (x), x ∈ X ) = (x∈X f (x))1/|X |
(38)
are described in Proposition 8. Proposition 8. Evaluation function G given by (38) has a lower T p -bound and an upper S D -bound with respect to the union of granules from G( f ), f ∈ F(U ). U we have that |X − X ∩ Y | = n , |X ∩ Y | = n and |Y − Y ∩ X | = n . Denote P = Proof. Let for X, Y ∈ 2˜ 1 2 3 1 f (x), P = f (x) and P = f (x). Then 2 3 x∈X −X ∩Y x∈X ∩Y x∈Y −X ∩Y
G( f X ∪Y ) = (P1 P2 P3 )1/(n 1 +n 2 +n 3 ) ⱖ (P1 P2 P2 P3 )1/(n 1 +n 2 +n 3 ) = (P1 P2 )1/(n 1 +n 2 +n 3 ) (P3 P2 )1/(n 1 +n 2 +n 3 ) ⱖ (P1 P2 )1/(n 1 +n 2 ) (P3 P2 )1/(n 2 +n 3 ) = T p (G( f X ), G( f Y )). If G( f X ) = 0 then there exists x ∈ X such that f (x) = 0. Therefore, for any Y ∈ 2˜ U G( f (x), x ∈ X ∪ Y ) = 0 ⱕ G( f (x), x ∈ Y ). Based on Lemma 4 we conclude that G has an upper SD -bound with respect to the union of nonempty finite sets. Recall that TP ⱕ TM and SL ⱕ S D . We will show that G has neither the lower TM -bound nor the upper SL -bound. Example 4. Let X = {x1 , x2 } and Y = {x1 , x3 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 ) = 1, f (x2 ) = 0.2 and f (x3 ) = 0.3. Then G ( f X ∪Y ) = (1(0.2)(0.3))1/3 = 0.391, G ( f X ) = (1(0.2))1/2 = 0.447 and G ( f Y ) = (1(0.3))1/2 = 0.547. However, 0.391 < min{0.447, 0.547} and therefore G does not have a lower TM -bound. Now let X = {x1 , x2 , x3 }, Y = {x2 , x3 , x4 }, f (x1 ) = 0.9, f (x2 ) = f (x3 ) = 0.01 and f (x4 ) = 0.8. Then G ( f X ∪Y ) = (0.9(0.01)(0.01)(0.8))1/4 = 0.092, G ( f X ) = (0.9(0.01)(0.01)1/3 = 0.0448, G ( f Y ) = (0.01(0.01)(0.8))1/3 = 0.043. However, 0.092 > min{0.0448+0.043, 1} = SL (G ( f X ), G ( f Y )) and therefore G does not have an upper SL -bound.
S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
45
Finally, we consider evaluation function med,c defined by med,c ( f X ) = Ac ( f (x), x ∈ X )
= med(0, c + ( f (x) − c), 1),
(39)
x∈X
where c ∈ [0, 1] and med stands for the median. Proposition 9. Evaluation function med,c given by (39) has neither an upper S-bound, nor a lower T-bound with respect to the union of granules from G( f ), f ∈ F(U ). Example 5. Let X = {x1 , x2 , x3 , x4 } and Y = {x2 , x3 , x4 , x5 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 ) = 0.5, f (x2 ) = f (x3 ) = f (x4 ) = 0 and f (x5 ) = 0.7. Choose c = 0.2. Then med,c ( f X ∪Y ) = med(0, 0.4, 1) = 0.4, med,c ( f X ) = med(0, −0.1, 1) = 0, med,c ( f Y ) = med(0, 0.1, 1) = 0.1. Because 0.4 > max{0, 0.1}, evaluation function med,c does not have an upper S D -bound. Now let X = {x1 , x2 , x3 }, Y = {x2 , x3 , x4 }, f (x1 ) = 0.4, f (x2 ) = f (x3 ) = 0.8 and f (x4 ) = 0.2. Choose c = 0.5. Then med,c ( f X ∪Y ) = med(0, 0.7, 1) = 0.7, med,c ( f X ) = med(0, 1, 1) = 1, med,c ( f Y ) = med(0, 0.8, 1) = 0.8. Because 0.7 < min{1, 0.8}, evaluation function med,c does not have a lower TD -bound. When we restrict evaluation functions only to aggregation functions, we can reformulate Definition 3 and introduce boundaries of aggregation functions as follows. Definition 4. Consider aggregation functions A and B. Let U be a nonempty finite set and ∗ be a binary operation on U. We say that A has a lower B-bound with respect to ∗ if and only if for all X, Y ∈ 2˜ U such that X ∗ Y ∅ and for all f : U → [0, 1] B(A( f (x), x ∈ X ), A( f (x), x ∈ Y )) ⱕ A( f (x), x ∈ X ∗ Y ).
(40)
We say that A has an upper B-bound with respect to ∗ if and only if B(A( f (x), x ∈ X ), A( f (x), x ∈ Y )) ⱖ A( f (x), x ∈ X ∗ Y ).
(41)
Boundaries of aggregation functions created by t-norms and/or t-conorms provide a T–S-characterization of aggregation functions with respect to ∗. The T–S-characterizations of aggregation functions mentioned in this paper with respect to the union of finite sets are presented in Table 1. Table 1 T–S-characterizations of selected aggregation functions. Function
Lower bound
Upper bound
T S M G M(TM , S M ) Ac
T SM TL TP TM None
TM S SL SD SM None
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5. Evaluation of granules of a fuzzy relation 5.1. Evaluations based on aggregation functions Let U and W be finite nonempty universes. A fuzzy set R ∈ F(U × W ) is referred to as binary fuzzy relation from U to W. The value of R(x, y) gives the strength of the relationship of x ∈ U to y ∈ W . A pair of sets (X, Y ) ∈ U × W induces granule R X Y of R, which is a fuzzy relation from U/ X to W/Y defined for all (x, y) ∈ X × Y by R X Y (x, y) = R(x, y).
(42)
An appropriate evaluation of R X Y provides information about the strength of the relationship of X to Y. Definition 5 (Bodjanova [1]). Let R be a fuzzy relation from U to W and A and B be aggregation functions. Then the U W U W AB-extension of R is a mapping AB R : 2 × 2 → [0, 1] defined for all (X, Y ) ∈ 2 × 2 by ⎧ ⎪ ⎨ Ax∈X (B y∈Y (R(x, y))) AB if X ∅, Y ∅, R (X, Y ) = (43) ⎪ ⎩ 0 otherwise. The value AB R (X, Y ) can be interpreted as the degree to which X is R-related to Y with respect to the evaluation based on A and B, or simply the degree to which X is RAB -related to Y. Lemma 5. For all (X, Y ) ∈ U × W (x, y) ∈ U × W , (P1) if X = {x} and Y = {y} then AB R (X, Y ) = R(x, y), (P2) if R(x, y) = 1 for all (x, y) ∈ X × Y then AB R (X, Y ) = 1, (P3) if R(x, y) = 0 for all (x, y) ∈ X × Y then AB R (X, Y ) = 0. Example 6. Consider the t-norm TM and the t-conorm SM . Then the TM SM -extension of R ∈ F(U × W ) is defined by TM SM R (X, Y ) = TM (SM (R(x, y), y ∈ Y ), x ∈ X ) = min max R(x, y) . (44) x∈X
y∈Y
Note that TRM SM (X, Y ) = means that each element x ∈ X is related to at least one element y ∈ Y to the degree R(x, y) ⱖ . The TM SM -extension of R should be used in situations where the strength of the relationship of X to Y is expected to decrease with an enlargement of X and to increase with an enlargement of Y. Example 7. The SM TM -extension of R is given by the formula SRM TM (X, Y ) = SM (TM (R(x, y), y ∈ Y ), x ∈ X ) = max min R(x, y) . x∈X
y∈Y
(45)
Then SRM TM (X, Y ) = means that there exist at least one element x ∈ X which is related to all elements y ∈ Y to the degree R(x, y) ⱖ . The SM TM -extension of R should be used in situations where the strength of the relationship of X to Y is expected to increase with an enlargement of X and to decrease with an enlargement of Y. Because in practice only granules R X Y created by nonempty sets are considered, further in this paper we assume that (X, Y ) ∈ 2˜ U × 2˜ W . Lemma 6. Consider a fuzzy relation R ∈ F(U × W ), a t-norm T and a t-conorm S. Then for all (X, Y ) ∈ 2˜ U × 2˜ W , all ∅Z ⊂ X and all ∅V ⊂ Y TS TS (1) TS R (X, V ) ⱕ R (X, Y ) ⱕ R (Z , Y ), ST ST (2) R (X, V ) ⱖ R (X, Y ) ⱖ ST R (Z , Y ),
S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
47
(3) SR (Z , V ) ⱕ SR (X, Y ), (4) TR (Z , V ) ⱖ TR (X, Y ). Clearly, boundaries of AB-extensions of a fuzzy relation R ∈ F(U × W ) with respect to a binary operation ∗ defined on U or W depend on the boundaries of aggregation functions A and B. Proposition 10. Let AB R be the AB-extension of R ∈ F(U × W ) and A has an upper P-bound and a lower Q-bound with respect to a binary operation ∗ defined on U. Then for all X, Z ∈ 2˜ U , X ∗ Z ∅ and all Y ∈ 2˜ W AB AB Q(AB R (X, Y ), R (Z , Y )) ⱕ R (X ∗ Z , Y )
(46)
AB AB AB R (X ∗ Z , Y ) ⱕ P( R (X, Y ), R (Z , Y )).
(47)
and
Proof. Define a fuzzy set g ∈ F(U ) such that g(x) = B y∈Y (R(x, y)) for all x ∈ U . Then Q(Ax∈X (g(x)), Ax∈Z (g(x))) ⱕ Ax∈X ∗Z (g(x))
(48)
Ax∈X ∗Z (g(x)) ⱕ P(Ax∈X (g(x)), Ax∈Z (g(x))).
(49)
and
AB Because Ax∈X (g(x)) = Ax∈X (B y∈Y (R(x, y))) = AB R (X, Y ) and Ax∈Z (g(x)) = Ax∈Z (B y∈Y (R(x, y))) = R (Z , Y ), inequality (48) proves (46) and inequality (49) proves (47).
