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Algorithms for solving fuzzy relational equations in a probabilistic setting
Fuzzy Sets and Systems 38 (1990) 313-327 North-Holland
313
ALGORITHMS FOR SOLVING F U Z Z Y RELATIONAL EQUATIONS IN A PROBABILISTIC SETI'ING W. PEDRYCZ Department of Electrical and Computer Engineering, University o f Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
Received March 1988 Revised March 1989 Abstract: We discuss methods suitable for finding approximate solutions to fuzzy relational equations
defined in finite universes of discouree and taking the form of either a system of equations X k • R = Yk, k = 1, 2 . . . . . K, (with R being unknown fuzzy relation) or a single equation X • R = Y (with X being looked for). The key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation (set). Numerical considerations support the established algorithms. Keywords: Fuzzy relational equations; approximate solutions; performance analysis.
1. Preliminaries The problem addressed in this paper concerns aspects of solving a system of fuzzy relational equations with max-min composition of the following type: XI.R=
Y1,
X2 " R = Y2, (1) X k . R = Yk,
X K ' R = YK, where Xk and Yk are fuzzy sets defined in finite universes of discourse, namely Xk~F(X),
X = ( X l , XE. . . . .
Xn},
Yk e F ( Y ) ,
Y = {Yl, Y2. . . . .
Ym}.
AS usual F(.) denotes a family of fuzzy sets (or relations) defined in a relevant universe of discourse (or a Cartesian product of some of them), while R stands for a fuzzy relation defined in the Cartesian product of X and ¥, R ~ F ( X x Y), and '-', refers to the max-min composition operator. In the further discussion we restrict ourselves to this type of composition, but all considerations might be immediately extended to e.g. max-t composition as well. The system of equations has to be solved with respect to R treating all pairs (Xk, Yk) as provided. 0165-0114/90/$03.50 (~ 1990--Elsevier Science Publishers B.V. (North-Holland)
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This kind of problem is often found in fuzzy modelling [4, 5, 7] where fuzzy relational equations are established as a formal means for fuzzy models. The problem refers to estimation of the parameters of the models (here summarized in the form of a fuzzy relation). Following this statement the fuzzy sets (Xk, Yk) are treated as input-output data collected for identification purposes. A direct theoretical finding refers to a maximal element of the family of fuzzy relations R fulfilling the system (1), under the obvious assumption it possesses any solution. Denote by ~ a family of fuzzy relations satisfying the k-th element of the above system,
~k = {R e F ( X × Y ) I X k " R = Yk}, and assure ~k 4: 0. Referring e.g. to [5] it has been proved that if the intersection of the families of equations forms a nonempty set, K
~= n ~4;0, k=l
then its greatest element, say/~, is taken as the intersection of a-compositions of the respective fuzzy sets (Xk, Yk), i.e. K
/~ = n (Xk a Yk) k=l
(2)
where/~k = Xk a Yk is the greatest element of the family ~t,. Recall that a stands for a-composition, namely
{lbifa<~b, ifa > b ,
aab=
a, b ~ [0, 1].
In fact, the above stated assumption is a very strong one. One could easily observe that it is enough to have one pair of sets (X~,0, Yko) for which the corresponding nonempty solution set ~k0 has an empty intersection with all the remaining families ~k, k 4: k0, namely K
~ ' = N ~,4:~, k ~k0
~kon~'=0,
and this fact implies the entire system of equations treated en bloc has no solution. Moreover bearing in mind the idea of solvability of the entire system of equations, cf. [1, 2] it would happen that just this single equation could lead to an extremely low value of the solvability index of the entire system. Therefore, if this equation were removed from (1), existence of the solution would be guaranteed. Below there are several observations deserving a certain caution and focussing some light on the way of accomplishing the task of reaching an approximate solution of the system of equations:
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(a) In solving the system of fuzzy relational equations we aggregate solutions of individual equations in a rather extremal way: here the minimal relation among maximal elements of ~i is selected. This aggregation having its theoretical background creates however very poor results when the relevant assumptions are violated. Thus we do not perform any filtering (e.g. averaging) of partial results and by default assume that there exist a fuzzy relation satisfying all equations. We can have yet another look at the problem being formulated as an interpolationlike task. Thus the pairs (Xk, Yk) form nodes over which the fuzzy relation is spanned. As for any interpolation-type problem one has to be aware that even some small disturbances of these nodes could cause significant changes in results which are generated. (b) Secondly we have to make a clear distinction between using fuzzy sets in description of our reality and further processing of membership functions of all data being gathered. While for the first area, set-theoretic operations are of significance, there cannot be any indication that at the second stage probability calculus or statistical methods are prohibited. These methods are used for dealing with processing numerical data which in turn represent fuzzy quantities. Additionally, performing some simple statistical analysis we can evaluate a precision of results that in some situations could lead to a more suitable representation expressed e.g. by interval-valued fuzzy sets or fuzzy sets of type 2. A particular analogy exists for polynomial-type of interpolation, cf. any book on numerical analysis. Then it is obvious how many constraints imposed on the problem could be satisfied in case of an interpolating polynomial of a given order. Generally, the problem (1) is of the same type. However, there it is much more difficult to establish a set of conditions under which the system of equations posesses a solution. Hence, instead of looking for a solution of the above interpolation-like problem (1) one has to try to reformulate it in an approximate-like scheme, thus searching the fuzzy relation for which a certain performance index is minimized. Some policies in gathering the fuzzy relation from the input-output fuzzy data have been investigated in [4, 5]. To get a certain view on the idea of solving the system of fuzzy relational equations let us discuss two fuzzy relations G and T with the following membership functions:
a, b ~ [0, 1]. Now take the pairs (Xk, Yk), k = 1, 2 . . . . . K - 1, K, such that the first k - 1 ones satisfy fuzzy relational equations with the relation G while the last pair (Ark, Yk) satisfies the fuzzy relational equation with the second relation T. Assume that the fuzzy sets X~, k = 1, 2 . . . . . K - 1, are equal to Xk = [1, ak] while XK=[1, bK]. Then Xk" G yields Yk with the membership function [1, a ^ ak]. For the last element of the data set we obtain YK = [b A bK, 1]. Having all the pairs collected we try to solve the entire system of equations
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simultaneously. This leads to the fuzzy relation
ak] OC[a lak])
(~i[1
~ 1 [~ k=l
17([1
ak](.,i[~Ab K
aA a A al,
br]tr[b lbr])^
~]
b A br
----
A br
('~ (a ^ a~) K-lk=l
b A br
['-'l (a
.
A ak)
k=l In case we put down b = el and a = 82 such that el
This in turn implies that for any Xk, k = 1, 2 . . . . .
Yk=[el
e2] ,
k=l,
K, one gets
2,...,K-1,
and the same holds for XK:
YK=[e,
e21.
There is no doubt that all of them differ from the corresponding fuzzy sets appearing in the original data set which differs significantly from the corresponding fuzzy sets Yk, k = 1, 2 . . . . . K, (being equal to [1, 0] for k = 1, 2 . . . . , K - 1 and to [0, 1] for k = K, respectively). In any case, when trying to solve the system of fuzzy relational equations with the empty solution set the following inequality becomes valid: K. This relationship does not cope, however, with a difference between both sides of the inclusion and as such could be misleading to a certain extent. In the above example the left-hand side of the inequality forms a membership function almost equal to zero. Hence the combination of solutions given by (2) leads to meaningless results. Referring to the interpolation-like character of the problem of solution of (1) under some extra assumptions, one could quite easily formulate conditions which are sufficient to have the entire system of equations solvable. For a single equation X . R = Y the situation is very evident: a necessary and sufficient condition of existence of the equation reads accordingly: ~#=~
iff hgt(X)>~hgt(Y),
where hgt(X) denotes the height of fuzzy set X, hgt(X)=
max X(xi). i=1,2 ..... n
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For the system of equations, one states: If ~k ~:0, k = 1, 2 . . . . . K, and Xk are degenerated fuzzy sets such that all their membership functions are equal to zero except one element of the universe of discourse and these elements are different from each other, say
XK(X,) 6ki
S 1 if i = k, 0 otherwise,
i, k = 1, 2 . . . . .
