Conditional entropy for the union of fuzzy and crisp partitions

Transcription

1 Conditional entropy for the union of fuzzy and crisp partitions Doretta Vivona Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate v.a.scarpa n ROMA(Italia) vivona@dmmm.uniroma1.it Maria Divari Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate v.a.scarpa n ROMA(Italia) Abstract In this paper we introduce the conditional entropy without a fuzzy measure for the union of fuzzy partitions. We recall its properties and we solve the system of functional equations which derives from these conditions, taking into account the locality principle and the independence axiom. Keywords: Fuzzy partitions, Entropy, Functional equations. 1 Introduction In 4 B.Forte has introduced the so called locality property in the crisp setting; later, in 2 Divari and Pandolfi have given some compositive laws for crisp partitions. Later, the authors have used the locality property in the fuzzy setting in 8. The aim of this paper is to find a class of conditional entropy for the union of fuzzy partitions, by using the locality property and the independence axiom. Finally, we present the crisp case: the system is analogous. 2 Preliminaires From algebraic point of view, similar concepts have been introduced in the setting of MV-algebras by Mundici in 5. Let X be an abstract space and F a family of fuzzy sets F 9 with membership function F (x), 0 F (x) 1 for all x X. Now we recall some definitions which we will use later. Let F 1 and F 2 be two fuzzy sets, with membership functions F 1 (x) and F 2 (x). We recall that (F 1 + F 2 )(x) = F 1 (x) F 2 (x), x X is the membership function of the intersection set F 1 + F 2 and (F 1 + F 2 )(x) = F 1 (x) F 2 (x), x X is the membership function of the union set F 1 + F 2. Moreover, we say that two fuzzy sets F 1 and F 2 are disjoint if F 1 + F 2 =. A family {F 1,..., F n } is a finite partition of a set F, called support, if F i, F i are disjoint sets and Σ n i=1 F i(x) = F (x) x X. We shall indicate with P(F) the fuzzy partition with support F, with {F } the fuzzy set thought as a partition and with K the family of all partitions of X. We consider the whole space X as a fuzzy partition, which shall be indicated by {X}. In 1 we have recognized that for every P(F) an algebra C P(F) of crisp sets is associated, whose elements are the inverse images of Borel sets of 0, 1 n through the map F 0, 1 n. Given two partitions P(F) and P (F) of the same set F : P(F) = {F 1,..., F i,..., F n /F i + F h =, i h, (1) Σ n i=1f i (x) = F (x) x X}, P (F) = {F 1,..., F j,..., F m/f j + F k =, j k, Σ m j=1f j(x) = F (x) x X} we say that P (F) is a less fine than P(F) (P (F) P(F)) if C P (F) C P(F). Later, as in, we shall use the operation algebraic joint between two partitions of X. Now, we consider two partitions P(F) as in (1) and P(G) in K: P(G) = {G 1,..., G j,..., G m /G j + G k =, j k, Σ m j=1g j (x) = G(x) x X}. The algebraic joint P(F) P(G) is P(F) P(G) = {F i G j /1 i n, i j m}

2 with lexicographic order, where (F i G j )(x) = F i (x) G j (x), x X. We say that two fuzzy partitions P and P(F) are algebraically independent if (F i G j )(x) 0, x X. (2) Now, we recall the definition of the union of two partitions, given in 6, 8. Given two partitions P(F) and P(G) K, the union P(F) + P(G) is defined by P(F) + P(G) = {F 1,..., F i,..., F n, G 1,..., G j,..., G m / x X, F i (x) G j (x) = 0 (i j), Σ n i=1f i (x) + Σ m j=1g j (x) 1}. () In this case, we call P(F) and P(G) compositive partitions. Let H K be the family of compositive partitions. As in 7, we introduce the entropy without a fuzzy measure for a fuzzy partition P(F) K conditionated by Q K with H(Q) + as a function ( ) : K 0, + with the following properties: (I) P (F) P(F) (P (F)) (P(F)), P (F) K. (II) (P(F)) = H (P(F)), if P(F) is not conditioned by Q. It follows that : HQ ({X}) = 0 and H Q ({ }) = +. (III) (P(F) P(G)) = (P(F)) + (P(G), if P(F) and P(G) are algebraically independent and P(G) K. Conditional entropy for the union of the partitions If P(F), P (F), P(G) H, Q, R K and H (Q) +, the conditional entropy HQ P(F) + P(G) for the union of the fuzzy partitions satisfies the following conditions: (α) 0 HQ P(F) + P(G) + as P(F) + P(G) = {X} P(F) + P(G) = 0 and P(F) + P(G) = { } H Q P(F) + P(G) = + ; (β) HQ P(F) + P(G) = HQ P(G) + P(F) ; (γ) P (F) P(F) P (F) + P(G) H Q P(F) + P(G), (δ) HQ (P(F) + P(G)) R = P(F) + P(G) + H (R), if R is algebraically independent from P(F) and P(G) and it is not conditioned by Q; the (δ) is a consequence of (II) and (III) seen above. 4 Statement of the problem Taking into account the locality principle fuzzy partitions 7: H P(F) + P(G) H P(F) + {G} = H {F } + P(G) H {F } + {G}. (4) we put P(F) + P(G) = Ψ H (P(F) + {G}) Q, H ({F } + P(G)) Q, H ({F } + {G}) Q, H (Q) where P(F), P (F), P(G) H, Q, R K, with H (Q) + and Ψ( ) = 0, + 0, + 0, + 0, + 0, +. From (α) (δ), we have: (A) Ψ H {X } Q, H {X } Q, H {X } Q, H (Q) = 0; (B)Ψ H (P(F) + {G}) Q, H ({F } + P(G)) Q, H ({F } + {G}) Q, H (Q) = (5)

