Some illustratios of possibilistic correlatio Robert Fullér IAMSR, Åbo Akademi Uiversity, Joukahaisekatu -5 A, FIN-252 Turku e-mail: rfuller@abofi József Mezei Turku Cetre for Computer Sciece, Joukahaisekatu -5 B, FIN-252 Turku e-mail: jmezei@abofi Péter Várlaki Budapest Uiversity of Techology ad Ecoomics, Bertala L u 2, H-1111 Budapest, Hugary ad Szécheyi Istvá Uiversity, Egyetem tér 1, H-926 Győr, Hugary e-mail: varlaki@kmebmehu Abstract: I this paper we will show some examples for possibilistic correlatio I particular, we will show (i) how the possibilistic correlatio coefficiet of two liear margial possibility distributios chages from zero to -1/2, ad from -1/2 to -/5 by takig out bigger ad bigger parts from the level sets of a their joit possibility distributio; (ii) how to compute the autocorrelatio coefficiet of fuzzy time series with liear fuzzy data 1 Itroductio A fuzzy umber A is a fuzzy set R with a ormal, fuzzy covex ad cotiuous membership fuctio of bouded support The family of fuzzy umbers is deoted by F Fuzzy umbers ca be cosidered as possibility distributios A fuzzy set C i R 2 is said to be a joit possibility distributio of fuzzy umbers A, B F, if it satisfies the relatioships maxx C(x, y)} = B(y) ad maxy C(x, y)} = A(x) for all x, y R Furthermore, A ad B are called
the margial possibility distributios of C Let A F be fuzzy umber with a γ- level set deoted by [A] γ = [a 1 (γ), a 2 (γ)], γ [, 1] ad let U γ deote a uiform probability distributio o [A] γ, γ [, 1] I possibility theory we ca use the priciple of expected value of fuctios o fuzzy sets to defie variace, covariace ad correlatio of possibility distributios Namely, we equip each level set of a possibility distributio (represeted by a fuzzy umber) with a uiform probability distributio, the apply their stadard probabilistic calculatio, ad the defie measures o possibility distributios by itegratig these weighted probabilistic otios over the set of all membership grades These weights (or importaces) ca be give by weightig fuctios A fuctio f : [, 1] R is said to be a weightig fuctio if f is o-egative, mootoe icreasig ad satisfies the followig ormalizatio coditio f(γ)dγ = 1 Differet weightig fuctios ca give differet (case-depedet) importaces to level-sets of possibility distributios I 29 we itroduced a ew defiitio of possibilistic correlatio coefficiet (see [2]) that improves the earlier defiitio itroduced by Carlsso, Fullér ad Majleder i 25 (see [1]) Defiitio ([2]) The f-weighted possibilistic correlatio coefficiet of A, B F (with respect to their joit distributio C) is defied by where ρ f (A, B) = ρ(x γ, Y γ ) = ρ(x γ, Y γ )f(γ)dγ (1) cov(x γ, Y γ ) var(xγ ) var(y γ ) ad, where X γ ad Y γ are radom variables whose joit distributio is uiform o [C] γ for all γ [, 1], ad cov(x γ, Y γ ) deotes their probabilistic covariace I other words, the f-weighted possibilistic correlatio coefficiet is othig else, but the f-weighted average of the probabilistic correlatio coefficiets ρ(x γ, Y γ ) for all γ [, 1] Cosider the case, whe A(x) = B(x) = (1 x) χ [,1] (x), for x R, that is [A] γ = [B] γ = [, 1 γ], for γ [, 1] Suppose that their joit possibility distributio is give by F (x, y) = (1 x y) χ T (x, y), where T = (x, y) R 2 x, y, x + y 1 } The we have [F ] γ = (x, y) R 2 x, y, x + y 1 γ }