K (observe K = n),
then the system of equations has a solution. In other words the number of interpolation nodes (viz. the pairs (Xk, Yk)) is restricted directly by the dimensionality of the universe of discourse X. In general, more than n interpolation nodes could make the problems unsolvable. Additionally, the interpolation nodes have been chosen in a very specific way, since they are just degenerated fuzzy sets (singletons). In many experimental tasks they would not be acceptable as being too precise. Originally, the way of constructing the fuzzy relation as it appeared in theory does not introduce any mechanisms protecting from an equal influence of a solution of the single equation on the solutions of the remaining equations. This way of aggregation leads to immediate deterioration of the quality of the solution of the entire system. Some algorithms for determination of approximate solutions of fuzzy relational equations have already been reported in existing literature. In [6] the structure of the system of equations has been investigated and a consistent subset of fuzzy sets (Xk, Yk) has been used to calculate the solution of the entire system. In [11, 12] interval-valued fuzzy sets have been studied where intervals of the grades of membership are considered as a kind of tolerances imposed on the values of the grades of membership, el. also some methods utilizing fuzzy sets of type 2. In [4] a notion of structured fuzzy models has been introduced where the fuzzy relation is associated to a probability matrix which specifies the underlying structure of the system and summarizes frequencies of occurrence of links between some elements of the fuzzy relation. The methods proposed in this paper have an obvious idea behind them. Their essence relies on the observation that the values of the membership function of the fuzzy relation should be affected by some collected statistics of the partial results (solutions of the respective equations of the system) rather than simply resulting from picking up extremal values of the membership function of R obtained from a~-compositions of the values of the membership functions of X~ and Yk. This also enables us to incorporate a level of accuracy to the derived approximate solution. When an equality index is employed for generating the approximate solution then the result of max-min composition is viewed as an interval-valued fuzzy set giving a transparent overview on the precision of the results produced. The approach developed here opens a new area in methods for estimating approximate solutions of fuzzy relational equations and could stimulate further studies in utilizing statistical methods in processing fuzzy data.
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318 2. Algorithms
Following the facts indicated above we investigate two ways of construction of the fuzzy relation of this system of equations and indicate in which way the obtained results have to be interpreted. The central idea of the first method is to gather statistics concerning the distribution of results of it-composition of the elements of the pairs of fuzzy sets (Xk, Yk) and look for their suitable representatives. The second method makes use of results of it-composition, transforms them into a unified form of the probabilistic set and furthermore extends the max-min composition to allow handling probability and fuzziness. Before moving into details let us observe that the equations (1) can be considered separately for each element of the space Y and therefore the values in each column of R are determined separately. In other words we are looking for the values R(xi, Yj) with j fixed that satisfy a set of equations
~/
[Xk(X,) ^ R ( x , , yj)] = Yk(Yj),
k = 1, 2 . . . . .
K, j = 1, 2 , . . . ,
m.
(3)
i=1
Omitting one of the indices that is fixed we rewrite the above system as follows:
~/ (aki^Ci)=bk,
k = l , 2 . . . . ,K,
i= 1
where now the values of ci are looked for. The algorithm below deals with approximate solutions of the system of equations of the type (3) and thus the index j has to be fixed.
Method 1. In this method for each pair of elements of X and Y, say (xi, yj), one collects the results of a~-composition of Xk(Xi) and Yk(Yj) for all the pairs, k = 1, 2 . . . . .
K. These results are arranged in the form of a probability function
P(xi, yj, w), w ~ [0, 1]. Hence P(xi, yj, w) describes the probability attached to the event that the value of the a~-composition (for the pair of elements of the universes of discourse already specified) is equal to w. Because of the finite number of fuzzy sets (K), the probability function has a finite number of discrete values lower than or equal to K. They could be distributed in a different way over the unit interval as illustrated in Figure 1. The shape of this function strongly depends on the distribution of all the empirical data. It is noticeable that a minimal value of the argument for which the probability function attains nonzero value corresponds just to the result obtained in the original approach (viz. the minimum of Xk(xi) tr Yk(Yj) taken over k and equal to w0). It is evident that in many cases the total mass of probability is distributed very far from this minimal value. Thus an idea is to replace this value by another one not so extremal in its character. Loosely speaking, we are looking for the value w' which is supported by some higher value of probability. In other words, one accepts the value w' as the (i, j)-th entry of the fuzzy relation if the cumulative probability (probability distribution function) is not lower than a certain threshold
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,_
W
I
I I IIII
: I
~
w
: 1
~
w
Fig. 1. Different s h a p e s of discrete probability function.
fl, fl~ (0, 1]. Then w' results from the inequality
P(xi, yj, v) >~fl,
(4)
more precisely we take a minimal value w' leading to the satisfaction of (4). Bearing in mind the diversity of shapes of the probability function, w' could significantly differ from the value w0. The choice of this value Wo (that may vary from a minimal result of the m-composition to a maximal one) has to be performed with respect to a certain optimization criterion. Let us denote it by Q(fl) with fl standing for the threshold value of cumulative probability as expressed by (4). Then starting our search from the lowest level of fl we summarize Q(fl) and select a value for which Q attains
320
w. Pedrycz
extremal value: ext Q(fl) = Q(fl*). # The lowest possible value of fl is equal to 1/K. The corresponding value w is taken as the minimal value of argument in [0, 1] for which this property is fulfilled. The higher the value of the threshold fl (in comparison to the first nonzero value), the stronger the inconsistency in the system of equations that are solved. We would like to comment on a suitable choice of the optimization criterion. One of them, being in common use, makes use of a sum of distances, between the grades of membership of the respective fuzzy sets, namely Yk(Yj) and (Xk" R)(yj) where the fuzzy relation is formed on the basis of a particular value of the level ft. Then Q (fl) has to be minimized with respect to ft. The second type of optimization criterion is based on a sum of equality indices between the relevant grades of membership Yk(Yj) =-(Xk" R)(yj), cf. [8], and therefore Q(fl) should be maximized to get the highest degree of equality. This approach is also advantageous with regard to ability of determination of confidence intervals, compared to the results of max-min composition. As introduced in [8], the length of the confidence interval is a decreasing function of the value of the equality index. In the situation that the equality index attains 1 (which specifies complete equality of the grades of membership), the relevant confidence interval reduces to one point while for the value 0 it covers the entire unit interval. In a following numerical example we will illustrate how the confidence intervals describe the precision of results obtained while solving a system of equations. Method 2. In this method for each entry of the fuzzy relation we take into
account the probability distribution functions resulting from the re-composition of the relevant grades of membership, say Xk(Xi) and Yk(Yj) for the (i, j)-th entry. Thus instead of the fuzzy relation as originated in the first method we get a probability-fuzzy relation, or a so-called probabilistic relation [3]. Then for any fuzzy set X the max-min composition is performed as if X and R were random variables. In sequel the result of the composition is again a probabilistic set specified by its probability distribution function. Thus the method aggregates the solutions of individual equations and generates probabilistic sets which give a closer look at the precision of the results. To come up with some closed formulas, let us denote by Fmxl.yj)(w), w e [0, 1], the probability distribution function of the grades of membership of the fuzzy relation at the specified pair of coordinates (xi, yj). It originates as a result of aggregation of the a~-compositions of Xk(Xi) and Yk(Yj) for all the equations of the system. In fact, R can be treated as a probabilistic relation. Now utilizing some basic formulas of probability calculus (e.g. [10] would form a good reference in this situation), and treating Xk as a probabilistic set (described by its distribution function) let us express the distribution function of Y. Referring to maximum and minimum functions of random variables and assuming additionally that X and
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R are independent variables (which is rather a critical point in these considerations), we get
Fyty~)(w) = ~ (Fx~x,)(w) + Fmx,.yj)(w) - Fxtx,)(w) " Fmx,.yj)(w)) i=1
where Fx~,) denotes the distribution function of grades of membership of the fuzzy set X at the point xl. In fact, since X(xi) has only one specified value of the membership function for each element of the universe of discourse, we obtain
Fx~x,)(w)={~
if w < x , if w >1xi.