According to Proposition 10, boundaries of AB R with respect to an arbitrary binary operation defined on U depend only on the boundaries of aggregation function A. However, in the case of a binary operation defined on W, dominance between boundaries of B and aggregation function A is an important factor. Let us recall that an aggregation function A dominates an aggregation function B, denoted by A?B, if for all [xi1 , . . . , xin ] ∈ [0, 1]n , i = 1, . . . , m, and m, n ∈ N A(B(x11 , . . . , x1n ), B(x21 , . . . , x2n ), . . . , B(xm1 , . . . , xmn )) ⱖ B(A(x11 , . . . , xm1 ), A(x12 , , . . . , xm2 ), . . . , A(x1n , . . . , xmn )).
(50)
See [16,20] for details. Proposition 11. Let AB R be the AB-extension of R ∈ F(U × W ). Assume that B has an upper P-bound with respect to a binary operation ∗ defined on W and that P dominates A. Then for all X ∈ 2˜ U and all Y, V ∈ 2˜ W , Y ∗ V ∅ AB AB AB R (X, Y ∗ V ) ⱕ P( R (X, Y ), R (X, V )).
(51)
Proof. Let X = {x1 , . . . , xn }. Then AB R (X, Y ∗ V ) = Ax∈X (B y∈Y ∗V (R(x, y))) ⱕ Ax∈X (P(B y∈Y (R(x, y)), B y∈V (R(x, y))) = A(P(B y∈Y (R(x1 , y)), B y∈V (R(x1 , y))), P(B y∈Y (R(x2 , y)), B y∈V (R(x2 , y))), . . . , (P(B y∈Y (R(xn , y)), B y∈V (R(xn , y))))), ⱕ P(A(B y∈Y (R(x1 , y)), . . . , B y∈Y (R(xn , y))), A(B y∈V (R(x1 , y)), . . . , B y∈V (R(xn , y)))), = P(Ax∈X B y∈Y (R(x, y)), Ax∈X B y∈V (R(x, y)) AB = P(AB R (X, Y ), R (X, V ).
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Analogously we can prove the next proposition. Proposition 12. Let AB R be the AB-extension of R ∈ F(U × W ). Assume that B has a lower Q-bound with respect to a binary operation ∗ defined on W and that Q is dominated by A. Then for all X ∈ 2˜ U and all Y, V ∈ 2˜ W ,Y ∗ V ∅ AB AB AB R (X, Y ∗ V ) ⱖ Q( R (X, Y ), R (X, V )).
(52)
Example 8. Let R ∈ F(U × W ). Because the arithmetic mean M has a lower TL -bound and an upper SL -bound with respect to the union of finite sets, referring to Proposition 10, for any aggregation function A, evaluation function MA R has a lower TL -bound and an upper SL -bound with respect to the union of sets from U. Because each t-norm T is dominated by TM and each t-conorm dominates SM , using Propositions 11 and 12 we conclude that SRM M has an upper SL -bound and TRM M has a lower TL -bound with respect to the union of sets from W. Now we will evaluate granule R X Y of R by counting proportions of elements from X which have evaluations of their R-neighborhoods on Y within the given thresholds. In the case of a fuzzy relation R, the R-neighborhood of x is fuzzy set r x ∈ F(W ) defined for all y ∈ W by r x (y) = R(x, y). We will use the notation r x /Y for R-neighborhood of x restricted to Y ⊂ W . Definition 6. Assume a fuzzy relation R ∈ F(U × W ), a mapping : [0, 1]n → [0, 1] for n ∈ N and ∈ [0, 1). Then the fuzzy relation R : 2˜ U × 2˜ W → [0, 1] defined for all (X, Y ) ∈ 2˜ U × 2˜ W by R (X, Y ) =
|x ∈ X : (r x /Y ) > | |X |
(53)
will be called the -extension of R. In many applications the relationship of x ∈ U to Y ∈ 2˜ W is considered meaningless if (r x /Y ) ⱕ . Note that R (X, Y ) is the proportion of elements from X which have the -evaluations of r x /Y greater than a given threshold . Gradual changes of parameter produce gradual evaluations of granules of R. Some examples of -evaluations are given below: ⎧ R(x, y) ⎪ ⎨ y∈Y if R(x, y) > 0 for some y ∈ Y, 1 (r x /Y ) = y∈W R(x, y) ⎪ ⎩ 0 otherwise, ⎧ ⎪ ⎨ max y∈Y R(x, y) if R(x, y) > 0 for some y ∈ Y, 2 (r x /Y ) = max y∈W R(x, y) ⎪ ⎩0 otherwise, 3 (r x /Y ) = A(R(x, y), y ∈ Y ),
(54)
(55) (56)
where A is an aggregation function. Observe that if A(R(x, y), y ∈ Y ) > 0 for all x ∈ X ∈ 2˜ U , and = 0 then 3 (X, Y ) = MA R (X, Y ), where M is the arithmetic mean. 5.2. Evaluations based on rough set approximation Let R be a crisp compatibility relation on U × W , i.e., for each x ∈ U there is y ∈ W such that R(x, y) = 1. Then, applying -evaluation from (54) we obtain 1 (r x /Y ) =
|Y ∩ r x | , |r x |
(57)
S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
49
which is the degree of inclusion of R-neighborhood of x in Y. From the point of view of rough sets theory, expression (57) is the rough membership function describing approximation of Y ∈ 2˜ W in the knowledge base (approximation space) created by R-neighborhoods of elements from U. Let us recall that in the classical rough set model introduced by Pawlak [18] we assume a nonempty universal set U and a crisp equivalence relation E ⊂ U × U . The pair (U, E) is called a knowledge base or an approximation space. Given an arbitrary set Z ⊂ U , it may not be possible to describe Z in (U, E) precisely. Instead, one may obtain a pair of lower and upper approximations of Z defined by Z E = {x ∈ U : [x] E ⊂ Z }
(58)
Z E = {x ∈ U : [x] E ∩ Z ∅},
(59)
and
where [x] E is the equivalence class containing x. The pair (Z E , Z E ) is called the rough set with a reference set Z. Pawlak’s rough set model may be extended by using an arbitrary binary relation R [24]. In the variable precision rough set model [25] associated with a crisp relation R on U × U and a parameter ∈ [0, 0.5) the lower and the upper approximations of Z ⊂ U are defined by |Z ∩ R(x)| ⱖ1 − (60) apr (Z ) = x ∈ U : |R(x)| and
|Z ∩ R(x)| apr (Z ) = x ∈ U V : > , |R(x)|
(61)
respectively, where R(x) = {y ∈ U : R(x, y) = 1}. Reviews of various rough set models can be found in [19,24]. Variable precision rough sets model can be extended to a crisp relation R between two different universes and we can obtain an interval-valued evaluation of granules of R. Definition 7. Let R ∈ U × W and ∈ [0, 0.5). Then the mapping R defined for all (X, Y ) ∈ 2˜ U × 2˜ W by a pair of coefficients x ∈ X : |Y ∩ r x | ⱖ 1 − |r x | R (X, Y ) = (62) |X | and
x ∈ X : |Y ∩ r x | > |r x | R (X, Y ) = |X |
(63)
will be called the probabilistic rough extension of R at level . Lemma 7. Let R be a crisp relation between U and W. Then for all (X, Y ) ∈ 2˜ U × 2˜ W and all ∈ [0, 0.5)
R (X, Y ) = 1 − R (X, Y c ), where Y c denote the complement of Y . Proof. Because x ∈ X : |Y ∩ r x | ⱖ 1 − = |X | − x ∈ X : |Y ∩ r x | < 1 − , |r x | |r x | we obtain that
x ∈ X : |Y ∩ r x | < 1 − |r x | R (X, Y ) = 1 − . |X |
(64)
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S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
Then |Y ∩ r x | |r x | − |Y c ∩ r x | |Y c ∩ r x | = =1− <1− |r x | |r x | |r x | is equivalent to |Y c ∩ r x | > . |r x | Therefore,
c x ∈ X : |Y ∩ r x | > |r x | R (X, Y ) = 1 − = 1 − R (X, Y c ). |X |
Using Lemma 7 we can say that the coefficient R (X, Y ) gives the proportion of elements from X which have at most 100 percent of their R-neighborhoods included in the complement of Y. Obviously, the coefficient R (X, Y ) is the proportion of elements from X which have more than 100 percent of their R-neighborhoods included in Y. In [4] we defined the rough extension of a fuzzy relation R ∈ F(U × W ) at a level ∈ (0, 1] using a rough set model over two universes [23,24] and -level-sets of R-neighborhoods of elements from U as follows. Definition 8. Let R ∈ F(U × W ) and ∈ (0, 1]. For each Y ∈ 2˜ W assume subsets from U R ,Y = {x ∈ U : (r x ) ⊂ Y } and R ,Y = {x ∈ U : (r x ) ∩ Y }.
(65)
Then the mapping R defined for all (X, Y ) ∈ 2˜ U × 2˜ W by a pair of coefficients R (X, Y ) =
|R ,Y | |R ,Y | and R (X, Y ) = |X | |X |
(66)
will be called the rough extension of R at level . Note that the rough extension of a fuzzy relation R at level from Definition 8 is actually the probabilistic rough extension of crisp relation R (which is a -level set of R), when parameter in Definition 7 is equal to zero. For an approximate evaluation of granules from R ∈ U ×W induced by composite sets from W we have the following result.
Proposition 13. Assume a crisp relation R between U and W and ∈ [0, 0.5). Then both extensions R and R of R have a lower SM -bound and an upper SL -bound with respect to the union of nonempty sets from W. Proof. For a threshold t ∈ [0, 1] we have that |(V ∪ Y ) ∩ r x | |V ∩ r x | >t ⊂ x ∈X : >t x∈X: |r x | |r x | and
|Y ∩ r x | |(V ∪ Y ) ∩ r x | x∈X: >t ⊂ x ∈X : >t . |r x | |r x |
Then for ∈ [0, 0.5) S M (R (X, V ), R (X, Y )) ⱕ R (X, V ∪ Y )
(67)
and
S M ( R (X, V ), R (X, Y )) ⱕ R (X, V ∪ Y ).
