Entropy of bi-capacities

Transcription

1 Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics Uit Uiv. of Luxembourg 162A, aveue de la Faïecerie L-1511 Luxembourg, G.D. Luxembourg jea-luc.marichal@ui.lu Abstract The otio of etropy, recetly geeralized to capacities, is exteded to bi-capacities ad its mai properties are studied. Keywords: Multicriteria decisio makig, bicapacity, Choquet itegral, etropy. 1 Itroductio The well-kow Shao etropy [10] is a fudametal cocept i probability theory ad related fields. I a geeral o probabilistic settig, it is merely a measure of the uiformity (eveess) of a discrete probability distributio. I a probabilistic cotext, it ca be aturally iterpreted as a measure of upredictability. By relaxig the additivity property of probability measures, requirig oly that they be mootoe, oe obtais Choquet capacities [1], also kow as fuzzy measures [11], for which a extesio of the Shao etropy was recetly defied [4, 5, 7, 8]. The cocept of capacity ca be further geeralized. I the cotext of multicriteria decisio makig, bi-capacities have bee recetly itroduced by Grabisch ad Labreuche [2, 3] to model i a flexible way the prefereces of a decisio maker whe the uderlyig scales are bipolar. Sice a bi-capacity ca be regarded as a geeralizatio of a capacity, the followig atural questio arises : how could oe appraise the uiformity or ucertaity associated with a bicapacity i the spirit of the Shao etropy? The mai purpose of this paper is to propose a defiitio of a extesio of the Shao etropy to bi-capacities. The iterpretatio of this cocept will be performed i the framework of multicriteria decisio makig based o the Choquet itegral. Hece, we cosider a set N := {1,..., } of criteria ad a set A of alteratives described accordig to these criteria, i.e., real-valued fuctios o N. The, give a alterative x A, for ay i N, x i := x(i) is regarded as the utility of x w.r.t. to criterio i. The utilities are further cosidered to be commesurate ad to lie either o a uipolar or o a bipolar scale. Compared to a uipolar scale, a biploar scale is characterized by the additioal presece of a eutral value (usually 0) such that values above this eutral referece poit are cosidered to be good by the decisio maker, ad values below it are cosidered to be bad. As i [2, 3], for simplicity reasos, we shall assume that the scale used for all utilities is [0, 1] if the scale is uipolar, ad [ 1, 1] with 0 as eutral value, if the scale is bipolar. This paper is orgaized as follows. The secod ad third sectios are devoted to a presetatio of the otios of capacity, bi-capacity ad Choquet itegral i the framework of multicriteria decisio makig. I the last sectio, after recallig the defiitios of the probabilistic Shao etropy ad of its extesio to capacities, we propose a geeralizatio of it to bi-capacities. We also give a iterpretatio of it i the cotext of multicriteria decisio makig ad we study its mai properties. 2 Capacities ad bi-capacities I the cotext of aggregatio, capacities [1] ad bi-capacities [2, 3] ca be regarded as geeraliza-