This implies that Fytyj) reduces to n
I
[ N i=l
Fy(yj)(W) -- ~ i:xi>w L 1 otherwise. So if w
Fy¢yj)(w) = [-[ FR¢~,.y~)(W) w < min xi, i = 1, 2 . . . . .
k.
i=1
Then one can easily determine the probability function of Y(yj) and applying a certain threshold fl to it (as in the first method) the grade of membership function of Y is determined. Observe that this approach is more elaborate; it needs processing of the distribution function of the fuzzy relation; to perform the final choice, moreover, the probability distribution function of X has to be specified. Additionally the above assumption of independency is rather artificial and difficult to satisfy. Taking X equal to X1, X 2 , . . . , XK, respectively (which is necessary to compare the resulting fuzzy set with the original one) we know that the distribution function of R has already been established with the aid of these particular fuzzy sets. Nevertheless, without this assumption no constructive calculations (and comparisons) are accessible. This approach is of significant interest in cases where X differs from X1, X2 . . . . , XK and is a genuine prohabilistic set. Then one has at one's disposal a valuable tool for performing sensitivity analysis.
Example 1. Consider a set of pairs of fuzzy sets (Xk, Yk) given in Table 1. Examine a set of induced equations X k " R = Yk. Each of them treated separately has a solution. When they are solved altogether a solution does not exist. The fuzzy relations are generated by it-composition and afterwards all results are aggregated by minimum. This yields 0.3 0.3 0.3 0.3
0.2 0.2 0.2 0.2
0.3 0.3 0.3 0.3
0.2] 0.2 0.2 ' 0.21
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Table 1. Example pair of fuzzy sets x~
1.0 0.4 0.5 016 0.9 0.5
l'k
0.4 1.0 0.5 0.4 0.6 0.6
0.3 0.6 0.8 0.7 0.4 0.8
0.1 0.4 0.9 1.0 0.3 1.0
0.5 0.3 0.4 0.5 0.4 1.0
0.5 0.3 0.2 0.6 0.3 0.6
0.7 1.0 0.2 1.0 0.5 0.3
1.0 0.9 0.6 0.3 0.7 0.2
so the values of the membership function of X • R do not exceed the level 0.3. Thus in fact we obtain results forming subsets of Yk. NOW let us apply the method proposed here. Two different performance indices are taken into account: (a) The equality index averaged over all pairs. We follow a particular form of equality index where the qg-operator is generated by product, cf. [8]. Then the equality index for two grades of membership a and b reads as a -= b = 0.5{(a q9 b) ^ (b
= 0.5[min(min(1, b / a ) , min(1, a / b ) )
+ min(min(1 x-b min/11-a a, b • [0, 1]. We discuss the averaged version of it, putting down
1 K Q = ~, ~ [(Xk" R ( y j ) - Yk(Y~)], k=l
J = 1, 2 . . . . .
m.
It is of interest to recall the so-called inverse problem in which an argument is looked for having the value of the equality index with one of the arguments, say a, not lower than y, y ~< 1. In a formal statement we get: Calculate x such that a=-x~y
is fulfilled. A set of solutions of the above inequality forms a confidence interval; cf. [81. Calculating the values of Q for different values of probability we have the results shown in Table 2. Notice that the lowest levels of probability are imposed by the number of data, e.g. 0 . 1 6 7 = 1 / 6 , 0 . 3 3 = 2 / 6 , etc. Optimal values of probability level fl are printed boldface. They are not equal to 0.167 which could indicate intersection of partial solutions. Rather higher values of the threshold are indicated as optimal ones, as e.g. 0.33 for the first coordinate and 0.5 for
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Table 2. Results for Q in ease (a) Probability
j
0.167
0.33
0.50
0.67
0.83
1
1 2 3 4
0.68 0.65 0.45 0.47
0.77 0.75 0.62 0.52
0.75 0.81 0.75 0.72
0.56 0.68 0.60 0.45
0,56 0,23 0,60 0.45
0.38 0.23 0.60 0.45
Max-rain composition
fuzzy set X
J
fuzzy
- [
Evaluation
proc~ure Y
relation
J
W-k]
T
[Q(Yl) Q(y2) ... Q(y m)] Fig. 2. A general scheme of utilization of the first approximation algorithm and determination of the confidence intervals.
remaining ones. This yields the fuzzy relation 0.4 0.4 0.4 0.4
0.3 0.7 0.7] 0.3 1.0 0.9 0.3 1.0 0.6 0.6 1.0 0.6
thus giving higher grades of membership in comparison to a single intersection of individual relations. Moreover the use of the values of the averaged equality index expands the entire analysis towards description of precision of generated results. This is schematically displayed in Figure 2. Thus following max-rain composition we get an evaluation procedure by which we replace simple membership grades by confidence intervals induced by the values of the equality indices for individual elements of 11. Following the procedure in Figure 2 the fuzzy sets X k • R with their confidence intervals are shown in Table 3. Following the values of the membership functions Yg it becomes clear that even though X k " R are not equal to them, in most cases Table 3. Fuzzy sets Xg • R with their confidence intervals
k
Yl
1 2 3 4 5 6
0.4 [0.28, 0.4 [0.28, 0.4 [0.28, 0.4 [0.28, 0.4 [0.28, 0.4 [0.28,
Y2
0.531(0.5) 0.53](0.3) 0.53](0.4) 0.53](0.5) 0.53](0.4) 0.53](1.0)
0.3 [0.22, 0.4 [0.31, 0.6 I0.50, 0.6 [0.50, 0.3 [0.22, 0.6 [0.50,
0.39](0.5) 0.50](0.3) 0.70](0.2) 0.70](0.6) 0.39](0.3) 0.70](0.6)
Y3
Y4
0.7 [0.57, 0.81](0.7) 1.0 [1.0, 1.0](1.0) 0.9 [0.83, 0.95](0.2) 1.0 [1.0, 1.0](1.0) 0.7 [0.57, 0.81](0.5) 1.0 [1.0, 1.0](0.3)
0.7 I0.54, 0.9 [0.81, 0.6 [0.44, 0.6 [0.44, 0.7 [0.54, 0.6 [0.44,
0.821(1.0) 0.95](0.9) 0.751(0.6) 0.751(0.3) 0.82](0.7) 0.75](0.2)
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Table 4. Results for Q in case (b) Probability j
0.167
0.33
0.50
0.67
0.83
1
1 2 3 4
0.22 0.22 0.42 0.42
0.15 0.15 0.22 0.35
0.15 0.12 0.17 0.17
0.32 0.23 0.25 0.35
0.32 0.55 0.25 0.35
0.45 0.55 0.25 0.35
Table 5. Distribution functions of the probabilistic relation R (lower numbers indicate the values of the distribution function while upper refer to discrete elements in [0, 1]) 0.3 0.4 0.17 0.5
0.5 1.0 0.83 1.0
0.3 0.4 0.17 0.5
1.0 1.0
0.3 0.4 0.5 0.17 0.33 0.5
0.2 0.3 0.17 0.5
0.5 1.0 0.67 1.0
0.2 0.3 0.17 0.5
1.0 1.0
0.2 0.3 0.17 0.5
1.0 1.0
0.3 0.17
0.3 0.5 0.7 0.17 0.33 0.5
1.0 1.0
0.3 0.5 0.17 0.33
0.2 0.3 0.7 0.17 0.33 0.5
1.0 1.0
0.2 0.3 0.7 1.0 0.17 0.33 0.5 1.0
1.0 1.0
0.6 1.0 0.83 1.0
1.0 1.0
0.3 0.4 0.5 0.17 0.33 0.5
1.0 1.0
0.2 0.3 0.6 1.0 0.17 0.33 0.67 1.0 0.3 0.8 1.0 0.17 0.33 1.0
0.2 0.3 0.6 0.17 0.33 0.5
1.0 1.0
0.2 0.3 0.6 0.17 0.33 0.5
1.0 1.0
t h e y a r e c o n t a i n e d in t h e c o n f i d e n c e i n t e r v a l s ; for c o m p a r i s o n t h e y a r e i n c l u d e d in t h e t a b l e ( n u m b e r s in b r a c k e t s ) . (b) N e x t we s t u d y t h e p e r f o r m a n c e i n d e x w h i c h is v i e w e d as an a v e r a g e d H a m m i n g d i s t a n c e b e t w e e n fuzzy sets:
1 r
a =g.k~l [Yk(Yi) -- (Xk"
R)(yj)I.
