Fuzzy Boundaries of Sample Selection Model

Transcription

1 Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) Fuzzy Boundares of Sample Selecton Model L. MUHMD SFIIH, NTON BDULBSH KMIL, M. T. BU OSMN 3,3 Mathematcs Department Faculty of Scence and Technology, KUSTEM 030 Kuala Terengganu, Terengganu, MLYSI. School of Mathematcal Scences Unverst Sans Malaysa 800 USM Penang, MLYSI. bstract-the ssue of uncertanty and ambguty are the major problem and become complcated n the modelng of econometrcs. In ths study, we wll focus n the context of fuzzy boundares of the sample selecton model. The best approach of accountng for uncertanty and ambguty s to take advantage of the tools provded by the theory of fuzzy sets. It seems partcularly approprate for modelng vague concepts. Fuzzy sets theory and ts propertes through the concept of fuzzy number and lngustc varables provde an deal framework n order to solve the problem of uncertanty data. In ths concept paper, we ntroduce a fuzzy membershp functon for solvng uncertanty data of a sample selecton model. Key-Words:- uncertanty, ambguty, sample selecton model, crsp data, membershp functon Introducton The purpose ths concept paper s to ntroduce a membershp functon of a sample selecton model that can be used to deal wth sample selecton model problems n whch hstorcal data contans some uncertanty. Wth ths concept provdes an deal framework to deal wth problems n whch there does not exst a defnte crteron for dscoverng what elements belongs or do not belongs to a gven set (Mcel, 998) Fuzzy set defnes by a fuzzy sets n a unverse of dscourse U s characterzed by a membershp functon denoted by the functon µ maps all elements of U that take the values n the nterval [0,] that s : X [0,] (Zadeh, 965) Clearly, a fuzzy set s a generalzaton of the concept of a set whose membershp functon takes only two values: {0,} and n prncple any functon of the form : X [0,] descrbe a membershp functon assocated wth a fuzzy sets. In ths paper, we use a monotoncally ncreasng S- Sharp (S-Curve) membershp functon as represented by the generc graphcal representaton n Fg..

2 Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) Degree of Membershp µ (0.5) Zero Membershp, α Flex Pont, β Fg. The S-Curve Functon Characterstcs From Fgure, we can see clearly the characterstc of the S-Curve. It s zero membershp value (α), t s complete membershp value ( and a thrd pece of α + γ nformaton, the parameter β, β, s the nflecton or crossover pont. The nflecton pont s chosen by the analyst accordng to the apparent, suspected, or known dstrbuton of the populaton (Cox, 998). Ths s the pont at whch the doman s 50% true. ). n S-Curve s defned usng three parameters, α, β and γ (Cox, 998). The membershp functon as proposed by Zadeh (965) can be express as: 0 (( x α)/( γ α)) F( x, α; β; (( x /( γ α)) x α α x β β x γ x γ One Membershp,γ Fuzzy Sets and Fuzzy Numbers () ccordng Fuzzy logc s the superset of classcal logc wth the ntroducton of "degree of membershp". The ntroducton of degree of membershp allows the nput to nterpolate between the crsp set (Jamshd, et.al., 993). Unlke classcal set theory that classfes the elements of the set nto crsp sets; fuzzy set has an ablty to classfy elements nto a contnuous set usng the concept of degree of membershp. The characterstc functon or membershp functon not only gves 0 or but can also gve values between 0 and. In other word, to express a number n words, we need a way to translate nput numbers nto confdences n a fuzzy set of word descrptors, the process of fuzzfcaton (or the process of convertng numercal measurements to confdences (grades of membershp) of fuzzy set members s called fuzzfcaton). In fuzzy math, that s done by membershp functons In ths paper we consder the sample selecton model proposed by Heckman (979). The conventonal sample selecton model has the form Y γ + Ζ ϑ + u 0 Y Y D ' 0 D f D X β + ε > 0 D 0 otherwse,..., n ' () where, D, Y are the dependent varables, X and Z are vectors of exogenous varables, γ 0, β and ϑ are unknown parameter vectors, and u and ε are zero mean error terms. The frst equaton s the outcome equaton and the second equaton s the partcpaton equaton. Wthout loss of generalty, assume all observatons are fuzzy numbers, snce crsp values can be represented by degenerated fuzzy number. Then, the fuzzy sample selecton model s gven by: ' Y γ 0 + Ζϑ0 + u ' D f D X β + ε > 0 D 0 otherwse,..., n Y Y D (3) where Y, ϑ o, β, u and ε are fuzzy numbers wth the membershp functons, µ, µ, ϖ, µ u and µ Y ϑ o β ε, respectvely. The problem s to fnd the estmates for γ 0 ϑ 0, β, u and ε consstently from the data { Y, D, X and Z ),,..n.