(68)
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51
On the other hand, |(Y ∪ V ) ∩ r x | |(V ∩ r x | |(Y ∩ r x | x∈X: >t ⊂ x ∈X : >t ∪ x ∈X : >t |r x | |r x | |r x | and therefore x ∈ X : |(Y ∪ V ) ∩ r x | > t ⱕ x ∈ X : |(V ∩ r x | > t + x ∈ X : |(Y ∩ r x | > t . |r x | |r x | |r x | Then for ∈ [0, 0.5) R (X, V ∪ Y ) ⱕ R (X, V ) + R (X, Y ), which means that R (X, V ∪ Y ) ⱕ min{1, R (X, V ) + R (X, Y )} = SL (R (X, V ), R (X, Y )). Analogously,
R (X, V ∪ Y ) ⱕ min{1, R (X, V ) + R (X, Y )} = SL ( R (X, V ), R (X, Y )).
Approximate evaluation of granules of a relation R between U and W induced by the union of subsets from U will be investigated for a special case when = 0. Then for all (X, Y ) ∈ 2˜ U × 2˜ W |Y ∩ r x | 0 >0 R (X, Y ) = x ∈ X : |r x | M (X, Y ). = Mx∈X (SM (R(x, y), y ∈ Y )) = MS R
Proposition 14. Let R ∈ U × W . Then for all X, Y ∈ 2˜ U , Y 2˜ W 0
0
0
0
0
TL ( R (X, Y ), R (V, Y ) ⱕ R (X ∪ V, Y ) ⱕ SL ( R (X, Y ), R (V, Y ))
(69)
TL (0R (X, Y ), 0R (V, Y ) ⱕ 0R (X ∪ V, Y ) ⱕ SL (0R (X, Y ), 0R (V, Y )).
(70)
and
0
M Proof. Because R (X, Y ) = MS (X, Y ), inequality (69) follows from Proposition 10 and the fact that the arithmetic R mean M has the upper SL -bound and the lower TL -bound with respect to the union of nonempty finite sets. Now, using Lemma 7, inequality (69) and duality between TL and SL , we will prove inequality (70):
0
0R (X ∪ V, Y ) = 1 − R (X ∪ V, Y c ) 0
0
ⱖ 1 − SL ( R (X, Y c ), R (V, Y c )) = 1 − SL (1 − 0R (X, Y ), 1 − 0R (X, Y )) = TL (0R (X, Y ), 0R (V, Y )). Analogously, 0
0R (X ∪ V, Y ) = 1 − R (X ∪ V, Y c ) 0
0
ⱕ 1 − TL ( R (X, Y c ), R (V, Y c )) = 1 − TL (1 − 0R (X, Y ), 1 − 0R (X, Y )) = SL (0R (X, Y ), 0R (V, Y )).
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6. Conclusion Approximation and approximate reasoning involving vague concepts have attracted the interest of numerous researchers and practitioners in psychology, medicine, finance, engineering and other areas of application. In our work we suggested a simple method for evaluation of composition of vague concepts described by fuzzy sets and fuzzy relations. We introduced the notion of boundaries of evaluation functions of composite concepts with respect to the binary operation used in their composition. T- and S-boundaries of evaluation functions based on frequently used aggregation functions (t-norms, t-conorms, the arithmetic mean, the geometric mean) with respect to the union of finite sets were specified. We showed that dominance of aggregation functions is an important property when boundaries of evaluation functions based on a pair of aggregation functions are considered. Evaluation and approximate evaluations of granules of a fuzzy relation by a pair of aggregation functions and an evaluation involving a generalization of the probabilistic rough set model were investigated. These evaluations allow researchers to assess the relationship between sets of related elements from two different universes. A further study of boundaries of evaluation functions may produce a more powerful and useful tool in approximate evaluation of composite concepts. In our future work we will explore T- and S-characterizations of evaluation functions taking into account classes of evaluation functions satisfying some specific properties while considering diverse binary operations. Acknowledgments The authors are grateful to the referees for their comments and suggestions for improving the paper. The work of Martin Kalina was supported by the Science and Technology Assistance Agency under the Contract no. APVV-0073-10 and by the grant agency VEGA, Grant no. 1/0143/11. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [23] [24] [25]
S. Bodjanova, Granules of a fuzzy relation, in: Proceedings of AGOP 2009, Palma de Mallorca, Spain, 2009, pp. 65–70. S. Bodjanova, Boundaries of evaluation functions, in: Proceedings of AGOP 2011, Benevento, Italy, 2011, pp. 159–164. S. Bodjanova, M. Kalina, T- evaluators and S-evaluators, Fuzzy Sets Syst. 160 (2009) 1965–1983. S. Bodjanova, M. Kalina, Gradual evaluation of granules of a fuzzy relation: R-related sets, in: E. Hullermeier, R. Kruse, F. Hoffmann (Eds.), IPMU 2010, Part I, CCIS 80, Springer-Verlag, Berlin, 2010, pp. 268–277. B. Bouchon-Meunier (Ed.), Aggregation and Fusion of Imperfect Information, Physica-Verlag, Heidelberg, 1998. T. Calvo, A. Kolesarova, M. Komornikova, R. Mesiar, Aggregation operators: properties, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators. New Trends and Applications, Physica-Verlag, Heidelberg, 2002, pp. 3–104. A. Di Nola, W. Pedrycz, S. Sessa, E. Sanchez, Fuzzy elation equations theory as a basis for fuzzy modelling: an overview, Fuzzy Sets Syst. 40 (1991) 414–429. D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, Dordrecht, 2000. G. Facchinetti, N. Pacchiarotti, Evaluation of fuzzy quantities, Fuzzy Sets Syst. 157 (2006) 892–903. J. Fodor, M. Roubens, Valued preference structures, Eur. J. Oper. Res. 79 (1994) 277–286. J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publishers, Dordrecht, 1994. M. Grabisch, E. Pap, J.L. Marichal, R. Mesiar, Aggregation Functions, University Press Cambridge, 2009. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. E.P. Klement, R. Mesiar (Eds.), Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, Elsevier, Amsterdam, 2005. G.J. Klir, T. Folger, Fuzzy Sets, Uncertainty and Information, Prentice Hall, Englewood Cliffs, 1988. R. Mesiar, S. Saminger, Domination of ordered weighted averaging operators over t-norms, Soft Comput. 8 (2004) 439–454. S. Ovchinnikov, An introduction to fuzzy relations, in: D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, Dordrecht, 2000, pp. 233–260. Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (1982) 341–356. Z. Pawlak, A. Skowron, Rough sets. Some extensions, Inf. Sci. 177 (2007) 28–40. S. Saminger, R. Mesiar, U. Bodenhofer, Domination of aggregation operators and preservation of transitivity, Int. J. Uncertainty Fuzziness Knowl. Based Syst. 10 (Suppl.) (2002) 11–35. Y.Y. Yao, S.K.M. Wong, L.S. Wang, A non-linear approach to uncertain reasoning, Int. J. General Syst. 23 (1995) 343–359. Y.Y. Yao, S.K.M. Wong, T.Y. Lin, A review of rough sets models, in: T.Y. Lin, N. Cercone, T.Y. Lin (Eds.), Rough Sets in Data Mining: Analysis for Imprecise Data, Kluwer Academic Publishers, Boston, 1997, pp. 47–75. W. Ziarko, Variable precision rough set model, J. Comput. Syst. Sci. 46 (1993) 35–59.
Fuzzy Sets and Systems 220 (2013) 34 – 52 www.elsevier.com/locate/fss
Approximate evaluations based on aggregation functions Slavka Bodjanovaa,∗ , Martin Kalinab a Department of Mathematics, Texas A&M University-Kingsville, MSC 172, Kingsville, TX 78363, USA b Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 81368 Bratislava, Slovakia
Received 29 January 2012; received in revised form 5 June 2012; accepted 25 July 2012 Available online 4 August 2012
Abstract Aggregation of real-valued evaluations of elements x,y from a set U is investigated in order to find an approximate evaluation of the composition of x and y performed by a binary operation defined on U. The notions of a lower bound and an upper bound of an evaluation function with respect to a binary operation are introduced and illustrated with examples. Approximate evaluations of composite concepts described by the union of granules of a finite fuzzy set are studied in detail. © 2012 Elsevier B.V. All rights reserved. Keywords: Evaluation; Aggregation functions; Triangular norms; Granules of a fuzzy set; Fuzzy relations; Approximation; Rough sets
1. Introduction The term “evaluation” is very broad and context dependent. To evaluate an element numerically means to assign a real number to it. This problem has been studied widely by many researchers, especially in the area of soft computing, where elements are described by fuzzy sets [5,8,9,15]. We assume a nonempty set U equipped with a binary operation ∗. The main goal of our study is to investigate how evaluations of elements x and y from U can be used to approximate evaluation of x ∗ y. We consider an evaluation function on U and we propose a lower and an upper bound of (x ∗ y) based on aggregation of (x) and (y) by an appropriate aggregation function [5,6,12]. When evaluation functions are aggregation functions and their boundaries are created by a t-norm T or a t-conorm S, we obtain a T–S-characterization of aggregation functions with respect to ∗. The T–S-characterizations of frequently used aggregation functions (e.g., triangular norms themselves, the arithmetic mean, the geometric mean, etc.) with respect to the union of finite sets will be provided. Knowing the boundaries of (x ∗ y) is beneficial in many areas of applications, especially in approximate reasoning and decision making. It allows researchers to identify elements whose composition will have a particular numerical evaluation within (or out of) given thresholds without performing the actual composition. We apply our theory to approximate evaluations of the union of granules of a finite fuzzy set and to approximate evaluations of the union of granules of a fuzzy relation between two finite universes. It is well known that a vague concept defined on a finite universal set U can be described by a fuzzy set f : U → [0, 1] [8,15]. A set Y ∈ 2U ∗ Corresponding author. Tel.: +1 361 593 3517.