T h e n t h e o b t a i n e d results a r e g i v e n in T a b l e 4. T h u s t h e m i n i m a l v a l u e s o f Q o c c u r for t h e s a m e v a l u e s o f t h e t h r e s h o l d so t h e fuzzy r e l a t i o n R is e x a c t l y t h e s a m e as b e f o r e . A d i s a d v a n t a g e o f this p e r f o r m a n c e i n d e x with r e s p e c t to t h e e q u a l i t y i n d e x is t h a t n o w w e d o n o t p o s s e s s a n y m e c h a n i s m i n v o l v e d with e v a l u a t i o n o f t h e p r e c i s i o n o f t h e results. F o l l o w i n g t h e s e c o n d m e t h o d t h e p r o b a b i l i s t i c r e l a t i o n is c a l c u l a t e d w h e r e e a c h e n t r y o f t h e r e l a t i o n is n o w r e p r e s e n t e d b y its p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n ; see T a b l e 5. C o n t r a r y to t h e first m e t h o d w h e r e t h e i n f o r m a t i o n d e a l i n g with p r e c i s i o n o f t h e results is a s s o c i a t e d to t h e s p a c e Y ( d i s t r i b u t e d b y (Q(YO, Q ( y 2 ) , . . . . Q(y,,,)) n o w this t y p e o f i n f o r m a t i o n is c o v e r e d b y t h e r e l a t i o n itself. T h e p r o c e s s o f c a l c u l a t i o n o f m a x - m i n c o m p o s i t i o n is s u m m a r i z e d in F i g u r e 3. Max-rain Composition fuzzy set X
J
probabilislic
-I
rclatitm
l~babilistic set Y v
Fig. 3. The process of calculation of max-min composition with probabilistic sets.
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3. Solving fuzzy relational equations X - R -----Y
In this section we indicate how, following the same way which has been used for solving a system of equations, a single equation
X.R=Y is solved with R and Y provided. Referring to well-known theoretical results [9] we know that if ~ = {X I X • R = Y} 4=0, viz. there exists a nonempty family of solutions, then the greatest element of it is calculated by means of m-composition of R and Y, R=Rmr, i.e. .~'(x~) =
min j=l,2,...,m
[R(x~, yj) mY(yj)],
i = 1, 2 , . . . , n.
(5)
To make use of the previous results we will reformulate the above problem and rewrite it in the same format as already discussed. Observe that the original equation has an equivalent representation in terms of a set of equations while each of them is indexed by consecutive elements of ¥, Yl, Y2. . . . . y,,. This yields . R 1 = Y(YO, X " RE = Y(Y2),
X
X " Rm = Y(Ym), where g l , R E , . . . , R,, are corresponding columns of the fuzzy relation R. Thus the max-min composition specified above is performed over two fuzzy sets (X and Rj) returning a scalar value Y(yj). To get a consistent picture with the system of equations being previously examined we rearrange them slightly: RI" X = Y(Y0, g2" X = Y(Y2),
(6)
R , , . X = y(ym), so now it is clear the X plays a role of the fuzzy relation from the system (1). For each equation studied separately the corresponding solution (maximal one) is given by
f((x,, Yi) = Rj(x,) m r(yj),
j = 1, 2 . . . . .
m,
where Ri(xi) = Ri(x i, yj). The second argument of X underlines the origin of this solution. Hence the formula (5) is obvious now: .~'(xi) is taken as intersection of results coming from separate equations, X(x,) = rain [Ri(x, ) m Y(y~)]. yj
Now the system (6) is solved with the aid of the method worked out before. Example 2. The fuzzy relation R and the fuzzy set Y are provided by means of
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326
Table 6. Results for Q Probability value
0.17
0.33
0.5
0.67
0.83
1.0
Q
0.37
0.44
0.61
0.49
0.39
0.34
the following membership function: 1.0 0.2 0.7 0.0 1.0 0.5q 0.6 0.8 0.9 0.2 0.6 0.3 R= 0.5 0.5 0.5 0.7 0.2 0.2 1.0 0.3 1.0 0.9 1.0 0.6 0.7 0.6 0.4 1.0 0.5 0.7 Y = 0.1 0.0 0.8 0.6 0.4 0.7].
t
Rewriting the equation X • R = Y as a system of equations we get [0.1
0.6
0.5
1.0
0.7].X=0.1,
[0.2
0.8
0.5
0.3
0.6]-X=0.0,
[0.7 0.9 0.5 1.0 0.41-x=0.8, [0.0
0.2
0.7
0.9
1.0].X=0.6,
[1.0
0.6
0.2
1.0
0.51 . X = 0 . 4 ,
[0.5
0.3
0.2
0.6
0.7]-X=0.7.
Each of the above equations has a solution: the condition maxx, R(xi, yj) >! Y(yj) is satisfied for each j. Also direct use of the theoretical result (5) gives a completely meaningless result, say X = R a~ Y = [0 0 0 0 0]. The first method with the performance index expressed by the equality index produced the results of Table 6, and the fuzzy set X is equal to [0.4 0.4 0.6 0.4 0.4]. Performing calculations with Q specified by the Hamming distance, the optimal value of fl is equal to 0.5 yielding the same fuzzy solution set.
4. Concluding remarks The methodology of deriving approximate solutions of fuzzy relational equations enables us to overcome shortcomings arising in a blind use of methods that are appropriate only in cases where the existence of solutions is assured. The main idea is to make use of statistical records covering partial results, namely results referring to individual equations. This statistical information is gathered and incorporated into a degree leading to extremal values of the performance index. Among two types of performance indices discussed here the equality index is more attractive due to its ability of determination of precision of the results of max-min composition.
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It is also of importance that statistical methods for coping with situations when there is no exact solution should be studied. This case generates an evident departure from the interpolation-like f o r m a t c o m m o n in fuzzy relational equations in favor of approximation-like formulation of those problems. T h e p a p e r dealt with m a x - m i n composition. H o w e v e r the same methodological frame applies for other types of relational equations as e.g. adjoint equations or m i n - m a x ones. T h e only difference is that now the adjustment of the direction of threshold values depends on the character of the solution. Thus if maximal solutions are to be summarized then the value of the threshold has to be increased; the opposite holds for minimal solutions.
Acknowledgement Support provided by the University of Manitoba (operating grant in aid of research) is highly appreciated.
References [1] S. Gottwald, On the existence of solutions of systems of fuzzy equations, Fuzzy Sets and Systems 12 (1984) 301-302. [2] S. Gottwald and W. Pedrycz, Solvability of fuzzy relational equations and manipulation of fuzzy data, Fuzzy Sets and Systems 18 (1986) 45-65. [3] K. Hirota, Concepts of probabilistic sets, Fuzzy Sets and Systems 1 (1981) 31-46. [4] W. Pedrycz, Structured fuzzy models. Cybernetics and Systems 16 (1985) 103-117. [5] W. Pedrycz, Applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data, Fuzzy Sets and Systems 16 (1985) 163-175. [6] W. Pedrycz, Approximate solutions of fuzzy relational equations, Fuzzy Sets and Systems 28 (1988) 183-202, [7] W. Pedrycz, Relevancy of fuzzy models, Inform. Sci. in press. [8] W. Pedrycz, Direct and inverse problem in comparison of fuzzy data, Fuzzy Sets and Systems 34 (1990) 223-235. [9] E. Sanchez, Resolution of composite relation equations, Inform. and Control 30 (1976) 38-48. [10] K.S. Trivedi, Probability and Statistics with Reliability in Querying and Computer Science Applications (Prentice-Hall, Englewood Cliffs, NJ, 1982). [11] M. Wagenknecht and K. Hartmann, Fuzzy modelling with tolerances, Fuzzy Sets and Systems 20 (1986) 325-332. [12] M. Wagenknecht and K. Hartman, On direct and inverse problems for fuzzy equation systems with tolerances, Fuzzy Sets and Systems, 24 (1987) 93-102.