2 tios of weightig vectors ivolved i the calculatios of weighted arithmetic meas. Let P(N) deote the power set of N ad let Q(N) := {(A, B) P(N) P(N) A B = }. Defiitio 2.1 A fuctio µ : P(N) [0, 1] is a capacity if it satisfies : (i) µ( ) = 0, µ(n) = 1, (ii) for ay S, T N, S T µ(s) µ(t ). A capacity µ o N is said to be additive if µ(s T ) = µ(s) µ(t ) for all disjoit subsets S, T N. A particular case of additive capacity is the uiform capacity o N. It is defied by µ (T ) = T /, T N. The dual (or cojugate) of a capacity µ o N is a capacity µ o N defied by µ(a) = µ(n) µ(n \ A), for all A N. Defiitio 2.2 A fuctio v : Q(N) R is a bi-capacity if it satisfies : (i) v(, ) = 0, v(n, ) = 1, v(, N) = 1, (ii) A B implies v(a, ) v(b, ) ad v(, A) v(, B). Furthermore, a bi-capacity v is said to be : of the Cumulative Prospect Theory (CPT) type [2, 3, 12] if there exist two capacities µ 1, µ 2 such that v(a, B) = µ 1 (A) µ 2 (B), (A, B) Q(N). Whe µ 1 = µ 2 the bi-capacity is further said to be symmetric, ad asymmetric whe µ 2 = µ 1 additive if it is of the CPT type with µ 1, µ 2 additive, i.e. for ay (A, B) Q(N) v(a, B) = µ 1 (i) µ 2 (i). i A i B Note that a additive bi-capacity with µ 1 = µ 2 is both symmetric ad asymmetric sice µ 1 = µ 1. As we cotiue, to idicate that a CPT type bicapacity v is costructed from two capacities µ 1, µ 2, we shall deote it by v µ1,µ 2 Let us also cosider a particular additive bicapacity o N : the uiform bi-capacity. It is defied by v (A, B) = A B, (A, B) Q(N). 3 The Choquet itegral Whe utilities are cosidered to lie o a uipolar scale, the importace of the subsets of (iteractig) criteria ca be modeled by a capacity. A suitable aggregatio operator that geeralizes the weighted arithmetic mea is the the Choquet itegral [6]. Defiitio 3.1 The Choquet itegral of a fuctio x : N R represeted by the profile (x 1,..., x ) w.r.t a capacity µ o N is defied by C µ (x) := x σ(i) [µ(a σ(i) ) µ(a σ(i1) )], i=1 where σ is a permutatio o N such that x σ(1) x σ(), A σ(i) := {σ(i),..., σ()}, for all i {1,..., }, ad A σ(1) :=. Whe the uderlyig utility scale is bipolar, Grabisch ad Labreuche proposed to substitute a bicapacity to the capacity ad proposed a atural geeralizatio of the Choquet itegral [3]. Defiitio 3.2 The Choquet itegral of a fuctio x : N R represeted by the profile (x 1,..., x ) w.r.t a bi-capacity v o N is defied by C v (x) := C ν v N ( x ) where ν v N is a game o N (i.e. a set fuctio o N vaishig at the empty set) defied by ν v N (C) = v(c N, C N ), C N, ad N := {i N x i 0}, N := N \ N.

3 As show i [3], a equivalet expressio of C v (x) is : C v (x) = [ x σ(i) v(a σ(i) N, A σ(i) N ) i N ] v(a σ(i1) N, A σ(i1) N ), (1) where A σ(i) := {σ(i),..., σ()}, A σ(1) := 0, ad σ is a permutatio o N so that x σ(1) x σ(). 4 Etropy of a bi-capacity 4.1 The cocept of probabilistic etropy The fudametal cocept of etropy of a probability distributio was iitially proposed by Shao [9, 10]. The Shao etropy of a probability distributio p defied o a oempty fiite set N := {1,..., } is defied by where h(x) := H S (p) := i N h[p(i)] { x l x, if x > 0, 0, if x = 0, The quatity H S (p) is always o egative ad zero if ad oly if p is a Dirac mass (decisivity property). As a fuctio of p, H S is strictly cocave. Furthermore, it reaches its maximum value (l ) if ad oly if p is uiform (maximality property). I a geeral o probabilistic settig, H S (p) is othig else tha a measure of the uiformity of p. I a probabilistic cotext, it ca be iterpreted as a measure of the iformatio cotaied i p. 4.2 Extesio to capacities Let µ be a capacity o N. The followig etropy was proposed by Marichal [5, 7] (see also [8]) as a extesio of the Shao etropy to capacities : H M (µ) := i N S N\i γ s ()h[µ(s i) µ(s)]. Regarded as a uiformity measure, H M has bee recetly axiomatized by meas