Figure 1: Illustratio of joit possibility distributio F This situatio is depicted o Fig 1, where we have shifted the fuzzy sets to get a better view of the situatio I this case the f-weighted possibilistic correlatio of A ad B is computed as (see [2] for details), ρ f (A, B) = 1 2 f(γ)dγ = 1 2 Cosider ow the case whe A(x) = B(x) = x χ [,1] (x) for x R, that is [A] γ = [B] γ = [γ, 1], for γ [, 1] Suppose that their joit possibility distributio is give by W (x, y) = maxx + y 1, } The we get 1 ρ f (A, B) = 2 f(γ)dγ = 1 2 We ote here that W is othig else but the Lukasiewitz t-orm, or i the statistical literature, W is geerally referred to as the lower Fréchet-Hoeffdig boud for copulas
2 A trasitio from zero to -1/2 Suppose that a family of joit possibility distributio of A ad B (where A(x) = B(x) = (1 x) χ [,1] (x), for x R) is defied by C (x, y) = 1 x 1 1 y, if x 1, x y, 1 1 y + x 1 1 x y, if y 1, y x, x + y 1, otherwise I the followig, for simplicity, we well write C istead of C A γ-level set of C is computed by [C] γ = (x, y) R 2 x } 1 (1 γ), y 1 γ 2 1 x } (x, y) R 2 2 1 1 (1 γ) x 1 γ, y 1 γ x The desity fuctio of a uiform distributio o [C] γ ca be writte as 1, if (x, y) [C]γ 2 1, if (x, y) [C]γ f(x, y) = [C] dxdy γ = (1 γ) 2 We ca calculate the margial desity fuctios: (2 1)(1 γ x) ( 1)(1 γ) 2, if (1 γ) x 1 γ 2 1 f 1 (x) = (2 1)(1 γ 1 x) (1 γ) 2, if x (1 γ) 2 1 ad, (2 1)(1 γ y) ( 1)(1 γ) 2, if (1 γ) y 1 γ 2 1 f 2 (y) = (2 1)(1 γ 1 y) (1 γ) 2, if y (1 γ) 2 1
We ca calculate the probabilistic expected values of the radom variables X γ ad Y γ, whose joit distributio is uiform o [C] γ for all γ [, 1] as, M(X γ ) = 2 1 (1 γ) 2 1 (1 γ) 2 + 2 1 ( 1)(1 γ) 2 x(1 γ 1 x)dx γ (1 γ) 2 1 x(1 γ x)dx = (1 γ)(4 1) 6(2 1) ad, M(Y γ ) = (1 γ)(4 1) 6(2 1) (We ca easily see that for = 1 we have M(X γ ) = 1 γ, ad for we 2 fid M(X γ ) 1 γ ) We calculate the variatios of X γ ad Y γ as, M(Xγ) 2 = 2 1 (1 γ) 2 1 (1 γ) 2 + 2 1 ( 1)(1 γ) 2 x 2 (1 γ 1 x)dx γ (1 γ) 2 1 x 2 (1 γ x)dx = (1 γ)2 ((2 1) + 8 6 2 + ) 12(2 1) (We ca easily see that for = 1 we get M(X 2 γ) = fid M(X 2 γ) (1 γ)2 ) Furthermore, 6 (1 γ)2, ad for we var(x γ ) = M(X 2 γ) M(X γ ) 2 = (1 γ)2 ((2 1) + 8 6 2 + ) 12(2 1) Ad similarly we obtai (1 γ)2 (4 1) 2 6(2 1) 2 = (1 γ)2 (2(2 1) 2 + ) 6(2 1) 2 var(y γ ) = (1 γ)2 (2(2 1) 2 + ) 6(2 1) 2 (We ca easily see that for = 1 we get var(x γ ) = (1 γ)2, ad for we 12