313
ALGORITHMS FOR SOLVING F U Z Z Y RELATIONAL EQUATIONS IN A PROBABILISTIC SETI'ING W. PEDRYCZ Department of Electrical and Computer Engineering, University o f Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
Received March 1988 Revised March 1989 Abstract: We discuss methods suitable for finding approximate solutions to fuzzy relational equations
defined in finite universes of discouree and taking the form of either a system of equations X k • R = Yk, k = 1, 2 . . . . . K, (with R being unknown fuzzy relation) or a single equation X • R = Y (with X being looked for). The key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation (set). Numerical considerations support the established algorithms. Keywords: Fuzzy relational equations; approximate solutions; performance analysis.
1. Preliminaries The problem addressed in this paper concerns aspects of solving a system of fuzzy relational equations with max-min composition of the following type: XI.R=
Y1,
X2 " R = Y2, (1) X k . R = Yk,
X K ' R = YK, where Xk and Yk are fuzzy sets defined in finite universes of discourse, namely Xk~F(X),
X = ( X l , XE. . . . .
Xn},
Yk e F ( Y ) ,
Y = {Yl, Y2. . . . .
Ym}.
AS usual F(.) denotes a family of fuzzy sets (or relations) defined in a relevant universe of discourse (or a Cartesian product of some of them), while R stands for a fuzzy relation defined in the Cartesian product of X and ¥, R ~ F ( X x Y), and '-', refers to the max-min composition operator. In the further discussion we restrict ourselves to this type of composition, but all considerations might be immediately extended to e.g. max-t composition as well. The system of equations has to be solved with respect to R treating all pairs (Xk, Yk) as provided. 0165-0114/90/$03.50 (~ 1990--Elsevier Science Publishers B.V. (North-Holland)
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314
This kind of problem is often found in fuzzy modelling [4, 5, 7] where fuzzy relational equations are established as a formal means for fuzzy models. The problem refers to estimation of the parameters of the models (here summarized in the form of a fuzzy relation). Following this statement the fuzzy sets (Xk, Yk) are treated as input-output data collected for identification purposes. A direct theoretical finding refers to a maximal element of the family of fuzzy relations R fulfilling the system (1), under the obvious assumption it possesses any solution. Denote by ~ a family of fuzzy relations satisfying the k-th element of the above system,
~k = {R e F ( X × Y ) I X k " R = Yk}, and assure ~k 4: 0. Referring e.g. to [5] it has been proved that if the intersection of the families of equations forms a nonempty set, K
~= n ~4;0, k=l
then its greatest element, say/~, is taken as the intersection of a-compositions of the respective fuzzy sets (Xk, Yk), i.e. K
/~ = n (Xk a Yk) k=l
(2)
where/~k = Xk a Yk is the greatest element of the family ~t,. Recall that a stands for a-composition, namely
{lbifa<~b, ifa > b ,
aab=
a, b ~ [0, 1].
In fact, the above stated assumption is a very strong one. One could easily observe that it is enough to have one pair of sets (X~,0, Yko) for which the corresponding nonempty solution set ~k0 has an empty intersection with all the remaining families ~k, k 4: k0, namely K
~ ' = N ~,4:~, k ~k0
~kon~'=0,
and this fact implies the entire system of equations treated en bloc has no solution. Moreover bearing in mind the idea of solvability of the entire system of equations, cf. [1, 2] it would happen that just this single equation could lead to an extremely low value of the solvability index of the entire system. Therefore, if this equation were removed from (1), existence of the solution would be guaranteed. Below there are several observations deserving a certain caution and focussing some light on the way of accomplishing the task of reaching an approximate solution of the system of equations:
Solving fuzzy relational equations
315
(a) In solving the system of fuzzy relational equations we aggregate solutions of individual equations in a rather extremal way: here the minimal relation among maximal elements of ~i is selected. This aggregation having its theoretical background creates however very poor results when the relevant assumptions are violated. Thus we do not perform any filtering (e.g. averaging) of partial results and by default assume that there exist a fuzzy relation satisfying all equations. We can have yet another look at the problem being formulated as an interpolationlike task. Thus the pairs (Xk, Yk) form nodes over which the fuzzy relation is spanned. As for any interpolation-type problem one has to be aware that even some small disturbances of these nodes could cause significant changes in results which are generated. (b) Secondly we have to make a clear distinction between using fuzzy sets in description of our reality and further processing of membership functions of all data being gathered. While for the first area, set-theoretic operations are of significance, there cannot be any indication that at the second stage probability calculus or statistical methods are prohibited. These methods are used for dealing with processing numerical data which in turn represent fuzzy quantities. Additionally, performing some simple statistical analysis we can evaluate a precision of results that in some situations could lead to a more suitable representation expressed e.g. by interval-valued fuzzy sets or fuzzy sets of type 2. A particular analogy exists for polynomial-type of interpolation, cf. any book on numerical analysis. Then it is obvious how many constraints imposed on the problem could be satisfied in case of an interpolating polynomial of a given order. Generally, the problem (1) is of the same type. However, there it is much more difficult to establish a set of conditions under which the system of equations posesses a solution. Hence, instead of looking for a solution of the above interpolation-like problem (1) one has to try to reformulate it in an approximate-like scheme, thus searching the fuzzy relation for which a certain performance index is minimized. Some policies in gathering the fuzzy relation from the input-output fuzzy data have been investigated in [4, 5]. To get a certain view on the idea of solving the system of fuzzy relational equations let us discuss two fuzzy relations G and T with the following membership functions:
a, b ~ [0, 1]. Now take the pairs (Xk, Yk), k = 1, 2 . . . . . K - 1, K, such that the first k - 1 ones satisfy fuzzy relational equations with the relation G while the last pair (Ark, Yk) satisfies the fuzzy relational equation with the second relation T. Assume that the fuzzy sets X~, k = 1, 2 . . . . . K - 1, are equal to Xk = [1, ak] while XK=[1, bK]. Then Xk" G yields Yk with the membership function [1, a ^ ak]. For the last element of the data set we obtain YK = [b A bK, 1]. Having all the pairs collected we try to solve the entire system of equations
W. Pedrycz
316
simultaneously. This leads to the fuzzy relation
ak] OC[a lak])
(~i[1
~ 1 [~ k=l
17([1
ak](.,i[~Ab K
aA a A al,
br]tr[b lbr])^
~]
b A br
----
A br
('~ (a ^ a~) K-lk=l
b A br
['-'l (a
.
A ak)
k=l In case we put down b = el and a = 82 such that el
This in turn implies that for any Xk, k = 1, 2 . . . . .
Yk=[el
e2] ,
k=l,
K, one gets
2,...,K-1,
and the same holds for XK:
YK=[e,
e21.
There is no doubt that all of them differ from the corresponding fuzzy sets appearing in the original data set which differs significantly from the corresponding fuzzy sets Yk, k = 1, 2 . . . . . K, (being equal to [1, 0] for k = 1, 2 . . . . , K - 1 and to [0, 1] for k = K, respectively). In any case, when trying to solve the system of fuzzy relational equations with the empty solution set the following inequality becomes valid: K. This relationship does not cope, however, with a difference between both sides of the inclusion and as such could be misleading to a certain extent. In the above example the left-hand side of the inequality forms a membership function almost equal to zero. Hence the combination of solutions given by (2) leads to meaningless results. Referring to the interpolation-like character of the problem of solution of (1) under some extra assumptions, one could quite easily formulate conditions which are sufficient to have the entire system of equations solvable. For a single equation X . R = Y the situation is very evident: a necessary and sufficient condition of existence of the equation reads accordingly: ~#=~
iff hgt(X)>~hgt(Y),
where hgt(X) denotes the height of fuzzy set X, hgt(X)=
max X(xi). i=1,2 ..... n
Solving fuzzy relational equations
317
For the system of equations, one states: If ~k ~:0, k = 1, 2 . . . . . K, and Xk are degenerated fuzzy sets such that all their membership functions are equal to zero except one element of the universe of discourse and these elements are different from each other, say
XK(X,) 6ki
S 1 if i = k, 0 otherwise,
i, k = 1, 2 . . . . .