3 Ψ H (P(G) + {F}) Q, H ({G} + P(F)) Q, H ({G} + {F }) Q, H (Q) ; (C) Ψ H (P (F) + {G}) Q, H ({F } + P(G)) Q, H ({F } + {G}) Q, H (Q) Ψ H (P(F) + {G}) Q, H ({F } + P(G)) Q, H ({F } + {G}) Q, H (Q) 5 Solution of the problem In this paragraph we shall solve the system of functional equations seen above. We look for a function Ψ continuous as a universal law, in the sense that the equations and the inequality about Ψ must be satisfied for all values of the variables in their proper spaces. Proposition 5.1 Let h be any bijettive and strictly increasing function, differentiable with its inverse, and with h(0) = 0; then every function Ψ of the type Ψ(x, y, z, t) = 1 h 1 h(x) h(t) + if P (F) P(F) ; (D) Ψ H ((P(F) + {G}) Q) R, H (({F } + P(G)) Q) R, H (({F } + {G}) Q) R, H (Q) Ψ H (P(F) + {G}) Q, H ({F } + P(G)) Q, H ({F } + {G}) Q, H (Q) + H (R), if R K is algebraically independent from P(F) and P(G) and it is not conditioned by Q. Setting: H ({F } + P(F)) Q = x, H (P(F) + {G}) Q = y, H ({F } + {G}) Q = z, H (P (F) + {G}) Q = y, H (R) = s, H (Q) = t, with x, y, z, y 0, +, x z, y z and s, t (0, + ). From (A) (D), we get the following system of functional equations: (a) Ψ(t, t, t, t) = 0, (b) Ψ(x, y, z, t) = Ψ(y, x, z, t), (c) Ψ(x, y, z, t) Ψ(x, y, z, t) if y y, (d) Ψ(x + s, y + s, z + s, t) = Ψ(x, y, z, t) + s. = h 1 h(y) h(t) + h 1 h(z) h(t), (6) enjoys the equations (a), (b) and (c); it satisfies also (d) if and only if Proof Ψ(x, y, z, t) = x + y + z t. (7) The conditions (a), (b), (c) are clearly satisfied by any function satisfying (6). Moreover, it is obvious that the function (7) satisfies all the conditions (a) (d), and is of the form (6), for h = identity map. So, it remains to prove that every function Ψ of the form (6), satisfying (d), must be of the type (7). By (6) the condition (δ) becomes 1 h 1 h(x + s) h(t) + h 1 h(y + s) h(t) +h 1 h(z + s) h(t) = 1 h 1 h(x) h(t) + h 1 h(y) h(t) + h 1 h(z) h(t) Putting then (8) becomes + s. (8) h 1 h(x) h(t) = ϕt (x), (9) 1 ϕ t (x + s) + ϕ t (y + s) + ϕ t (z + s)

4 = 1 ϕ t (x) + ϕ t (y) + ϕ t (z) + s, i.e. ϕ t (x + s) + ϕ t (y + s) + ϕ t (z + s) = ϕ t (x) + ϕ t (y) + ϕ t (z) + s. (10) Choosing y = x + s, z = y + s, (10) gives rise to ϕ t (z + s) = ϕ t (x) + s, i.e. ϕ t (x + s) = ϕ t (x) + s. (11) Now, we calculate the derivative of (11) with respect to s : ϕ t ϕ t (x + s) =, s s = 1 i.e., by arbitrariness of x and s, ϕ t (s) = s + c(t), (12) where c(t) is an arbitrary function, which we can calculate by the derivative of the function (11) with respect to t. We get ϕ t t (x + s) = ϕ t t (x), x, s so ϕ t t = (cost) = c 1 and from (12) c (t) = c 1. The last equality implies c(t) = c 1 t + c 2, where c 2 is another constant. So, from (12), we have obtained the function ϕ t (x) = x + c 1 t + c 2. (1) By substituting (1) in (9), we get: h(x) h(t) = h(x + c 1 t + c 2 ), x; (14) in particular for x = t we obtain 0 = h((1 + c 1 )t + c 2 ) t. (15) Now (15) implies two possibilities: 1) h(t) = 0, t which we eliminate, 2) c 1 = 1 and c 2 = 0. From (14) we deduce h(x) h(t) = h(x t), so the function h is linear: h(x) = αx. In conclusion, from (6) it follows (7): Ψ(x, y, z, t) = x + y + z t. 6 Conditional entropy for crisp partitions For crisp partitions 2, we can use the following locality property: H πa π B H πa {B} = H {A} π B H {A} {B}, (16) and we put H π (π A π B ) = Φ H (πa {B}) π, H ({A} π B ) π, H ({A} {B}) π, H(π ) (17) π A, π B, π E, E is the family of all partitions of A, B X, π is the partition which conditiones π A π B and H(π ) +, and Φ( ) = 0, + 0, + 0, + 0, + 0, +. The system of functional equations are the same, so, also in the crisp setting, we get the same solution (7). 7 Conclusion Taking into account the locality principle and the independence axiom, the measures of conditional entropy are: - from (5) and (7), for fuzzy partitions: H Q (P(F) + P(G)) = 1 H ({F } + P(F)) Q + H (P(F) + {G}) Q +H ({F } + {G}) Q H(Q) - from (17) and (7), for crisp partitions: H π (π A π B )) = 1 H ({A} πb ) π +H (πa {B}) π + H ({A} {B}) π H(π ). Acknowledgement This research was supported by GNFM of MIUR (ITALY) References 1 P.BENVENUTI - D.VIVONA - M.DIVARI, Fuzzy partitions and entropy, Proc. III Linz Seminar on Fuzzy Set Theory: Uncertainty Measures, (Ed. P.Klement, S.Weber ) (1991),

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