E-mail addresses: [email protected] (S. Bodjanova), [email protected] (M. Kalina). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.07.014
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35
induces the granule f Y of f. A numerical evaluation of f Y can be interpreted as the degree to which the collection of elements in Y satisfies the vague concept described by f. A fuzzy set R defined on the Cartesian product of two finite universes U × W is referred to as a binary fuzzy relation from U to W [7,10,11,17]. A pair of sets (X, Y ) ∈ 2U × 2W induces the granule R X Y of R. An appropriate numerical evaluation of R X Y provides information about the strength of the relationship of X to Y. This contribution is a continuation and an extension of our results from [3,2]. It is organized as follows. In Section 2 we present the basic definitions, notations and properties of evaluation functions. Section 3 is devoted to evaluation and approximate evaluation of granules of a finite fuzzy set. In Section 4 we focus on evaluation functions based on aggregation functions. Evaluation and approximate evaluation of granules of a fuzzy relation are studied in Section 5. Concluding remarks are in Section 6. 2. Evaluation functions Throughout this paper we will only deal with quantitative evaluations (real numbers) from the unit interval. Definition 1. Let U be a nonempty set. Then a mapping : U → [0, 1] will be called an evaluation function on U. Further properties of an evaluation function depend on the structure of U and on the purpose of evaluation. For example, if U is partially ordered by a relation , we say that is nondecreasing, if for all x, y ∈ U , x y ⇒ (x) ⱕ (y), and is nonincreasing if for all x, y ∈ U , x y ⇒ (x) ⱖ (y). If ⊥ and are the least and the greatest elements of U, respectively, and (⊥) = 0, while () = 1, we say that is bounded. In [3] we introduced the notions of T- and S-evaluators as follows. Definition 2. Consider a lattice (L , , ⊥, ). A function : L → [0, 1] is said to be an evaluator on L if and only if it satisfies the following properties: • (⊥) = 0, () = 1, • for all x, y ∈ L : x y ⇒ (x) ⱕ (y). For a t-norm T and a t-conorm S, an evaluator on L is called a T-evaluator if and only if for all x, y ∈ L T ((x), (y)) ⱕ (x ∧ y),
(1)
and it is called an S-evaluator if and only if S((x), (y)) ⱖ (x ∨ y).
(2)
Recall that the lattice operations x ∧ y = inf{x, y} and x ∨ y = sup{x, y} are known also as the meet and the join, respectively. Further in this paper we will consider the four basic t-norms, i.e., TM (x, y) = min{x, y} (the minimum), TP (x, y) = x · y (the product), TL (x, y) = max{x + y − 1, 0} (the Łukasiewicz t-norm) and the drastic product 0 if max{x, y} < 1, TD (x, y) = min{x, y} if max{x, y} = 1. The basic t-conorms (dual of the four basic t-conorms) are: S M (x, y) = max{x, y} (the maximum), S P (x, y) = x + y − x · y (the probabilistic sum), SL (x, y) = min{x + y, 1} (the Łukasiewicz t-conorm) and the drastic sum 1 if min{x, y} > 0, S D (x, y) = max{x, y} if min{x, y} = 0.
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t-Norms and t-conorms [14,13] are special types of aggregation functions. Let us recall that an aggregation function [5,6,12] is a function A : n∈N [0, 1]n → [0, 1], where N denotes the set of all positive integers, such that A(x1 , . . . , xn ) ⱕ A(y1 , . . . , yn ) whenever xi ⱕ yi for all i = 1, . . . , n, A(x) = x for all x ∈ [0, 1], A(0, . . . , 0) = 0, and A(1, . . . , 1) = 1. Now we will generalize the statements (1) and (2) from Definition 2. Definition 3. Consider an aggregation function A and an evaluation function on a nonempty set U. Let ∗ be a binary operation on U. We say that has a lower A-bound with respect to ∗ if and only if for all x, y ∈ U A((x), (y)) ⱕ (x ∗ y),
(3)
and it has an upper A-bound with respect to ∗ if and only if A((x), (y)) ⱖ (x ∗ y).
(4)
Based on the definition above, a T-evaluator on L is a nondecreasing bounded evaluation function which has a lower T-bound with respect to the meet of elements from L. On the other hand, an S-evaluator on L is a nondecreasing bounded evaluation function which has an upper S-bound with respect to the join of elements from L. We will focus on boundaries created by t-norms and t-conorms. For the best approximation of (x ∗ y) we want to identify the greatest lower bound and the smallest upper bound. We say that an aggregation function A is stronger than and aggregation function B with the notation B ⱕ A, if B(x, y) ⱕ A(x, y) for all (x, y) ∈ [0, 1]2 . It is well known that TD ⱕ TL ⱕ TP ⱕ TM and TD ⱕ T ⱕ TM for any t-norm T. Analogously, S D ⱖ SL ⱖ S P ⱖ S M and S D ⱖ S ⱖ S M for any t-conorm S. Then • • • •
if has a lower TM -bound then it has a lower T-bound for any t-norm T, if has an upper S M -bound then it has an upper S-bound for any t-conorm S, if does not have a lower TD -bound then a lower T-bound of does not exist, if does not have an upper S D -bound then an upper S-bound of does not exist.
Lemma 1. Let U be a partially ordered set and for all x, y ∈ U, x ∨ y = sup{x, y} and x ∧ y = inf{x, y}. Then each nondecreasing evaluation function on U has a lower S M -bound with respect to ∨ and an upper TM -bound with respect to ∧. Each nonincreasing evaluation function on U has a lower S M -bound with respect to ∧ and an upper TM -bound with respect to ∨. Corollary 1. If is a T-evaluator on a lattice L then for all x, y ∈ L T ((x), (y)) ⱕ (x ∧ y) ⱕ TM ((x), (y)).
(5)
If is an S-evaluator on L then for all x, y ∈ L S M ((x), (y)) ⱕ (x ∨ y) ⱕ S((x), (y)).
(6)
Lemma 2. Assume that an evaluation function on a lattice L has the valuation property, i.e., for all x, y ∈ L (x ∨ y) = (x) + (y) − (x ∧ y). Then has an upper SL -bound and a lower TL -bound with respect to ∨ and also with respect to ∧. In applications, only the nontrivial bounds (i.e., the upper bound less than 1 and the lower bound more than 0) provide an useful information. However, some combinations of aggregation functions do not yield both nontrivial boundaries. Lemma 3. Let has an upper SL -bound and a lower TL -bound with respect to a binary operation ∗ defined on U. Then either the upper bound is equal to 1 or the lower bound is equal to 0. Because a lower T-bound (an upper S-bound) of exists only if has a lower TD -bound (an upper S D -bound), checking the existence of TD and S D boundaries is an important starting point in searching for boundaries of .
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Lemma 4. An evaluation function has an upper S D -bound with respect to a binary operation ∗ defined on U if an only if for all x, y ∈ U (x) = 0 ⇒ (x ∗ y) ⱕ (y),
(7)
and has a lower TD -bound with respect to ∗ if and only if for all x, y ∈ U (x) = 1 ⇒ (x ∗ y) ⱖ (y).
(8)
In general, we may have evaluation functions whose both boundaries are based either on t-norms or on t-conorms. Clearly, if an evaluation function has a lower S-bound than its upper bound, if it exists, is an S1 -bound such that S1 ⱖ S. Analogously, if an evaluation function has an upper T-bound, its lower bound, if it exists, is a T1 -bound such that T1 ⱕ T . The diversity of evaluation functions is very broad. In the next section we will investigate boundaries of some evaluation functions of granules of finite fuzzy sets. 3. Evaluation of fuzzy set granules A vague concept defined on a finite universal set U can be described by a fuzzy set f : U → [0, 1] [8,15]. Note that a fuzzy set itself is an evaluation function on U, where f (x) evaluates the degree to which an element x from U satisfies the concept described by f. We will use the notation F(U ) for the family of all fuzzy sets defined on U. A set Y ∈ 2U induces the granule f Y of f, which is a fuzzy set from F(Y ) given for all x ∈ Y by f Y (x) = f (x).
(9)
We will denote the family of all granules of a fuzzy set f ∈ F(U ) by G( f ). Hence G( f ) = { f Y : Y ∈ 2U }.
(10)
Obviously, fU = f and the granule of f induced by the empty set is the empty set. For f X , f Y ∈ G( f ) we say that f X is included in f Y , denoted by f X ⊂ f Y , if and only if X ⊂ Y . Analogously, the union and the intersection of granules are defined by f X ∪ f Y = f X ∪Y
(11)
f x ∩ f Y = f X ∩Y ,
(12)
and
respectively. An appropriate numerical evaluation of f Y can be interpreted as the degree to which the collection of elements in Y satisfies the vague concept described by f. From a practical point of view only evaluations of f Y where Y ∅ are useful. We will denote by 2˜ U the family of all nonempty subsets from U. Various evaluation functions are used in different areas of application. We will explore just a few of them. A natural way of evaluation of a fuzzy set granule is an aggregation of its membership grades. Then, using an aggregation function A, we define evaluation function A as follows: for all f Y ∈ G( f ), Y ∈ 2˜ U A ( f Y ) = A( f (x), x ∈ Y ).
(13)
An evaluation function on G( f ) can also be based on evaluation of -level sets of f, i.e., crisp sets f = {x ∈ U : f (x) ⱖ },
(14)
∈ (0, 1], by a fuzzy measure m on 2U . Recall that m is a nondecreasing and bounded evaluation function on 2U . Then, for f Y ∈ G( f ), Y ∈ 2˜ U we define m, ( f Y ) =
m( f ∩ Y ) . m(Y )
(15)
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When | · | denotes the cardinality of a set, formula (15) can be rewritten as |·|, ( f Y ) =
| f ∩ Y | |Y |
(16)
and one may interpret it as a degree of inclusion of Y in f . Many evaluation functions of f Y are based on an evaluation of the individual membership grades of f by nonnegative real numbers. Assume a mapping h : [0, 1] → [0, b], where b is a positive real number and Y ∅. Then ⎧ ⎪ ⎨ x∈Y h(x) if h(x)0, (17) h ( f Y ) = x∈U x∈U h(x) ⎪ ⎩ 0 otherwise. For example, if h(x) = −x ln(x) − (1 − x) ln(1 − x) with the convention h(0) = 0, then h ( f Y ) evaluates the relative fuzziness of f Y . If h(x) = x, then h ( f Y ) evaluates the relative cardinality of f Y . If we want to evaluate the range of membership grades of granule f Y , we can create an evaluation function which compares the extreme values of f (x), x ∈ Y . For example, 1 ( f Y ) = max f (x) − min f (x)
(18)
⎧ ⎨ min x∈Y f (x) if max f (x) > 0, x∈Y 2 ( f Y ) = maxx∈Y f (x) ⎩ 1 otherwise.