K (observe K = n),
then the system of equations has a solution. In other words the number of interpolation nodes (viz. the pairs (Xk, Yk)) is restricted directly by the dimensionality of the universe of discourse X. In general, more than n interpolation nodes could make the problems unsolvable. Additionally, the interpolation nodes have been chosen in a very specific way, since they are just degenerated fuzzy sets (singletons). In many experimental tasks they would not be acceptable as being too precise. Originally, the way of constructing the fuzzy relation as it appeared in theory does not introduce any mechanisms protecting from an equal influence of a solution of the single equation on the solutions of the remaining equations. This way of aggregation leads to immediate deterioration of the quality of the solution of the entire system. Some algorithms for determination of approximate solutions of fuzzy relational equations have already been reported in existing literature. In [6] the structure of the system of equations has been investigated and a consistent subset of fuzzy sets (Xk, Yk) has been used to calculate the solution of the entire system. In [11, 12] interval-valued fuzzy sets have been studied where intervals of the grades of membership are considered as a kind of tolerances imposed on the values of the grades of membership, el. also some methods utilizing fuzzy sets of type 2. In [4] a notion of structured fuzzy models has been introduced where the fuzzy relation is associated to a probability matrix which specifies the underlying structure of the system and summarizes frequencies of occurrence of links between some elements of the fuzzy relation. The methods proposed in this paper have an obvious idea behind them. Their essence relies on the observation that the values of the membership function of the fuzzy relation should be affected by some collected statistics of the partial results (solutions of the respective equations of the system) rather than simply resulting from picking up extremal values of the membership function of R obtained from a~-compositions of the values of the membership functions of X~ and Yk. This also enables us to incorporate a level of accuracy to the derived approximate solution. When an equality index is employed for generating the approximate solution then the result of max-min composition is viewed as an interval-valued fuzzy set giving a transparent overview on the precision of the results produced. The approach developed here opens a new area in methods for estimating approximate solutions of fuzzy relational equations and could stimulate further studies in utilizing statistical methods in processing fuzzy data.
W. Pedrycz
318 2. Algorithms
Following the facts indicated above we investigate two ways of construction of the fuzzy relation of this system of equations and indicate in which way the obtained results have to be interpreted. The central idea of the first method is to gather statistics concerning the distribution of results of it-composition of the elements of the pairs of fuzzy sets (Xk, Yk) and look for their suitable representatives. The second method makes use of results of it-composition, transforms them into a unified form of the probabilistic set and furthermore extends the max-min composition to allow handling probability and fuzziness. Before moving into details let us observe that the equations (1) can be considered separately for each element of the space Y and therefore the values in each column of R are determined separately. In other words we are looking for the values R(xi, Yj) with j fixed that satisfy a set of equations
~/
[Xk(X,) ^ R ( x , , yj)] = Yk(Yj),
k = 1, 2 . . . . .
K, j = 1, 2 , . . . ,
m.
(3)
i=1
Omitting one of the indices that is fixed we rewrite the above system as follows:
~/ (aki^Ci)=bk,
k = l , 2 . . . . ,K,
i= 1
where now the values of ci are looked for. The algorithm below deals with approximate solutions of the system of equations of the type (3) and thus the index j has to be fixed.
Method 1. In this method for each pair of elements of X and Y, say (xi, yj), one collects the results of a~-composition of Xk(Xi) and Yk(Yj) for all the pairs, k = 1, 2 . . . . .
K. These results are arranged in the form of a probability function
P(xi, yj, w), w ~ [0, 1]. Hence P(xi, yj, w) describes the probability attached to the event that the value of the a~-composition (for the pair of elements of the universes of discourse already specified) is equal to w. Because of the finite number of fuzzy sets (K), the probability function has a finite number of discrete values lower than or equal to K. They could be distributed in a different way over the unit interval as illustrated in Figure 1. The shape of this function strongly depends on the distribution of all the empirical data. It is noticeable that a minimal value of the argument for which the probability function attains nonzero value corresponds just to the result obtained in the original approach (viz. the minimum of Xk(xi) tr Yk(Yj) taken over k and equal to w0). It is evident that in many cases the total mass of probability is distributed very far from this minimal value. Thus an idea is to replace this value by another one not so extremal in its character. Loosely speaking, we are looking for the value w' which is supported by some higher value of probability. In other words, one accepts the value w' as the (i, j)-th entry of the fuzzy relation if the cumulative probability (probability distribution function) is not lower than a certain threshold
Solving fuzzy relational equations
Ill
319
,_
W
I
I I IIII
: I
~
w
: 1
~
w
Fig. 1. Different s h a p e s of discrete probability function.
fl, fl~ (0, 1]. Then w' results from the inequality
P(xi, yj, v) >~fl,
(4)
more precisely we take a minimal value w' leading to the satisfaction of (4). Bearing in mind the diversity of shapes of the probability function, w' could significantly differ from the value w0. The choice of this value Wo (that may vary from a minimal result of the m-composition to a maximal one) has to be performed with respect to a certain optimization criterion. Let us denote it by Q(fl) with fl standing for the threshold value of cumulative probability as expressed by (4). Then starting our search from the lowest level of fl we summarize Q(fl) and select a value for which Q attains
320
w. Pedrycz
extremal value: ext Q(fl) = Q(fl*). # The lowest possible value of fl is equal to 1/K. The corresponding value w is taken as the minimal value of argument in [0, 1] for which this property is fulfilled. The higher the value of the threshold fl (in comparison to the first nonzero value), the stronger the inconsistency in the system of equations that are solved. We would like to comment on a suitable choice of the optimization criterion. One of them, being in common use, makes use of a sum of distances, between the grades of membership of the respective fuzzy sets, namely Yk(Yj) and (Xk" R)(yj) where the fuzzy relation is formed on the basis of a particular value of the level ft. Then Q (fl) has to be minimized with respect to ft. The second type of optimization criterion is based on a sum of equality indices between the relevant grades of membership Yk(Yj) =-(Xk" R)(yj), cf. [8], and therefore Q(fl) should be maximized to get the highest degree of equality. This approach is also advantageous with regard to ability of determination of confidence intervals, compared to the results of max-min composition. As introduced in [8], the length of the confidence interval is a decreasing function of the value of the equality index. In the situation that the equality index attains 1 (which specifies complete equality of the grades of membership), the relevant confidence interval reduces to one point while for the value 0 it covers the entire unit interval. In a following numerical example we will illustrate how the confidence intervals describe the precision of results obtained while solving a system of equations. Method 2. In this method for each entry of the fuzzy relation we take into
account the probability distribution functions resulting from the re-composition of the relevant grades of membership, say Xk(Xi) and Yk(Yj) for the (i, j)-th entry. Thus instead of the fuzzy relation as originated in the first method we get a probability-fuzzy relation, or a so-called probabilistic relation [3]. Then for any fuzzy set X the max-min composition is performed as if X and R were random variables. In sequel the result of the composition is again a probabilistic set specified by its probability distribution function. Thus the method aggregates the solutions of individual equations and generates probabilistic sets which give a closer look at the precision of the results. To come up with some closed formulas, let us denote by Fmxl.yj)(w), w e [0, 1], the probability distribution function of the grades of membership of the fuzzy relation at the specified pair of coordinates (xi, yj). It originates as a result of aggregation of the a~-compositions of Xk(Xi) and Yk(Yj) for all the equations of the system. In fact, R can be treated as a probabilistic relation. Now utilizing some basic formulas of probability calculus (e.g. [10] would form a good reference in this situation), and treating Xk as a probabilistic set (described by its distribution function) let us express the distribution function of Y. Referring to maximum and minimum functions of random variables and assuming additionally that X and
Solving fuzzy relational equations
321
R are independent variables (which is rather a critical point in these considerations), we get
Fyty~)(w) = ~ (Fx~x,)(w) + Fmx,.yj)(w) - Fxtx,)(w) " Fmx,.yj)(w)) i=1
where Fx~,) denotes the distribution function of grades of membership of the fuzzy set X at the point xl. In fact, since X(xi) has only one specified value of the membership function for each element of the universe of discourse, we obtain
Fx~x,)(w)={~
if w < x , if w >1xi.