fid var(x γ ) (1 γ)2 ) Ad, 18 cov(x γ, Y γ ) = M(X γ Y γ ) M(X γ )M(Y γ ) = (1 γ)2 (4 1) 12(2 1) 2 (1 γ)2 (1 )(4 1) 6(2 1) 2 (We ca easily see that for = 1 we have cov(x γ, Y γ ) =, ad for we (1 γ)2 fid cov(x γ, Y γ ) ) We ca calculate the probabilisctic correlatio 6 of the radom variables, ρ(x γ, Y γ ) = cov(x γ, Y γ ) var(xγ ) (1 )(4 1) = var(y γ ) 2(2 1) 2 + (We ca easily see that for = 1 we have ρ(x γ, Y γ ) =, ad for we fid ρ(x γ, Y γ ) 1 ) Ad fially the f-weighted possibilistic correlatio of A ad 2 B is computed as, ρ f (A, B) = ρ(x γ, Y γ )f(γ)dγ = (1 )(4 1) 2(2 1) 2 + We obtai, that ρ f (A, B) = for = 1 ad if the ρ f (A, B) 1 2 A trasitio from -1/2 to -/5 Suppose that the joit possibility distributio of A ad B (where A(x) = B(x) = (1 x) χ [,1] (x), for x R) is defied by C (x, y) = (1 x y) χ T (x, y), where T = } (x, y) R 2 1 x, y, x + y 1, 1 x y (x, y) R 2 x, y, x + y 1, ( 1)x y } I the followig, for simplicity, we well write C istead of C A γ-level set of C is computed by [C] γ = (x, y) R 2 x 1 } (1 γ), ( 1)x y 1 γ x
(x, y) R 2 y 1 (1 γ), ( 1)y x 1 γ y } The desity fuctio of a uiform distributio o [C] γ ca be writte as f(x, y) = 1, if (x, y) [C]γ [C] dxdy γ, if (x, y) [C]γ = (1 γ) 2 We ca calculate the margial desity fuctios: (1 γ x + x 1 ) (1 γ) 2, if x 1 γ x (1 γ) ( 1)(1 γ) f 1 (x) = (1 γ) 2, if x ( 1) (1 γ x) ( 1)(1 γ) (1 γ) 2, if x 1 γ ad, f 2 (y) = (1 γ y + y 1 ) (1 γ) 2, if y 1 γ y (1 γ) ( 1)(1 γ) (1 γ) 2, if y ( 1) (1 γ y) ( 1)(1 γ) (1 γ) 2, if y 1 γ We ca calculate the probabilistic expected values of the radom variables X γ ad Y γ, whose joit distributio is uiform o [C] γ for all γ [, 1] as M(X γ ) = (1 γ) 2 + (1 γ) 2 = 1 γ γ ( 1)(1 γ) 1 γ x(1 γ x + x 1 )dx x 2 1 dx + (1 γ) 2 γ) ( 1)(1 γ) x(1 γ x)dx
That is, M(Y γ ) = 1 γ We calculate the variatios of X γ ad Y γ as, ad, M(X 2 γ) = + + γ (1 γ) 2 (1 γ) 2 (1 γ) 2 ( 1)(1 γ) 1 γ γ) ( 1)(1 γ) x 2 (1 γ x + = (1 γ)2 ( 2 + 2) 12 2 x 1 dx x 2 (1 γ x)dx x 1 )dx var(x γ ) = M(Xγ) 2 M(X γ ) 2 = (1 γ)2 ( 2 + 2) (1 γ)2 12 2 9 = (1 γ)2 (5 2 9 + 6) 6 2 Ad, similarly, we obtai var(y γ ) = (1 γ)2 (5 2 9 + 6) 6 2 From cov(x γ, Y γ ) = M(X γ Y γ ) M(X γ )M(Y γ ) = (1 γ)2 ( 2) (1 γ)2 12 2, 9 we ca calculate the probabilisctic correlatio of the readom variables: ρ(x γ, Y γ ) = cov(x γ, Y γ ) var(xγ ) var(y γ ) = 2 + 7 6 5 2 9 + 6 Ad fially the f-weighted possibilistic correlatio of A ad B: ρ f (A, B) = We obtai, that for = 2 ad if, the ρ(x γ, Y γ )f(γ)dγ = 2 + 7 6 5 2 9 + 6 ρ f (A, B) = 1 2, ρ f (A, B) 5 We ote that i this extremal case the joit possibility distributio is othig else but the margial distributios themselves, that is, C (x, y) =, for ay iterior poit (x, y) of the uit square
4 A trapezoidal case Cosider the case, whe x, if x 1 1, if 1 x 2 A(x) = B(x) = x, if 2 x, otherwise for x R, that is [A] γ = [B] γ = [γ, γ], for γ [, 1] Suppose that their joit possibility distributio is give by: y, if x, y 1, x y, x y 1, if 1 x 2, 1 y 2, x y C(x, y) = x, if x 1, y, y x, x y, otherwise The [C] γ = (x, y) R 2 γ x γ, γ y x } The desity fuctio of a uiform distributio o [F ] γ ca be writte as 1, if (x, y) [C]γ 2, if (x, y) [C]γ f(x, y) = [C] dxdy γ = ( 2γ) 2 The margial fuctios are obtaied as 2( γ x) f 1 (x) = ( 2γ) 2, if γ x γ ad, f 2 (y) = 2( γ y) ( 2γ) 2, if γ y γ We ca calculate the probabilistic expected values of the radom variables X γ ad Y γ, whose joit distributio is uiform o [C] γ for all γ [, 1] : ad, M(X γ ) = M(Y γ ) = 2 γ ( 2γ) 2 x( γ x)dx = γ + γ 2 γ ( 2γ) 2 y( γ y)dx = γ + γ