(19)
x∈Y
or
x∈Y
Now we will examine boundaries of evaluation functions presented above with respect to the union of granules of a finite fuzzy set. Proposition 1. Let m be a probability measure on 2U . Then for all f ∈ F(U ) and all ∈ (0, 1] evaluation function m, given by (15) has an upper S P -bound and a lower TP -bound with respect to the union of granules from G( f ). Proof. First we show that m, has a lower TP -bound with respect to the union of granules from G( f ). Let us take an arbitrary couple of granules f X and f Y and choose ∈ (0, 1]. Using the notation n 1 = m(X \ f ∩ X ), n 2 = m( f ∩ X ), n 3 = m(Y \ f ∩ Y ), n 4 = m( f ∩ Y ),
(20)
we obtain that m, ( f X ) = n 2 /(n 1 + n 2 ), m, ( f Y ) = n 4 /(n 3 + n 4 ). Let us denote m, ( f X ∪Y ) = M /M. Obviously there exist nonnegative constants 1 , 2 such that 1 + 2 = n 4 and M = n 2 + 1 .
(21)
Clearly, n 1 + M ≤ M ≤ n 1 + n 3 + M .
(22)
Assume that n 4 ≤ n 2 . Since for all a, b, c > 0, a ≤ b we have a a+c ≤ , b b+c
(23)
we get a lower bound and an upper bound of m, ( f X ∪Y ) by (24) and (25), respectively. M n 2 + 1 n2 ≥ ≥ , M n 1 + n 3 + n 2 + 1 n1 + n2 + n3
(24)
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M n 2 + 1 + 2 n 2 + 1 ≤ ≤ M n 1 + n 2 + 1 n 1 + n 2 + 1 + 2 n2 + n4 ≤ . n1 + n2 + n4
39
(25)
Now we will show that m, ( f X ∪Y ) ≥ m, ( f X ) · m, ( f Y ). Assume that n 4 ≤ n 2 , i.e., n 4 = n 2 + c, c ≥ 0. Then n4 + c n4 n2 ≥ = n3 + n4 + c n3 + n2 n3 + n4 and for m, ( f X ) · m, ( f Y ) we have n4 n2 m, ( f X ) · m, ( f Y ) = n1 + n2 n4 + n3 n2 n2 ≤ n1 + n2 n2 + n3 =
n 22 . (n 1 + n 2 )(n 2 + n 3 )
(26)
Suppose that n 1 + n 2 + n 3 = 1, since this is just a matter of normalization. Then, using (24) and (26) it is enough to show that n2 ≥
n 22 , (n 1 + n 2 )(n 2 + n 3 )
or equivalently (n 1 + n 2 )(n 2 + n 3 ) ≥ n 2 .
(27)
At the left-hand-side we have n 1 n 2 + n 22 + n 1 n 3 + n 2 n 3 = n 2 (n 1 + n 2 + n 3 ) +n 1 n 3 ,
=1
and hence the inequality (27) is valid. Now, we show that m, has an upper S P -bound with respect to the union of granules from G( f ). We take an arbitrary couple of granules f X and f Y and we keep that notation (20). Then the following yields: S P (m, ( f X ), m, ( f Y )) =
n2 n4 n2n4 + − . n1 + n2 n3 + n4 (n 1 + n 2 )(n 3 + n 4 )
Assume that n 1 ≥ n 3 . Then n4 n4 ≤ n1 + n4 n3 + n4 and the monotonicity of S P implies S P (m, ( f X ), m, ( f Y )) ≥
n2 n4 n2n4 . + − n1 + n2 n1 + n4 (n 1 + n 2 )(n 1 + n 4 )
(28)
Using (25) and (28) it is enough to show that n2 + n4 n2 n4 n2n4 . ≤ + − n1 + n2 + n4 n1 + n2 n3 + n4 (n 1 + n 2 )(n 1 + n 4 )
(29)
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As a matter of normalization we will assume that n 1 + n 2 + n 4 = 1. Then inequality (29) can be reformulated into n2 + n4 ≤
n 2 (n 1 + n 4 ) + n 4 (n 1 + n 2 ) − n 2 n 4 , (n 1 + n 2 )(n 1 + n 4 )
or equivalently (n 2 + n 4 )(n 1 + n 2 )(n 1 + n 4 ) ≤ n 1 n 2 + n 2 n 4 + n 1 n 4 .
(30)
At the left-hand-side of (30) we have the following expression: n 1 n 2 (n 1 + n 2 + n 4 ) +n 2 n 4 (n 1 + n 2 + n 4 ) +n 1 n 4 (n 1 + n 4 ),
=1
=1
≤1
which proves inequality (30). The next example shows that m, has neither the lower TM -bound nor the upper S M -bound with respect to the union of granules from G( f ). Example 1. Let X = {x1 , x2 }, Y = {x2 , x3 }. Assume f ∈ F(U ) and ∈ (0, 1] such that f (x1 ) = f (x3 ) = 0 and f (x2 ) = 1. Then by formula (16) we get |·|, ( f X ) = |·|, ( f Y ) =
1 2
and |·|, ( f X ∪Y ) = 13 .
However, |·|, ( f X ∪Y ) < min{|·|, ( f X ), |·|, ( f Y )}, therefore |·|, does not have a lower TM -bound. Now assume g ∈ F(U ) and ∈ (0, 1] such that g (x1 ) = g (x3 ) = 1 and f (x2 ) = 0. Then |·|, (g X ) = |·|, (gY ) =
1 2
and |·|, (g X ∪Y ) = 23 .
However, |·|, (g X ∪Y ) > max{|·|, (g X ), |·|, (gY )}, therefore |·|, does not have an upper S M -bound. When we consider only disjoint granules of a fuzzy set, we get the following result. Proposition 2. Let m be a probability measure on 2U . Then for all f ∈ F(U ) and all ∈ (0, 1] evaluation function m, has an upper S M -bound and a lower TM -bound with respect to the union of disjoint granules from G( f ). Proof. We will use the notations from the proof of Proposition 1. Assume that, e.g., m, ( f X ) ⱕ m, ( f Y ). Then, because X ∩ Y = ∅, n2 + n4 m, ( f X ∪Y ) = n1 + n2 + n3 + n4 =
(n 1 + n 2 )m, ( f X ) + (n 3 + n 4 )m, ( f Y ) n1 + n2 + n3 + n4
≤
(n 1 + n 2 )m, ( f Y ) + (n 3 + n 4 )m, ( f Y ) n1 + n2 + n3 + n4
= m, ( f Y ) = max{m, ( f Y ), m, ( f X )}.
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On the other hand, m, ( f X ∪Y ) =
(n 1 + n 2 )m, ( f X ) + (n 3 + n 4 )m, ( f Y ) n1 + n2 + n3 + n4
ⱖ
(n 1 + n 2 )m, ( f X ) + (n 3 + n 4 )m, ( f X ) n1 + n2 + n3 + n4
= m, ( f X ) = min{m, ( f Y ), m, ( f X )}.
Proposition 3. For an arbitrary f ∈ F(U ), with respect to the union of granules from G( f ), • the evaluation function h given by (17) has a lower S M -bound and an upper SL -bound, • the evaluation function 1 given by (54) has a lower S M -bound and it does not have an upper S-bound for any t-conorm S, • the evaluation function 2 given by (19) has an upper TM -bound and it does not have a lower T-bound for any t-norm T. Proof. (i) Evaluation function h is increasing and it has the valuation property. Therefore, according to Lemma 1 it has a lower S M -bound and according to Lemma 2 it has an upper SL -bound. (ii) Evaluation function 1 is increasing and therefore it has a lower S M -bound. In Example 2 we will show that 1 does not have an upper S D -bound, and therefore it does not have an upper S-bound for any t-conorm S. (iii) Evaluation function 2 is nonincreasing and therefore it has an upper TM -bound. In Example 2 we will show that 2 does not have a lower TD -bound, and therefore it does not have a lower T-bound for any t-norm T. Example 2. Let X = {x1 , x2 } and Y = {x3 , x4 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 ) = f (x2 ) = 0.6 and f (x3 ) = f (x4 ) = 0.4. Then 1 ( f X ) = 1 ( f Y ) = 0 and 1 ( f X ∪Y ) = 0.6 − 0.4 = 0.2. Clearly, 0.2 > max{0, 0}, and therefore 1 does not have an upper S D -bound. 2 When we use the evaluation function 2 , we obtain that 2 ( f X ) = 2 ( f Y ) = 1 and 2 (FX ∪Y ) = 0.4 0.6 = 3 . However, 2 3 < min{1, 1} and hence 2 does not have a lower TD -bound. Boundaries of the evaluation function A given by (13) will be studied in the next section. 4. Evaluation functions based on aggregation functions First of all we will examine evaluation functions based on t-norms and t-conorms. The next proposition is obvious. Proposition 4. When A is an arbitrary t-norm T, the evaluation function A given by (13) has an upper TM -bound and a lower T-bound with respect to the union of granules from G( f ). When A is an arbitrary t-conorm S, then A has an upper S-bound and a lower S M -bound with respect to the union of granules from G( f ). Now we will analyze evaluation of f X ∈ G( f ) by the arithmetic mean M. Hence f (x) M ( f X ) = M( f (x), x ∈ X ) = x∈X . |X |
(31)
Proposition 5. Evaluation function M has a lower TL -bound and an upper SL -bound with respect to the union of granules from G( f ), f ∈ F(U ). Proof. For X, Y ∈ 2U we have that
f (x) ⱕ f (x) + f (x) x∈X ∪Y
x∈X
x∈Y
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and |X ∪ Y | ⱖ max{|X |, |Y |}. Therefore f (x) f (x) x∈X ∪Y f (x) M( f X ∪Y ) = ⱕ x∈X + x∈Y |X ∪ Y | |X ∪ Y | |X ∪ Y | f (x) x∈X f (x) ⱕ + x∈Y |X | |Y | ⱕ min{M( f X ) + M( f Y ), 1} = SL (M( f X ), M( f Y ). Now denote n 1 = |X \ X ∩ Y |, n 2 = |X ∩ Y | and n 3 = |Y \ X ∩ Y |. Obviously, n 1 ⱖ n 1 M( f Y ), n 3 ⱖ n 3 M( f X ) and n 2 ⱖ X ∩Y f (x). Therefore
n 1 + n 2 + n 3 ⱖ n 1 M( f Y ) + f (x) + n 3 M( f X ). (32) X ∩Y
We will add to both sides of (32) the expression
f (x) = f (x) + f (x) + f (x). x∈X ∪Y
x∈X \X ∩Y
x∈Y \X ∩Y
x∈X ∩Y
Then on the left side of (32) we obtain
f (x) = (n 1 + n 2 + n 3 )(1 + M( f X ∪Y )) n1 + n2 + n3 +
(33)
x∈X ∪Y
and on the right side of (32) we have
n 1 M( f Y ) + n 3 M( f X ) + f (x) + X ∩Y
= n 1 M( f Y ) + n 3 M( f X ) +
x∈X
x∈X \X ∩Y
f (x) +
f (x) +
f (x) +
x∈Y \X ∩Y
f (x)
x∈X ∩Y
f (x)
x∈Y
= n 1 M( f Y ) + n 3 M( f X ) + (n 1 + n 2 )M( f X ) + (n 3 + n 2 )M( f Y ) = (n 1 + n 2 + n 3 )M( f X ) + (n 1 + n 2 + n 3 )M( f Y ).