This implies that Fytyj) reduces to n
I
[ N i=l
Fy(yj)(W) -- ~ i:xi>w L 1 otherwise. So if w
Fy¢yj)(w) = [-[ FR¢~,.y~)(W) w < min xi, i = 1, 2 . . . . .
k.
i=1
Then one can easily determine the probability function of Y(yj) and applying a certain threshold fl to it (as in the first method) the grade of membership function of Y is determined. Observe that this approach is more elaborate; it needs processing of the distribution function of the fuzzy relation; to perform the final choice, moreover, the probability distribution function of X has to be specified. Additionally the above assumption of independency is rather artificial and difficult to satisfy. Taking X equal to X1, X 2 , . . . , XK, respectively (which is necessary to compare the resulting fuzzy set with the original one) we know that the distribution function of R has already been established with the aid of these particular fuzzy sets. Nevertheless, without this assumption no constructive calculations (and comparisons) are accessible. This approach is of significant interest in cases where X differs from X1, X2 . . . . , XK and is a genuine prohabilistic set. Then one has at one's disposal a valuable tool for performing sensitivity analysis.
Example 1. Consider a set of pairs of fuzzy sets (Xk, Yk) given in Table 1. Examine a set of induced equations X k " R = Yk. Each of them treated separately has a solution. When they are solved altogether a solution does not exist. The fuzzy relations are generated by it-composition and afterwards all results are aggregated by minimum. This yields 0.3 0.3 0.3 0.3
0.2 0.2 0.2 0.2
0.3 0.3 0.3 0.3
0.2] 0.2 0.2 ' 0.21
322
W. Pedrycz
Table 1. Example pair of fuzzy sets x~
1.0 0.4 0.5 016 0.9 0.5
l'k
0.4 1.0 0.5 0.4 0.6 0.6
0.3 0.6 0.8 0.7 0.4 0.8
0.1 0.4 0.9 1.0 0.3 1.0
0.5 0.3 0.4 0.5 0.4 1.0
0.5 0.3 0.2 0.6 0.3 0.6
0.7 1.0 0.2 1.0 0.5 0.3
1.0 0.9 0.6 0.3 0.7 0.2
so the values of the membership function of X • R do not exceed the level 0.3. Thus in fact we obtain results forming subsets of Yk. NOW let us apply the method proposed here. Two different performance indices are taken into account: (a) The equality index averaged over all pairs. We follow a particular form of equality index where the qg-operator is generated by product, cf. [8]. Then the equality index for two grades of membership a and b reads as a -= b = 0.5{(a q9 b) ^ (b
= 0.5[min(min(1, b / a ) , min(1, a / b ) )
+ min(min(1 x-b min/11-a a, b • [0, 1]. We discuss the averaged version of it, putting down
1 K Q = ~, ~ [(Xk" R ( y j ) - Yk(Y~)], k=l
J = 1, 2 . . . . .
m.
It is of interest to recall the so-called inverse problem in which an argument is looked for having the value of the equality index with one of the arguments, say a, not lower than y, y ~< 1. In a formal statement we get: Calculate x such that a=-x~y
is fulfilled. A set of solutions of the above inequality forms a confidence interval; cf. [81. Calculating the values of Q for different values of probability we have the results shown in Table 2. Notice that the lowest levels of probability are imposed by the number of data, e.g. 0 . 1 6 7 = 1 / 6 , 0 . 3 3 = 2 / 6 , etc. Optimal values of probability level fl are printed boldface. They are not equal to 0.167 which could indicate intersection of partial solutions. Rather higher values of the threshold are indicated as optimal ones, as e.g. 0.33 for the first coordinate and 0.5 for
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323
Table 2. Results for Q in ease (a) Probability
j
0.167
0.33
0.50
0.67
0.83
1
1 2 3 4
0.68 0.65 0.45 0.47
0.77 0.75 0.62 0.52
0.75 0.81 0.75 0.72
0.56 0.68 0.60 0.45
0,56 0,23 0,60 0.45
0.38 0.23 0.60 0.45
Max-rain composition
fuzzy set X
J
fuzzy
- [
Evaluation
proc~ure Y
relation
J
W-k]
T
[Q(Yl) Q(y2) ... Q(y m)] Fig. 2. A general scheme of utilization of the first approximation algorithm and determination of the confidence intervals.
remaining ones. This yields the fuzzy relation 0.4 0.4 0.4 0.4
0.3 0.7 0.7] 0.3 1.0 0.9 0.3 1.0 0.6 0.6 1.0 0.6
thus giving higher grades of membership in comparison to a single intersection of individual relations. Moreover the use of the values of the averaged equality index expands the entire analysis towards description of precision of generated results. This is schematically displayed in Figure 2. Thus following max-rain composition we get an evaluation procedure by which we replace simple membership grades by confidence intervals induced by the values of the equality indices for individual elements of 11. Following the procedure in Figure 2 the fuzzy sets X k • R with their confidence intervals are shown in Table 3. Following the values of the membership functions Yg it becomes clear that even though X k " R are not equal to them, in most cases Table 3. Fuzzy sets Xg • R with their confidence intervals
k
Yl
1 2 3 4 5 6
0.4 [0.28, 0.4 [0.28, 0.4 [0.28, 0.4 [0.28, 0.4 [0.28, 0.4 [0.28,
Y2
0.531(0.5) 0.53](0.3) 0.53](0.4) 0.53](0.5) 0.53](0.4) 0.53](1.0)
0.3 [0.22, 0.4 [0.31, 0.6 I0.50, 0.6 [0.50, 0.3 [0.22, 0.6 [0.50,
0.39](0.5) 0.50](0.3) 0.70](0.2) 0.70](0.6) 0.39](0.3) 0.70](0.6)
Y3
Y4
0.7 [0.57, 0.81](0.7) 1.0 [1.0, 1.0](1.0) 0.9 [0.83, 0.95](0.2) 1.0 [1.0, 1.0](1.0) 0.7 [0.57, 0.81](0.5) 1.0 [1.0, 1.0](0.3)
0.7 I0.54, 0.9 [0.81, 0.6 [0.44, 0.6 [0.44, 0.7 [0.54, 0.6 [0.44,
0.821(1.0) 0.95](0.9) 0.751(0.6) 0.751(0.3) 0.82](0.7) 0.75](0.2)
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Table 4. Results for Q in case (b) Probability j
0.167
0.33
0.50
0.67
0.83
1
1 2 3 4
0.22 0.22 0.42 0.42
0.15 0.15 0.22 0.35
0.15 0.12 0.17 0.17
0.32 0.23 0.25 0.35
0.32 0.55 0.25 0.35
0.45 0.55 0.25 0.35
Table 5. Distribution functions of the probabilistic relation R (lower numbers indicate the values of the distribution function while upper refer to discrete elements in [0, 1]) 0.3 0.4 0.17 0.5
0.5 1.0 0.83 1.0
0.3 0.4 0.17 0.5
1.0 1.0
0.3 0.4 0.5 0.17 0.33 0.5
0.2 0.3 0.17 0.5
0.5 1.0 0.67 1.0
0.2 0.3 0.17 0.5
1.0 1.0
0.2 0.3 0.17 0.5
1.0 1.0
0.3 0.17
0.3 0.5 0.7 0.17 0.33 0.5
1.0 1.0
0.3 0.5 0.17 0.33
0.2 0.3 0.7 0.17 0.33 0.5
1.0 1.0
0.2 0.3 0.7 1.0 0.17 0.33 0.5 1.0
1.0 1.0
0.6 1.0 0.83 1.0
1.0 1.0
0.3 0.4 0.5 0.17 0.33 0.5
1.0 1.0
0.2 0.3 0.6 1.0 0.17 0.33 0.67 1.0 0.3 0.8 1.0 0.17 0.33 1.0
0.2 0.3 0.6 0.17 0.33 0.5
1.0 1.0
0.2 0.3 0.6 0.17 0.33 0.5
1.0 1.0
t h e y a r e c o n t a i n e d in t h e c o n f i d e n c e i n t e r v a l s ; for c o m p a r i s o n t h e y a r e i n c l u d e d in t h e t a b l e ( n u m b e r s in b r a c k e t s ) . (b) N e x t we s t u d y t h e p e r f o r m a n c e i n d e x w h i c h is v i e w e d as an a v e r a g e d H a m m i n g d i s t a n c e b e t w e e n fuzzy sets:
1 r
a =g.k~l [Yk(Yi) -- (Xk"
R)(yj)I.