of three axioms [4] : the symmetry property, a boudary coditio for which H M reduces to the Shao etropy, ad a geeralized versio of the wellkow recursivity property. A fudametal property of H M is that it ca be rewritte i terms of the maximal chais of the Hasse diagram of N [4], which is equivalet to : H M (µ) = 1! σ Π N H S (p µ σ), (2) where Π N deotes the set of permutatios o N ad, for ay σ Π N, p µ σ (i) := µ({σ(i),..., σ()}) µ({σ(i 1),..., σ()}), i N. The quatity H M (µ) ca therefore simply be see as a average over Π N of the uiformity values of the probability distributios p µ σ calculated by meas of the Shao etropy. As show i [4], i the cotext of aggregatio by a Choquet itegral w.r.t a capacity µ o N, H M (µ) ca be iterpreted as a measure of the average value over all x [0, 1] of the degree to which the argumets x 1,..., x cotribute to the calculatio of the aggregated value C µ (x). To stress o the fact that H M is a average of Shao etropies, we shall equivaletly deote it by H S as we go o. It has also bee show that H M = H S satisfies may properties that oe would ituitively require from a etropy measure [4, 7]. The most importat oes are : 1. Boudary property for additive measures. For ay additive capacity µ o N, we have H S (µ) = H S (p), where p is the probability distributio o N defied by p(i) = µ(i) for all i N. 2. Boudary property for cardialitybased measures. For ay cardiality-based capacity µ o N (i.e. such that, for ay T N, µ(t ) depeds oly o T ), we have H S (µ) = H S (p µ ),

4 where p µ is the probability distributio o N defied by p µ (i) = µ({1,..., i}) µ({1,..., i 1}) for all i N. 3. Decisivity. For ay capacity µ o N, H S (µ) 0. Moreover, H S (µ) = 0 if ad oly if µ is a biary-valued capacity, that is, such that µ(t ) {0, 1} for all T N. 4. Maximality. For ay capacity µ o N, we have H S (µ) l. with equality if ad oly if µ is the uiform capacity µ o N. 5. Icreasig mootoicity toward µ. Let µ be a capacity o N such that µ µ ad, for ay λ [0, 1], defie the capacity µ λ o N as µ λ := µ λ(µ N µ). The for ay 0 λ 1 < λ 2 1 we have H S (µ λ1 ) < H S (µ λ2 ). 6. Strict cocavity. For ay two capacities µ 1, µ 2 o N ad ay λ ]0, 1[, we have H S (λ µ 1 (1 λ) µ 2 ) > λ H S (µ 1 )(1 λ) H S (µ 2 ). 4.3 Geeralizatio to bi-capacities For ay bi-capacity v o N ad ay N N, as i [3], we defie the game νn v o N by ν v N (C) := v(c N, C N ), C N, where N := N \ N. Furthermore, for ay N N, let p v σ,n be the probability distributio o N defied, for ay i N, by p v σ,n (i) := νn v (A σ(i) ) νn v (A σ(i1) ) j N νv N (A σ(j) ) νn v (A σ(j1) ) (3) where A σ(i) := {σ(i),..., σ()}, for all i N, ad A σ(1) := We the propose the followig simple defiitio of the extesio of the Shao etropy to a bicapacity v o N : H S (v) := 1 2 N N 1! H S (p v σ,n ) (4) σ Π N As i the case of capacities, the exteded Shao etropy H S (v) is othig else tha a average of the uiformity values of the probability distributios p v σ,n calculated by meas of H S. I the cotext of aggregatio by a Choquet itegral w.r.t a bi-capacity v o N, let us show that, as previously, H S (v) ca be iterpreted as a measure of the average value over all x [ 1, 1] of the degree to which the argumets x 1,..., x cotribute to the calculatio of the aggregated value C v (x). I order to do so, cosider a alterative x [ 1, 1] ad deote by N N the subset of criteria for which x 0. The, from Eq. (1), we see that the Choquet itegral of x w.r.t v is simply