We calculate the variatios of X γ ad Y γ from the formula var(x) = M(X 2 ) M(X) 2 as ad, M(X 2 γ) = 2 γ ( 2γ) 2 x 2 ( γ x)dx = 2γ2 + 9 γ 6 var(x γ ) = M(X 2 γ) M(X γ ) 2 = 2γ2 + 9 6 (γ + )2 9 = ( 2γ)2 18 Ad similarly we obtai Usig the relatioship, var(y γ ) = ( 2γ)2 18 cov(x γ, Y γ ) = M(X γ Y γ ) M(X γ )M(Y γ ) = ( 2γ)2, 6 we ca calculate the probabilisctic correlatio of the radom variables: ρ(x γ, Y γ ) = cov(x γ, Y γ ) var(xγ ) var(y γ ) = 1 2 Ad fially the f-weighted possibilistic correlatio of A ad B is, 1 ρ f (A, B) = 2 f(γ)dγ = 1 2 5 Time Series With Fuzzy Data A time series with fuzzy data is referred to as fuzzy time series (see []) Cosider a fuzzy time series idexed by t (, 1], A t (x) = 1 x t, if x t It is easy to see that i this case, ad A (x) = [A t ] γ = [, t(1 γ)], γ [, 1] 1, if x =
If we have t 1, t 2 [, 1], the the joit possibility distributio of the correspodig fuzzy umbers is give by: ) C(x, y) = (1 xt1 yt2 χ T (x, y), where The [C] γ = T = (x, y) R 2 x, y, x + y } 1 t 1 t 2 (x, y) R 2 x, y, x + y } 1 γ t 1 t 2 The desity fuctio of a uiform distributio o [C] γ ca be writte as That is, f(x, y) = 1, if (x, y) [C]γ [C] dxdy γ 2 f(x, y) = t 1 t 2 (1 γ) 2, if (x, y) [C]γ The margial fuctios are obtaied as 2(1 γ x t 1 ) f 1 (x) = t 1 (1 γ) 2, if x t 1 (1 γ) ad, 2(1 γ y t 2 ) f 2 (y) = t 2 (1 γ) 2, if y t 2 (1 γ) We ca calculate the probabilistic expected values of the radom variables X γ ad Y γ, whose joit distributio is uiform o [C] γ for all γ [, 1] : ad M(X γ ) = M(Y γ ) = 2 t 1 (1 γ) 2 2 t 2 (1 γ) 2 t1 (1 γ) t2 (1 γ) x(1 γ x t 1 )dx = t 1(1 γ) y(1 γ y )dx = t 2(1 γ) t 2
We calculate ow the variatios of X γ ad Y γ as, ad, M(X 2 γ) = 2 t 1 (1 γ) 2 t1 (1 γ) var(x γ ) = M(Xγ) 2 M(X γ ) 2 = t2 1 (1 γ)2 6 Ad, i a similar way, we obtai, From, x 2 (1 γ x )dx = t2 1 (1 γ)2 t 1 6 var(y γ ) = t2 2 (1 γ)2 18 t2 1 (1 γ)2 9 = t2 1 cov(x γ, Y γ ) = t 1t 2 (1 γ) 2, 6 we ca calculate the probabilisctic correlatio of the readom variables, ρ(x γ, Y γ ) = cov(x γ, Y γ ) var(xγ ) var(y γ ) = 1 2 The f-weighted possibilistic correlatio of A t1 ad A t2, ρ f (A t1, A t2 ) = 1 2 f(γ)dγ = 1 2 (1 γ)2 18 So, the autocorrelatio fuctio of this fuzzy time series is costat Namely, for all t 1, t 2 [, 1] R(t 1, t 2 ) = 1 2, Refereces [1] C Carlsso, R Fullér ad P Majleder, O possibilistic correlatio, Fuzzy Sets ad Systems, 155(25) 425-445 [2] R Fullér, J Mezei ad P Várlaki, A New Look at Possibilistic Correlatio, Fuzzy Sets ad Systems (submitted) [] B Möller, M Beer ad U Reuter, Theoretical Basics of Fuzzy Radomess - Applicatio to Time Series with Fuzzy Data, I: Safety ad Reliability of Egieerig Systems ad Structures - Proceedigs of the 9th It Coferece o Structural Safety ad Reliability, edited by G Augusti ad GI Schueller ad M Ciampoli Millpress, Rotterdam, pages 171-177