T h e n t h e o b t a i n e d results a r e g i v e n in T a b l e 4. T h u s t h e m i n i m a l v a l u e s o f Q o c c u r for t h e s a m e v a l u e s o f t h e t h r e s h o l d so t h e fuzzy r e l a t i o n R is e x a c t l y t h e s a m e as b e f o r e . A d i s a d v a n t a g e o f this p e r f o r m a n c e i n d e x with r e s p e c t to t h e e q u a l i t y i n d e x is t h a t n o w w e d o n o t p o s s e s s a n y m e c h a n i s m i n v o l v e d with e v a l u a t i o n o f t h e p r e c i s i o n o f t h e results. F o l l o w i n g t h e s e c o n d m e t h o d t h e p r o b a b i l i s t i c r e l a t i o n is c a l c u l a t e d w h e r e e a c h e n t r y o f t h e r e l a t i o n is n o w r e p r e s e n t e d b y its p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n ; see T a b l e 5. C o n t r a r y to t h e first m e t h o d w h e r e t h e i n f o r m a t i o n d e a l i n g with p r e c i s i o n o f t h e results is a s s o c i a t e d to t h e s p a c e Y ( d i s t r i b u t e d b y (Q(YO, Q ( y 2 ) , . . . . Q(y,,,)) n o w this t y p e o f i n f o r m a t i o n is c o v e r e d b y t h e r e l a t i o n itself. T h e p r o c e s s o f c a l c u l a t i o n o f m a x - m i n c o m p o s i t i o n is s u m m a r i z e d in F i g u r e 3. Max-rain Composition fuzzy set X
J
probabilislic
-I
rclatitm
l~babilistic set Y v
Fig. 3. The process of calculation of max-min composition with probabilistic sets.
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325
3. Solving fuzzy relational equations X - R -----Y
In this section we indicate how, following the same way which has been used for solving a system of equations, a single equation
X.R=Y is solved with R and Y provided. Referring to well-known theoretical results [9] we know that if ~ = {X I X • R = Y} 4=0, viz. there exists a nonempty family of solutions, then the greatest element of it is calculated by means of m-composition of R and Y, R=Rmr, i.e. .~'(x~) =
min j=l,2,...,m
[R(x~, yj) mY(yj)],
i = 1, 2 , . . . , n.
(5)
To make use of the previous results we will reformulate the above problem and rewrite it in the same format as already discussed. Observe that the original equation has an equivalent representation in terms of a set of equations while each of them is indexed by consecutive elements of ¥, Yl, Y2. . . . . y,,. This yields . R 1 = Y(YO, X " RE = Y(Y2),
X
X " Rm = Y(Ym), where g l , R E , . . . , R,, are corresponding columns of the fuzzy relation R. Thus the max-min composition specified above is performed over two fuzzy sets (X and Rj) returning a scalar value Y(yj). To get a consistent picture with the system of equations being previously examined we rearrange them slightly: RI" X = Y(Y0, g2" X = Y(Y2),
(6)
R , , . X = y(ym), so now it is clear the X plays a role of the fuzzy relation from the system (1). For each equation studied separately the corresponding solution (maximal one) is given by
f((x,, Yi) = Rj(x,) m r(yj),
j = 1, 2 . . . . .
m,
where Ri(xi) = Ri(x i, yj). The second argument of X underlines the origin of this solution. Hence the formula (5) is obvious now: .~'(xi) is taken as intersection of results coming from separate equations, X(x,) = rain [Ri(x, ) m Y(y~)]. yj
Now the system (6) is solved with the aid of the method worked out before. Example 2. The fuzzy relation R and the fuzzy set Y are provided by means of
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326
Table 6. Results for Q Probability value
0.17
0.33
0.5
0.67
0.83
1.0
Q
0.37
0.44
0.61
0.49
0.39
0.34
the following membership function: 1.0 0.2 0.7 0.0 1.0 0.5q 0.6 0.8 0.9 0.2 0.6 0.3 R= 0.5 0.5 0.5 0.7 0.2 0.2 1.0 0.3 1.0 0.9 1.0 0.6 0.7 0.6 0.4 1.0 0.5 0.7 Y = 0.1 0.0 0.8 0.6 0.4 0.7].
t
Rewriting the equation X • R = Y as a system of equations we get [0.1
0.6
0.5
1.0
0.7].X=0.1,
[0.2
0.8
0.5
0.3
0.6]-X=0.0,
[0.7 0.9 0.5 1.0 0.41-x=0.8, [0.0
0.2
0.7
0.9
1.0].X=0.6,
[1.0
0.6
0.2
1.0
0.51 . X = 0 . 4 ,
[0.5
0.3
0.2
0.6
0.7]-X=0.7.
Each of the above equations has a solution: the condition maxx, R(xi, yj) >! Y(yj) is satisfied for each j. Also direct use of the theoretical result (5) gives a completely meaningless result, say X = R a~ Y = [0 0 0 0 0]. The first method with the performance index expressed by the equality index produced the results of Table 6, and the fuzzy set X is equal to [0.4 0.4 0.6 0.4 0.4]. Performing calculations with Q specified by the Hamming distance, the optimal value of fl is equal to 0.5 yielding the same fuzzy solution set.
4. Concluding remarks The methodology of deriving approximate solutions of fuzzy relational equations enables us to overcome shortcomings arising in a blind use of methods that are appropriate only in cases where the existence of solutions is assured. The main idea is to make use of statistical records covering partial results, namely results referring to individual equations. This statistical information is gathered and incorporated into a degree leading to extremal values of the performance index. Among two types of performance indices discussed here the equality index is more attractive due to its ability of determination of precision of the results of max-min composition.
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It is also of importance that statistical methods for coping with situations when there is no exact solution should be studied. This case generates an evident departure from the interpolation-like f o r m a t c o m m o n in fuzzy relational equations in favor of approximation-like formulation of those problems. T h e p a p e r dealt with m a x - m i n composition. H o w e v e r the same methodological frame applies for other types of relational equations as e.g. adjoint equations or m i n - m a x ones. T h e only difference is that now the adjustment of the direction of threshold values depends on the character of the solution. Thus if maximal solutions are to be summarized then the value of the threshold has to be increased; the opposite holds for minimal solutions.
Acknowledgement Support provided by the University of Manitoba (operating grant in aid of research) is highly appreciated.
References [1] S. Gottwald, On the existence of solutions of systems of fuzzy equations, Fuzzy Sets and Systems 12 (1984) 301-302. [2] S. Gottwald and W. Pedrycz, Solvability of fuzzy relational equations and manipulation of fuzzy data, Fuzzy Sets and Systems 18 (1986) 45-65. [3] K. Hirota, Concepts of probabilistic sets, Fuzzy Sets and Systems 1 (1981) 31-46. [4] W. Pedrycz, Structured fuzzy models. Cybernetics and Systems 16 (1985) 103-117. [5] W. Pedrycz, Applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data, Fuzzy Sets and Systems 16 (1985) 163-175. [6] W. Pedrycz, Approximate solutions of fuzzy relational equations, Fuzzy Sets and Systems 28 (1988) 183-202, [7] W. Pedrycz, Relevancy of fuzzy models, Inform. Sci. in press. [8] W. Pedrycz, Direct and inverse problem in comparison of fuzzy data, Fuzzy Sets and Systems 34 (1990) 223-235. [9] E. Sanchez, Resolution of composite relation equations, Inform. and Control 30 (1976) 38-48. [10] K.S. Trivedi, Probability and Statistics with Reliability in Querying and Computer Science Applications (Prentice-Hall, Englewood Cliffs, NJ, 1982). [11] M. Wagenknecht and K. Hartmann, Fuzzy modelling with tolerances, Fuzzy Sets and Systems 20 (1986) 325-332. [12] M. Wagenknecht and K. Hartman, On direct and inverse problems for fuzzy equation systems with tolerances, Fuzzy Sets and Systems, 24 (1987) 93-102.