(34)
After dividing (33) and (34) by (n 1 + n 2 + n 3 ) we obtain that M( f X ∪Y ) + 1 ⱖ M( f X ) + M( f Y ), and therefore M( f X ∪Y ) ⱖ max{M( f X ) + M( f Y ) − 1, 0} = TL (M( f X ), M( f Y )).
The next example shows that M has neither an upper S M -bound nor a lower TM -bound with respect to the union of fuzzy set granules. Example 3. Let X = {x1 , x2 } and Y = {x2 , x3 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 = f (x3 ) = 1 and f (x2 ) = 0.9. Then M ( f X ∪Y ) = (1 + 0.9 + 1)/3 = 0.967, M ( f X ) = M ( f Y ) = (1 + 0.9)/2 = 0.95. Because 0.967 > max{0.95, 0.95}, evaluation function M does not have an upper S M -bound. Now let g(x1 ) = g(x3 ) = 0.1 and g(x2 ) = 0.2. Then M ( f X ∪Y ) = (0.1 + 0.2 + 0.1)/3 = 0.13, M ( f X ) = M ( f Y ) = (0.1 + 0.2)/2 = 0.15. Because 0.13 < min{0.15, 0.15}, evaluation function M does not have a lower TM -bound. However, if we consider only disjoint granules, we have the following result. Proposition 6. Evaluation function M has a lower TM -bound and an upper S M -bound with respect to the union of disjoint granules from G( f ), f ∈ F(U ).
S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
43
Proof. Consider X, Y ∈ 2˜ U such that X ∩ Y = ∅. Let n 1 = |X | and n 2 = |Y |. Assume that M( f X ) ⱕ M( f Y ). Then n 1 M( f X ) + n 2 M( f Y ) M( f X ∪Y ) = n1 + n2 ≤
n 1 M( f Y ) + n 2 M( f Y ) n1 + n2
= M( f Y ) = SM (M( f X ), M( f Y )). On the other hand, n 1 M( f X ) + n 2 M( f Y ) M( f X ∪Y ) = n1 + n2 ≥
n 1 M( f X ) + n 2 M( f X ) n1 + n2
= M( f X ) = TM (M( f X ), M( f Y )).
The arithmetic mean can be used to aggregate outputs of other aggregation functions. We will analyze evaluation function min x∈X f (x) + maxx∈X f (x) 3 ( f X ) = 2 = M(TM ( f (x), x ∈ X ), S M ( f (x), x ∈ X )). (35) Proposition 7. Evaluation function 3 given by (35) has an upper S M -bound and a lower TM -bound with respect to the union of granules from G( f ), f ∈ F(U ). Proof. For X, Y ∈ 2˜ U and a mapping f : U → [0, 1], let min x∈X f (x) = a, min x∈Y f (x) = b, maxx∈X f (x) = c and maxx∈Y f (x) = d. Then a ⱕ c, b ⱕ d and we have that K( f X ) =
a+c b+d and K( f Y ) = . 2 2
We will show that max{K( f X ), K( f Y )} ⱖ K( f X ∪Y )
(36)
min{K( f X ), K( f Y )} ⱕ K( f X ∪Y ).
(37)
and
The following four cases may occur: Case 1: If a ⱕ b and d ⱕ c then a ⱕ b ⱕ d ⱕ c and we obtain a+c b+d a+c max , ⱖ = K( f X ∪Y ) 2 2 2 a+c b+d , . ⱖ min 2 2 Case 2: If a ⱕ b and c ⱕ d then a+c b+d b+d a+d max , = > 2 2 2 2
a+c b+d , = K( f X ∪Y ) ⱖ min 2 2
=
a+c . 2
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Case 3: If b ⱕ a and c ⱕ d then b ⱕ a ⱕ c ⱕ d and b+d a+d a+c b+d , = > = K( f X ∪Y ) max 2 2 2 2 a+c b+d , . ⱖ min 2 2 Case 4: If b ⱕ a and d ⱕ c then a+c b+d a+c b+c max , = ⱖ 2 2 2 2
a+c b+d , = K( f X ∪Y ) ⱖ min 2 2
=
b+d . 2
Boundaries of the evaluation function G based on the geometric mean defined by G ( f X ) = G( f (x), x ∈ X ) = (x∈X f (x))1/|X |
(38)
are described in Proposition 8. Proposition 8. Evaluation function G given by (38) has a lower T p -bound and an upper S D -bound with respect to the union of granules from G( f ), f ∈ F(U ). U we have that |X − X ∩ Y | = n , |X ∩ Y | = n and |Y − Y ∩ X | = n . Denote P = Proof. Let for X, Y ∈ 2˜ 1 2 3 1 f (x), P = f (x) and P = f (x). Then 2 3 x∈X −X ∩Y x∈X ∩Y x∈Y −X ∩Y
G( f X ∪Y ) = (P1 P2 P3 )1/(n 1 +n 2 +n 3 ) ⱖ (P1 P2 P2 P3 )1/(n 1 +n 2 +n 3 ) = (P1 P2 )1/(n 1 +n 2 +n 3 ) (P3 P2 )1/(n 1 +n 2 +n 3 ) ⱖ (P1 P2 )1/(n 1 +n 2 ) (P3 P2 )1/(n 2 +n 3 ) = T p (G( f X ), G( f Y )). If G( f X ) = 0 then there exists x ∈ X such that f (x) = 0. Therefore, for any Y ∈ 2˜ U G( f (x), x ∈ X ∪ Y ) = 0 ⱕ G( f (x), x ∈ Y ). Based on Lemma 4 we conclude that G has an upper SD -bound with respect to the union of nonempty finite sets. Recall that TP ⱕ TM and SL ⱕ S D . We will show that G has neither the lower TM -bound nor the upper SL -bound. Example 4. Let X = {x1 , x2 } and Y = {x1 , x3 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 ) = 1, f (x2 ) = 0.2 and f (x3 ) = 0.3. Then G ( f X ∪Y ) = (1(0.2)(0.3))1/3 = 0.391, G ( f X ) = (1(0.2))1/2 = 0.447 and G ( f Y ) = (1(0.3))1/2 = 0.547. However, 0.391 < min{0.447, 0.547} and therefore G does not have a lower TM -bound. Now let X = {x1 , x2 , x3 }, Y = {x2 , x3 , x4 }, f (x1 ) = 0.9, f (x2 ) = f (x3 ) = 0.01 and f (x4 ) = 0.8. Then G ( f X ∪Y ) = (0.9(0.01)(0.01)(0.8))1/4 = 0.092, G ( f X ) = (0.9(0.01)(0.01)1/3 = 0.0448, G ( f Y ) = (0.01(0.01)(0.8))1/3 = 0.043. However, 0.092 > min{0.0448+0.043, 1} = SL (G ( f X ), G ( f Y )) and therefore G does not have an upper SL -bound.
S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
45
Finally, we consider evaluation function med,c defined by med,c ( f X ) = Ac ( f (x), x ∈ X )
= med(0, c + ( f (x) − c), 1),
(39)
x∈X
where c ∈ [0, 1] and med stands for the median. Proposition 9. Evaluation function med,c given by (39) has neither an upper S-bound, nor a lower T-bound with respect to the union of granules from G( f ), f ∈ F(U ). Example 5. Let X = {x1 , x2 , x3 , x4 } and Y = {x2 , x3 , x4 , x5 }. Consider a fuzzy set f ∈ F(U ) such that f (x1 ) = 0.5, f (x2 ) = f (x3 ) = f (x4 ) = 0 and f (x5 ) = 0.7. Choose c = 0.2. Then med,c ( f X ∪Y ) = med(0, 0.4, 1) = 0.4, med,c ( f X ) = med(0, −0.1, 1) = 0, med,c ( f Y ) = med(0, 0.1, 1) = 0.1. Because 0.4 > max{0, 0.1}, evaluation function med,c does not have an upper S D -bound. Now let X = {x1 , x2 , x3 }, Y = {x2 , x3 , x4 }, f (x1 ) = 0.4, f (x2 ) = f (x3 ) = 0.8 and f (x4 ) = 0.2. Choose c = 0.5. Then med,c ( f X ∪Y ) = med(0, 0.7, 1) = 0.7, med,c ( f X ) = med(0, 1, 1) = 1, med,c ( f Y ) = med(0, 0.8, 1) = 0.8. Because 0.7 < min{1, 0.8}, evaluation function med,c does not have a lower TD -bound. When we restrict evaluation functions only to aggregation functions, we can reformulate Definition 3 and introduce boundaries of aggregation functions as follows. Definition 4. Consider aggregation functions A and B. Let U be a nonempty finite set and ∗ be a binary operation on U. We say that A has a lower B-bound with respect to ∗ if and only if for all X, Y ∈ 2˜ U such that X ∗ Y ∅ and for all f : U → [0, 1] B(A( f (x), x ∈ X ), A( f (x), x ∈ Y )) ⱕ A( f (x), x ∈ X ∗ Y ).