a weighted sum of x σ(1),..., x σ(), where each x σ(i) is weighted by ν v N (A σ(i)) ν v N (A σ(i1)). Clearly, these weights are ot always positive, or do they sum up to oe. From the mootoicity coditios of a bi-capacity, it follows that the weight correspodig to x σ (i) is positive if ad oly if σ(i) N. Depedig o the eveess of the distributio of the absolute values of the weights, the utilities x 1,..., x will cotribute more or less evely i the calculatio of C v (x). A straightforward way to measure the eveess of the cotributio of x 1,..., x to C v (x) cosists i measurig the uiformity of the probability distributio p v σ,n defied by Eq. (3). Note that p v σ,n is simply obtaied by ormalizig the distributio of the absolute values of the weights ivolved i the calculatio of C v (x). Clearly, the uiformity of p v σ,n ca be measured by the Shao etropy. Should H S (p v σ,n ) be close to l, the distributio p v σ,n will be approximately uiform ad all the partial evalua- tios x 1,..., x will be ivolved almost equally i the calculatio of C v (x). O the cotrary, should H S (p v σ,n ) be close to zero, oe p v σ,n (i) will be very close to oe ad C v (x) will be almost proportioal to the correspodig partial evaluatio. Let us ow go back to the defiitio of the exteded Shao etropy. From Eq. (4), we clearly

5 see that H S (v) is othig else tha a measure of the average of the behavior we have just discussed, i.e. takig ito accout all the possibilities for σ ad N with uiform probability. More formally, for ay N N, ad ay σ Π N, defie the set O σ,n := {x [ 1, 1] i N, x i [0, 1], i N, x i [ 1, 0[, x σ(1) x σ() }. We clearly have N N σ Π N O σ,n = [ 1, 1]. Let x [ 1, 1] be fixed. The there exist N N ad σ Π N such that x O σ,n ad hece C v (x) is proportioal to i N x σ(i) p v σ,n (i). Startig from Eq. (4) ad usig the fact that x O dx = 1/!, the etropy H σ,n S(v) ca be rewritte as H M (µ) = 1 2 = 1 2 N N σ Π N x O σ,n H S (p v [ 1,1] σ x,n x ) dx, H S (p v σ,n ) dx where N x N ad σ x Π N are defied such that x O σx,n x. We thus observe that H S (v) measures the average value over all x [ 1, 1] of the degree to which the argumets x 1,..., x cotribute to the calculatio of C v (x). I probabilistic terms, it correspods to the expectatio over all x [ 1, 1], with uiform distributio, of the degree of cotributio of argumets x 1,..., x i the calculatio of C v (x). 4.4 Properties of H S We first preset two lemmas givig the form the probability distributios p v σ,n for CPT type bicapacities. Lemma 4.1 For ay bi-capacity v µ1,µ 2 of the CPT type o N, ay N N, ad ay σ Π N, we have [ p vµ 1,µ 2 σ,n (i) = µ 1 (A σ(i) N ) µ 1 (A σ(i1) N ) ] µ 2 (A σ(i) N ) µ 2 (A σ(i1) N ) / [ µ 1 (N ) µ 2 (N ) ], i N. Lemma 4.2 For ay CPT type asymmetric bicapacity v µ1,µ 2 o N, ay N N, ad ay σ Π N, we have p vµ 1,µ 2 σ,n (i) = µ 1 (A σ(i) N ) µ 1 (A σ(i1) N ) for all i N. µ 1 (A σ(i) N ) µ 1 (A σ(i1) N ), We ow state four importat properties of H S. Property 4.1 (Additive bi-capacity) For ay additive bi-capacity v µ1,µ 2 o N, H S (v µ1,µ 2 ) equals [ 1 µ1 (i N ) µ 2 (i N ] ) 2 h N N i N j N µ 1(j) j N µ 2(j) Proof. Let v µ1,µ 2 be a additive bi-capacity o N. The, usig Lemma 4.1, for ay N N, ay σ Π N, ay i N, we obtai that ν vµ 1,µ 2 N (A σ(i) ) ν vµ 1,µ 2 N (A σ(i1) ) = µ 1 (σ(i) N ) µ 2 (σ(i) N ). It follows that, for ay N N, [ H S (p vµ 1,µ 2 σ,n ) = i N h µ 1 (i N ) µ 2 (i N ) j N µ 1(j) j N µ 2(j) for all σ Π N, from which we get the desired result. Property 4.2 (Add. sym./asym. bi-capacity) For ay additive asymmetric/symmetric bicapacity v µ1,µ 2 o N, H S (v µ1,µ 2 ) = H S (p), where p is the probability distributio o N defied by p(i) := µ 1 (i) for all i N. Proof. The result follows from Property 4.1. Property 4.3 (Decisivity) For ay bi-capacity v o N, H S (v) 0. Moreover, H S (v) = 0 if ad oly, for ay x [ 1, 1], oly oe partial evaluatio is used i the calculatio of C v (x). ],