3 Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34). rthmetc operatons on FN The Zadeh s extenson prncple method and the α- cut method are two methods that can be appled to defne arthmetcal operatons wth fuzzy numbers. Both methods are very mportant concept n the relatonshp between fuzzy sets and crsp sets. Zadeh extenson prncple provdes a general method for extendng the classcal (tradtonal) and crsp mathematcal concept and arthmetc operatons to the fuzzy envronment whle n the α-cut method, the arthmetc for fuzzy number can be reduced to nterval arthmetc... Lngustc Varable Lngustcs varables are concept whch plays an mportant role n applcatons of fuzzy sets. Informally, a lngustc varable, ζ s a varable whose values are words or sentences whch serve as names of fuzzy subsets of a unverse of dscourse or varable whose values are not numbers but words or sentences n a natural or artfcal language (Zmmermann, 985). ccordng to Zadeh (974), n general, verbal characterzatons are less precse then numercal ones, and thus serve the functon of provdng a means of approxmate descrpton of phenomena whch are too complex or too ll-defned to admt of analyss n conventonal quanttatve terms. The motvaton for ths concept derves from the observaton that n most of our commonsense reasonng, we employ words rather than numbers to descrbed the values of varables (Zadeh, 987) 3 Fuzzy Boundares of Sample Selecton Model Defnton 3. Let U be the unverse of dscourse, U { u, u,, u n }. fuzzy set of U s defned by f ( u ) f ( u ) f ( u ) u u u n n (4) where f s the membershp functon of, : U 0, f ndcates the grade of f [ ], and ( ) u membershp of n. u n, where ( ) [ 0, ] f and u Instead of usng crsp value.e., sngle-valued parameters on whch statstcal nference may be done n the case of classcal sample selecton model, fuzzy parameter n the form of membershp functon are used. From the above defnton we know that the membershp functon s constructed from the quadratc functon. In order to construct the fuzzy boundares (mnmum and maxmum), we consder the followng functon: 0, f : ax + bx + c, F( x, α, β, g : a x + bx + c,, 0 x α α x β β x γ x γ (5) Then we need to determne the value of a, b and c where,. nd t s should meet and satsfy the condtons () g ( α) 0, () g ( β) /, (3) ' g ( α) 0, (4) g (, (5) g ( β) /, and (6) g '( 0. The functon F should contnue and followng S curve term. Here, we propose that the functons g and g meet at the pont (β, ). wα + wγ Defne β, where w and w are the w + w weghted. The weghted values are chosen, t s depend on the values of maxmum and mnmum boundares we need. When, w w, we get the mddle membershp functon, as n (4). When, w > w, we get the upper (maxmum) membershp functon, µ '( x), and when, w <, we get the w lower (mnmum) membershp functon, µ (x). We consder w and w 4 to construct the mnmum membershp functon µ (x). Ths follows that α + 4γ β. We then substtute 6

4 Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) α + 4γ β nto the functon F whch satsfes 6 the sx condtons and solve them to get a, b and c. fter smplfyng we get the mnmum membershp functon µ (x) as follows (6) 0, 9 u α, 8 α γ F mn ( u; α, β, 9 u γ +, α γ, 0 u α α u β β u γ u γ Fg. The maxmum boundary of S- Curve membershp functon We consder w 4 and w to construct the maxmum membershp functon µ '( x). Here, we 4α + γ have β. fter dong smlar procedure as 6 above, we get maxmum membershp functon µ '( x) as follows: F 0, 9 u α, α γ ( u; α, β, 9 u γ +, 8 α γ, 0 u α α u β max (7) β u γ u γ Fg.3 The mnmum boundary of S-Curve membershp functon 4 Result The maxmum and mnmum membershp functons depend on the chosen value of w and w. When the value of the weghted s qute smlar, then the membershp functon s close to the mddle membershp functon. In the followng graph of (4), (5), and (6), we take α, γ 5.

5 Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) 4 Concluson Fg.4 The crsp, mnmum and maxmum boundares of S-Curve membershp functon In ths research, we ntroduced a fuzzy membershp functon of a sample selecton model. In a sample selecton model, the major problems of uncertanty data can be solved by usng set of fuzzy number as a fuzzy boundares. Instead of usng crsp value.e. sngle-valued parameters on whch statstcal nference n the case of classcal sample selecton model, the quadratc term n term of membershp functon are used. fuzzy boundary of mnmum and maxmum for Fuzzy Sample Selecton Model S-Curve (FBSSM S-Curve) s defne and clearly represented by usng graphcal.

6 Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) References: [] Cox, E., The Fuzzy System Handbook, nd ed. P Profesonal, New York, 998. [] Heckman, J.J., Sample selecton as a specfcaton error, Econometrca, Vol.47, 979 pp [3] Jamshd, M., Vadee, N., Ross, T. J., Fuzzy Logc and Control, Prentce Hall, 993 [4] Mcel, D., Measurng Poverty Usng Fuzzy Sets, Dscusson Paper no. 38, 998, Unversty of Canberra. [5] Zmmermann, H.J.. Fuzzy Sets Theory and ts pplcatons, M. : Kluwer cademc, Boston, 985. [6] Zadeh, L.., Fuzzy Sets, Informaton and Control, Vol.8(3), 965, pp [7] Zadeh, L.., On the analyss of large-scale system. In: Gottnger, H., ed., Systems approaches and Envronment Problems. Vandenhoeck and Ruprecht, Gottngen, 974. [8] Zadeh, L.., Fuzzy Sets: Usualty and Commonsense Reasonng, In: Vana, L., ed., Matters of Intellgence, D. Redel Publshng Company, Dordrecht, 987.