(40)
We say that A has an upper B-bound with respect to ∗ if and only if B(A( f (x), x ∈ X ), A( f (x), x ∈ Y )) ⱖ A( f (x), x ∈ X ∗ Y ).
(41)
Boundaries of aggregation functions created by t-norms and/or t-conorms provide a T–S-characterization of aggregation functions with respect to ∗. The T–S-characterizations of aggregation functions mentioned in this paper with respect to the union of finite sets are presented in Table 1. Table 1 T–S-characterizations of selected aggregation functions. Function
Lower bound
Upper bound
T S M G M(TM , S M ) Ac
T SM TL TP TM None
TM S SL SD SM None
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5. Evaluation of granules of a fuzzy relation 5.1. Evaluations based on aggregation functions Let U and W be finite nonempty universes. A fuzzy set R ∈ F(U × W ) is referred to as binary fuzzy relation from U to W. The value of R(x, y) gives the strength of the relationship of x ∈ U to y ∈ W . A pair of sets (X, Y ) ∈ U × W induces granule R X Y of R, which is a fuzzy relation from U/ X to W/Y defined for all (x, y) ∈ X × Y by R X Y (x, y) = R(x, y).
(42)
An appropriate evaluation of R X Y provides information about the strength of the relationship of X to Y. Definition 5 (Bodjanova [1]). Let R be a fuzzy relation from U to W and A and B be aggregation functions. Then the U W U W AB-extension of R is a mapping AB R : 2 × 2 → [0, 1] defined for all (X, Y ) ∈ 2 × 2 by ⎧ ⎪ ⎨ Ax∈X (B y∈Y (R(x, y))) AB if X ∅, Y ∅, R (X, Y ) = (43) ⎪ ⎩ 0 otherwise. The value AB R (X, Y ) can be interpreted as the degree to which X is R-related to Y with respect to the evaluation based on A and B, or simply the degree to which X is RAB -related to Y. Lemma 5. For all (X, Y ) ∈ U × W (x, y) ∈ U × W , (P1) if X = {x} and Y = {y} then AB R (X, Y ) = R(x, y), (P2) if R(x, y) = 1 for all (x, y) ∈ X × Y then AB R (X, Y ) = 1, (P3) if R(x, y) = 0 for all (x, y) ∈ X × Y then AB R (X, Y ) = 0. Example 6. Consider the t-norm TM and the t-conorm SM . Then the TM SM -extension of R ∈ F(U × W ) is defined by TM SM R (X, Y ) = TM (SM (R(x, y), y ∈ Y ), x ∈ X ) = min max R(x, y) . (44) x∈X
y∈Y
Note that TRM SM (X, Y ) = means that each element x ∈ X is related to at least one element y ∈ Y to the degree R(x, y) ⱖ . The TM SM -extension of R should be used in situations where the strength of the relationship of X to Y is expected to decrease with an enlargement of X and to increase with an enlargement of Y. Example 7. The SM TM -extension of R is given by the formula SRM TM (X, Y ) = SM (TM (R(x, y), y ∈ Y ), x ∈ X ) = max min R(x, y) . x∈X
y∈Y
(45)
Then SRM TM (X, Y ) = means that there exist at least one element x ∈ X which is related to all elements y ∈ Y to the degree R(x, y) ⱖ . The SM TM -extension of R should be used in situations where the strength of the relationship of X to Y is expected to increase with an enlargement of X and to decrease with an enlargement of Y. Because in practice only granules R X Y created by nonempty sets are considered, further in this paper we assume that (X, Y ) ∈ 2˜ U × 2˜ W . Lemma 6. Consider a fuzzy relation R ∈ F(U × W ), a t-norm T and a t-conorm S. Then for all (X, Y ) ∈ 2˜ U × 2˜ W , all ∅Z ⊂ X and all ∅V ⊂ Y TS TS (1) TS R (X, V ) ⱕ R (X, Y ) ⱕ R (Z , Y ), ST ST (2) R (X, V ) ⱖ R (X, Y ) ⱖ ST R (Z , Y ),
S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
47
(3) SR (Z , V ) ⱕ SR (X, Y ), (4) TR (Z , V ) ⱖ TR (X, Y ). Clearly, boundaries of AB-extensions of a fuzzy relation R ∈ F(U × W ) with respect to a binary operation ∗ defined on U or W depend on the boundaries of aggregation functions A and B. Proposition 10. Let AB R be the AB-extension of R ∈ F(U × W ) and A has an upper P-bound and a lower Q-bound with respect to a binary operation ∗ defined on U. Then for all X, Z ∈ 2˜ U , X ∗ Z ∅ and all Y ∈ 2˜ W AB AB Q(AB R (X, Y ), R (Z , Y )) ⱕ R (X ∗ Z , Y )
(46)
AB AB AB R (X ∗ Z , Y ) ⱕ P( R (X, Y ), R (Z , Y )).
(47)
and
Proof. Define a fuzzy set g ∈ F(U ) such that g(x) = B y∈Y (R(x, y)) for all x ∈ U . Then Q(Ax∈X (g(x)), Ax∈Z (g(x))) ⱕ Ax∈X ∗Z (g(x))
(48)
Ax∈X ∗Z (g(x)) ⱕ P(Ax∈X (g(x)), Ax∈Z (g(x))).
(49)
and
AB Because Ax∈X (g(x)) = Ax∈X (B y∈Y (R(x, y))) = AB R (X, Y ) and Ax∈Z (g(x)) = Ax∈Z (B y∈Y (R(x, y))) = R (Z , Y ), inequality (48) proves (46) and inequality (49) proves (47).
According to Proposition 10, boundaries of AB R with respect to an arbitrary binary operation defined on U depend only on the boundaries of aggregation function A. However, in the case of a binary operation defined on W, dominance between boundaries of B and aggregation function A is an important factor. Let us recall that an aggregation function A dominates an aggregation function B, denoted by A?B, if for all [xi1 , . . . , xin ] ∈ [0, 1]n , i = 1, . . . , m, and m, n ∈ N A(B(x11 , . . . , x1n ), B(x21 , . . . , x2n ), . . . , B(xm1 , . . . , xmn )) ⱖ B(A(x11 , . . . , xm1 ), A(x12 , , . . . , xm2 ), . . . , A(x1n , . . . , xmn )).
(50)
See [16,20] for details. Proposition 11. Let AB R be the AB-extension of R ∈ F(U × W ). Assume that B has an upper P-bound with respect to a binary operation ∗ defined on W and that P dominates A. Then for all X ∈ 2˜ U and all Y, V ∈ 2˜ W , Y ∗ V ∅ AB AB AB R (X, Y ∗ V ) ⱕ P( R (X, Y ), R (X, V )).
(51)
Proof. Let X = {x1 , . . . , xn }. Then AB R (X, Y ∗ V ) = Ax∈X (B y∈Y ∗V (R(x, y))) ⱕ Ax∈X (P(B y∈Y (R(x, y)), B y∈V (R(x, y))) = A(P(B y∈Y (R(x1 , y)), B y∈V (R(x1 , y))), P(B y∈Y (R(x2 , y)), B y∈V (R(x2 , y))), . . . , (P(B y∈Y (R(xn , y)), B y∈V (R(xn , y))))), ⱕ P(A(B y∈Y (R(x1 , y)), . . . , B y∈Y (R(xn , y))), A(B y∈V (R(x1 , y)), . . . , B y∈V (R(xn , y)))), = P(Ax∈X B y∈Y (R(x, y)), Ax∈X B y∈V (R(x, y)) AB = P(AB R (X, Y ), R (X, V ).
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S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
Analogously we can prove the next proposition. Proposition 12. Let AB R be the AB-extension of R ∈ F(U × W ). Assume that B has a lower Q-bound with respect to a binary operation ∗ defined on W and that Q is dominated by A. Then for all X ∈ 2˜ U and all Y, V ∈ 2˜ W ,Y ∗ V ∅ AB AB AB R (X, Y ∗ V ) ⱖ Q( R (X, Y ), R (X, V )).