6 Proof. From the decisivity property satisfied by the Shao etropy, we have that, for ay probability distributio p o N, H S (p) 0 with equality if ad oly if p is Dirac. Let v be a bi-capacity o N. If follows that H S (v) 0 with equality if ad oly if, for ay N N, ay σ Π N, p v σ,n is Dirac, which is clearly equivalet to havig, for ay x [ 1, 1], oly oe partial evaluatio cotributig i the calculatio of C v (x). Property 4.4 (Maximality) For ay bicapacity v o N, we have H S (v) l. with equality if ad oly if v is the uiform capacity v o N. Proof. From the maximality property satisfied by the Shao etropy, we have that, for ay probability distributio p o N, H S (p) l with equality if ad oly if p is uiform. Let v be a bi-capacity o N. It follows that H S (v) l with equality if ad oly if, for ay N N, ad ay σ Π N, p v σ,n is uiform. It is easy to see that if v = v, the H S (v) = l. Let us show that if H S (v) = l, the ecessarily v = v. To do so, cosider first the case where N {, N}. From the ormalizatio coditio v(n, ) = 1 = v(, N), it easy to verify that, for ay σ Π N, νn v (A σ(j)) νn v (A σ(j1)) = 1. j N It follows that, if, for ay σ Π N, p v σ,n is uiform, the, for ay σ Π N, νn v (A σ(i)) νn v (A σ(i1)) = 1, i N. This implies that, v(i, ) = 1 = v(, i), i N. (5) Cosider ow the case where N 2 N \ {, N}. If H S (v) = l, we kow that, for ay σ Π N, p v σ,n is uiform. From Eq. (5), we have that, for ay σ Π N, ν v N (A σ()) ν v N (A σ(1)) = 1. Sice, for ay σ Π N, p v σ,n is uiform, we obtai that νn v (A σ(i)) νn v (A σ(i1)) = 1, i N. Refereces [1] G. Choquet. Theory of capacities. Aales de l Istitut Fourier, 5: , [2] M. Grabisch ad C. Labreuche. Bi-capacities I : defiitio, Möbius trasform ad iteractio. Fuzzy Sets ad Systems, 151: , [3] M. Grabisch ad C. Labreuche. Bi-capacities II : the Choquet itegral. Fuzzy Sets ad Systems, 151: , [4] I. Kojadiovic, J.-L. Marichal, ad M. Roubes. A axiomatic approach to the defiitio of the etropy of a discrete Choquet capacity. Iformatio Scieces, I press. [5] J.-L. Marichal. Aggregatio operators for multicriteria deciso aid. PhD thesis, Uiversity of Liège, Liège, Belgium, [6] J.-L. Marichal. A axiomatic approach of the discrete Choquet itegral as a tool to aggregate iteractig criteria. IEEE Trasactios o Fuzzy Systems, 8(6): , [7] J.-L. Marichal. Etropy of discrete Choquet capacities. Europea Joural of Operatioal Research, 3(137): , [8] J.-L. Marichal ad M. Roubes. Etropy of discrete fuzzy measures. Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge-Based Systems, 8(6): , [9] C. Shao ad W. Weaver. A mathematical theory of commuicatio. Uiversity of Illiois, Urbaa, [10] C. E. Shao. A mathematical theory of commuicatio. Bell Systems Techical Joural, 27: , [11] M. Sugeo. Theory of fuzzy itegrals ad its applicatios. PhD thesis, Tokyo Istitute of Techology, Tokyo, Japa, [12] A. Tversky ad D. Kahema. Advaces i prospect theory : cumulative represetatio of ucertaity. Joural of Risk ad Ucertaity, 5: , 1992.