Stability of Aggregation Operators T. Calvo Department of Computer Science University of Alcala 28871 Alcala de Henares, Spain
R. Mesiar Department of Mathematics 81368 Bratislava, Slovak Republic Academy of Sciences of the Czech Republic 18208 Praha 8, Czech Republic Department of Mathematics and Computer Science
[email protected] University of Balearic Islands 07071 Palma de Mallorca, Spain tomasa.calvo@{uah.es,uib.es)
Abstract Stability of aggregation operators is characterized and discussed. 1-stability turns to coincide with 1-Lipschitz p r o p erty. Several examples and basic results are given.
Keywords: aggregation operator, Lipschitz property, norm, stability.
1
Introduction
Stability of any mathematical model of engineering problems means, roughly speaking, that the "small inputs errors" do not result t o a "big output error". In the framework of aggregation operators, i.e., non-decreasing map[O, 11, A(0, . . . , 0) = 0, pings U [0, lIn nEN
-
A(1,. . . ,1) = 1 and A(x) = x for all x E [O, 11, see [13], this desirable effect corresponds to the continuity of A. More, because of the compactness of [0, I.]", n E N , for any continuous aggregation operator A, n e N, E > 0 there is = 6 such that if the input errors do not exceed 6, the output error do not exceed E, i.e., ) A(x1,. .. , x n ) - A(yl, . . . ,yn) I < E whenever m q I xi - yi 15 6. For instance, in the case of the geometric mean G, i.e.7 G(x, Y) = (n = 2) for a given E > 0 the smallest relevant 6 is equal to E'. However, for ensuring good accuracy of the result, say with E = 0.01, we need to ensure extremely good accuracy of input values, b = 0.0001 in our case, what is not realistic. To avoid such stability problems, we have to require stronger properties than simple continuity. The
aim of this paper is the proposal of pstability property of aggregation operators p E [l,m ] , and the discussion of p s t a b l e aggregation operators. Several examples will be given.
2
pstable aggregation operators
Recall the standard p m e t r i c in R n , n E N , p E n
[ I , m ] , given by (((XI,..., x n ) l J p = ( C I xi I ~ )if~ ' ~ i=l p > 1, and 11(~1,.. . ,xn)IIas = m y I xi I-
-
Each n-ary aggregation operator A(,) : [0, I.In [O, 11 can be understood as a mapping from a subspace of Banach space (Rn, )) . )Ip) t o the Banach space (R, )I - (Ip) (observe that on R all metrics (1 - 11, coincide with the standard absolute value, 11 . )Ip =(. I for all p E [ l , m ] ) . Then the norm of this mapping )IA(,)llp can be introduced as
-
For the for all x , y E [0, lIn,x # y. global aggregation operator A : IJ [0, lIn nEN
[0,1] the p n o r m will be given by l)Allp = s u ~ ( l l ~ ( n ) lI l ~pE N ) Evidently, IIA(n)111 = 1 for any aggregation operator A and hence 11 A.llp 2 1.
Definition 1 Let A : aggregation operator.
U
[0, :l.In
nEN
1) A will be called n llA(n)llp 5 1.
-
-
p-stable
[O,:L] be a n
whenever
2) A will be called p-stable if llAllp = 1, i.e., if A i s n - p-stable for all n E N .
The class of all n - p-stable aggregation operators will be denoted as Sn,pwhile the class of all pstable aggregation operators will be denoted as Sp.
3
Basic properties of p-stable aggregation operators
Similarly it can be shown that the weakest and the strongest members of Sp, p E [I, m ] , are the respectively, Yager t-norm TpYand t-onorm see [19, 91 where = min, = m a , and for lIp