(52)
Example 8. Let R ∈ F(U × W ). Because the arithmetic mean M has a lower TL -bound and an upper SL -bound with respect to the union of finite sets, referring to Proposition 10, for any aggregation function A, evaluation function MA R has a lower TL -bound and an upper SL -bound with respect to the union of sets from U. Because each t-norm T is dominated by TM and each t-conorm dominates SM , using Propositions 11 and 12 we conclude that SRM M has an upper SL -bound and TRM M has a lower TL -bound with respect to the union of sets from W. Now we will evaluate granule R X Y of R by counting proportions of elements from X which have evaluations of their R-neighborhoods on Y within the given thresholds. In the case of a fuzzy relation R, the R-neighborhood of x is fuzzy set r x ∈ F(W ) defined for all y ∈ W by r x (y) = R(x, y). We will use the notation r x /Y for R-neighborhood of x restricted to Y ⊂ W . Definition 6. Assume a fuzzy relation R ∈ F(U × W ), a mapping : [0, 1]n → [0, 1] for n ∈ N and ∈ [0, 1). Then the fuzzy relation R : 2˜ U × 2˜ W → [0, 1] defined for all (X, Y ) ∈ 2˜ U × 2˜ W by R (X, Y ) =
|x ∈ X : (r x /Y ) > | |X |
(53)
will be called the -extension of R. In many applications the relationship of x ∈ U to Y ∈ 2˜ W is considered meaningless if (r x /Y ) ⱕ . Note that R (X, Y ) is the proportion of elements from X which have the -evaluations of r x /Y greater than a given threshold . Gradual changes of parameter produce gradual evaluations of granules of R. Some examples of -evaluations are given below: ⎧ R(x, y) ⎪ ⎨ y∈Y if R(x, y) > 0 for some y ∈ Y, 1 (r x /Y ) = y∈W R(x, y) ⎪ ⎩ 0 otherwise, ⎧ ⎪ ⎨ max y∈Y R(x, y) if R(x, y) > 0 for some y ∈ Y, 2 (r x /Y ) = max y∈W R(x, y) ⎪ ⎩0 otherwise, 3 (r x /Y ) = A(R(x, y), y ∈ Y ),
(54)
(55) (56)
where A is an aggregation function. Observe that if A(R(x, y), y ∈ Y ) > 0 for all x ∈ X ∈ 2˜ U , and = 0 then 3 (X, Y ) = MA R (X, Y ), where M is the arithmetic mean. 5.2. Evaluations based on rough set approximation Let R be a crisp compatibility relation on U × W , i.e., for each x ∈ U there is y ∈ W such that R(x, y) = 1. Then, applying -evaluation from (54) we obtain 1 (r x /Y ) =
|Y ∩ r x | , |r x |
(57)
S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
49
which is the degree of inclusion of R-neighborhood of x in Y. From the point of view of rough sets theory, expression (57) is the rough membership function describing approximation of Y ∈ 2˜ W in the knowledge base (approximation space) created by R-neighborhoods of elements from U. Let us recall that in the classical rough set model introduced by Pawlak [18] we assume a nonempty universal set U and a crisp equivalence relation E ⊂ U × U . The pair (U, E) is called a knowledge base or an approximation space. Given an arbitrary set Z ⊂ U , it may not be possible to describe Z in (U, E) precisely. Instead, one may obtain a pair of lower and upper approximations of Z defined by Z E = {x ∈ U : [x] E ⊂ Z }
(58)
Z E = {x ∈ U : [x] E ∩ Z ∅},
(59)
and
where [x] E is the equivalence class containing x. The pair (Z E , Z E ) is called the rough set with a reference set Z. Pawlak’s rough set model may be extended by using an arbitrary binary relation R [24]. In the variable precision rough set model [25] associated with a crisp relation R on U × U and a parameter ∈ [0, 0.5) the lower and the upper approximations of Z ⊂ U are defined by |Z ∩ R(x)| ⱖ1 − (60) apr (Z ) = x ∈ U : |R(x)| and
|Z ∩ R(x)| apr (Z ) = x ∈ U V : > , |R(x)|
(61)
respectively, where R(x) = {y ∈ U : R(x, y) = 1}. Reviews of various rough set models can be found in [19,24]. Variable precision rough sets model can be extended to a crisp relation R between two different universes and we can obtain an interval-valued evaluation of granules of R. Definition 7. Let R ∈ U × W and ∈ [0, 0.5). Then the mapping R defined for all (X, Y ) ∈ 2˜ U × 2˜ W by a pair of coefficients x ∈ X : |Y ∩ r x | ⱖ 1 − |r x | R (X, Y ) = (62) |X | and
x ∈ X : |Y ∩ r x | > |r x | R (X, Y ) = |X |
(63)
will be called the probabilistic rough extension of R at level . Lemma 7. Let R be a crisp relation between U and W. Then for all (X, Y ) ∈ 2˜ U × 2˜ W and all ∈ [0, 0.5)
R (X, Y ) = 1 − R (X, Y c ), where Y c denote the complement of Y . Proof. Because x ∈ X : |Y ∩ r x | ⱖ 1 − = |X | − x ∈ X : |Y ∩ r x | < 1 − , |r x | |r x | we obtain that
x ∈ X : |Y ∩ r x | < 1 − |r x | R (X, Y ) = 1 − . |X |
(64)
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S. Bodjanova, M. Kalina / Fuzzy Sets and Systems 220 (2013) 34 – 52
Then |Y ∩ r x | |r x | − |Y c ∩ r x | |Y c ∩ r x | = =1− <1− |r x | |r x | |r x | is equivalent to |Y c ∩ r x | > . |r x | Therefore,
c x ∈ X : |Y ∩ r x | > |r x | R (X, Y ) = 1 − = 1 − R (X, Y c ). |X |
Using Lemma 7 we can say that the coefficient R (X, Y ) gives the proportion of elements from X which have at most 100 percent of their R-neighborhoods included in the complement of Y. Obviously, the coefficient R (X, Y ) is the proportion of elements from X which have more than 100 percent of their R-neighborhoods included in Y. In [4] we defined the rough extension of a fuzzy relation R ∈ F(U × W ) at a level ∈ (0, 1] using a rough set model over two universes [23,24] and -level-sets of R-neighborhoods of elements from U as follows. Definition 8. Let R ∈ F(U × W ) and ∈ (0, 1]. For each Y ∈ 2˜ W assume subsets from U R ,Y = {x ∈ U : (r x ) ⊂ Y } and R ,Y = {x ∈ U : (r x ) ∩ Y }.
(65)
Then the mapping R defined for all (X, Y ) ∈ 2˜ U × 2˜ W by a pair of coefficients R (X, Y ) =
|R ,Y | |R ,Y | and R (X, Y ) = |X | |X |
(66)
will be called the rough extension of R at level . Note that the rough extension of a fuzzy relation R at level from Definition 8 is actually the probabilistic rough extension of crisp relation R (which is a -level set of R), when parameter in Definition 7 is equal to zero. For an approximate evaluation of granules from R ∈ U ×W induced by composite sets from W we have the following result.
Proposition 13. Assume a crisp relation R between U and W and ∈ [0, 0.5). Then both extensions R and R of R have a lower SM -bound and an upper SL -bound with respect to the union of nonempty sets from W. Proof. For a threshold t ∈ [0, 1] we have that |(V ∪ Y ) ∩ r x | |V ∩ r x | >t ⊂ x ∈X : >t x∈X: |r x | |r x | and
|Y ∩ r x | |(V ∪ Y ) ∩ r x | x∈X: >t ⊂ x ∈X : >t . |r x | |r x |
Then for ∈ [0, 0.5) S M (R (X, V ), R (X, Y )) ⱕ R (X, V ∪ Y )
(67)
and
S M ( R (X, V ), R (X, Y )) ⱕ R (X, V ∪ Y ).
(68)
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51
On the other hand, |(Y ∪ V ) ∩ r x | |(V ∩ r x | |(Y ∩ r x | x∈X: >t ⊂ x ∈X : >t ∪ x ∈X : >t |r x | |r x | |r x | and therefore x ∈ X : |(Y ∪ V ) ∩ r x | > t ⱕ x ∈ X : |(V ∩ r x | > t + x ∈ X : |(Y ∩ r x | > t . |r x | |r x | |r x | Then for ∈ [0, 0.5) R (X, V ∪ Y ) ⱕ R (X, V ) + R (X, Y ), which means that R (X, V ∪ Y ) ⱕ min{1, R (X, V ) + R (X, Y )} = SL (R (X, V ), R (X, Y )). Analogously,
R (X, V ∪ Y ) ⱕ min{1, R (X, V ) + R (X, Y )} = SL ( R (X, V ), R (X, Y )).
Approximate evaluation of granules of a relation R between U and W induced by the union of subsets from U will be investigated for a special case when = 0. Then for all (X, Y ) ∈ 2˜ U × 2˜ W |Y ∩ r x | 0 >0 R (X, Y ) = x ∈ X : |r x | M (X, Y ). = Mx∈X (SM (R(x, y), y ∈ Y )) = MS R
Proposition 14. Let R ∈ U × W . Then for all X, Y ∈ 2˜ U , Y 2˜ W 0
0
0
0
0
TL ( R (X, Y ), R (V, Y ) ⱕ R (X ∪ V, Y ) ⱕ SL ( R (X, Y ), R (V, Y ))
(69)
TL (0R (X, Y ), 0R (V, Y ) ⱕ 0R (X ∪ V, Y ) ⱕ SL (0R (X, Y ), 0R (V, Y )).
(70)
and
0
M Proof. Because R (X, Y ) = MS (X, Y ), inequality (69) follows from Proposition 10 and the fact that the arithmetic R mean M has the upper SL -bound and the lower TL -bound with respect to the union of nonempty finite sets. Now, using Lemma 7, inequality (69) and duality between TL and SL , we will prove inequality (70):
0
0R (X ∪ V, Y ) = 1 − R (X ∪ V, Y c ) 0
0
ⱖ 1 − SL ( R (X, Y c ), R (V, Y c )) = 1 − SL (1 − 0R (X, Y ), 1 − 0R (X, Y )) = TL (0R (X, Y ), 0R (V, Y )). Analogously, 0
0R (X ∪ V, Y ) = 1 − R (X ∪ V, Y c ) 0
0
ⱕ 1 − TL ( R (X, Y c ), R (V, Y c )) = 1 − TL (1 − 0R (X, Y ), 1 − 0R (X, Y )) = SL (0R (X, Y ), 0R (V, Y )).
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6. Conclusion Approximation and approximate reasoning involving vague concepts have attracted the interest of numerous researchers and practitioners in psychology, medicine, finance, engineering and other areas of application. In our work we suggested a simple method for evaluation of composition of vague concepts described by fuzzy sets and fuzzy relations. We introduced the notion of boundaries of evaluation functions of composite concepts with respect to the binary operation used in their composition. T- and S-boundaries of evaluation functions based on frequently used aggregation functions (t-norms, t-conorms, the arithmetic mean, the geometric mean) with respect to the union of finite sets were specified. We showed that dominance of aggregation functions is an important property when boundaries of evaluation functions based on a pair of aggregation functions are considered. Evaluation and approximate evaluations of granules of a fuzzy relation by a pair of aggregation functions and an evaluation involving a generalization of the probabilistic rough set model were investigated. These evaluations allow researchers to assess the relationship between sets of related elements from two different universes. A further study of boundaries of evaluation functions may produce a more powerful and useful tool in approximate evaluation of composite concepts. In our future work we will explore T- and S-characterizations of evaluation functions taking into account classes of evaluation functions satisfying some specific properties while considering diverse binary operations. Acknowledgments The authors are grateful to the referees for their comments and suggestions for improving the paper. The work of Martin Kalina was supported by the Science and Technology Assistance Agency under the Contract no. APVV-0073-10 and by the grant agency VEGA, Grant no. 1/0143/11. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [23] [24] [25]
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