LONG-TERM PREDICTIVE VALUE INTERVAL WITH THE FUZZY TIME SERIES

Transcription

1 Journal of Marine Science and Technology, Vol 19, No 5, pp (2011) 509 LONG-TERM PREDICTIVE VALUE INTERVAL WITH THE FUZZY TIME SERIES Ming-Tao Chou* Key words: fuzzy tie series, fuzzy forecasting, long-ter predictive value interval ABSTRACT This paper presents a new atheatical ethod to construct a fuzzy tie series odel of a syste where the fuzzy support and interval values are used This paper further presents an iproved fuzzy tie series odel with a long-ter predictive value interval and shows that the proposed definition can be used for long-ter forecasting The enrollent data of the University of Alabaa (adopted by Song and Chisso, 1993) are used to deonstrate the proposed forecast odel I INTRODUCTION Since its creation in 1965 by Zadeh [15], fuzzy set theory has enjoyed successful achieveents both in theory advanceent [15-17] and practical applications [1, 4, 6, 10] In previous papers, the inadequacy of fuzzy tie series odels is found when fuzzy iplications are used to express forecasting rules [1-17] Several fuzzy tie series odels have been developed since Chen s [5] paper was published The ain purpose of this paper is to present a atheatical ethod to construct a fuzzy tie series odel for long-ter prediction This paper proposes a new ethod for long-ter forecasting using the fuzzy tie series based on Chou and Lee s ethod [5] According to Chou and Lee s ethod [5], a long-ter predictive value interval definition is added to iprove attribution based on the definition of the fuzzy tie series This paper focuses on the enhanceent of increasing and decreasing cases in Chou and Lee s fraework The current paper tackles the issues of iproving the forecasting accuracy by controlling the uncertainties, and deterining the support [7] of the fuzzy nubers The university enrollent data used by Song and Chisso [2-5, 11-14] is used to deonstrate the proposed forecasting odel The reainder of this paper is organized as follows Section 2 presents the fuzzy Paper subitted 11/28/08; revised 06/28/09; accepted 05/25/10 Author for correspondence: Ming-Tao Chou (e-ail: tchou@gailco) *Departent of Aviation and Maritie Transportation Manageent, Chang Jung Christian University, Tainan County, Taiwan, ROC tie series definition In Section 3, the long-ter predictive value interval using the fuzzy tie series is presented The forecasting of enrollent data is shown in Section 4 Finally, the conclusions are ade in Section 5 II FUZZY SETS AND FUZZY TIME SERIES Song and Chisso [11-14] use discrete fuzzy sets to define the fuzzy tie series Chen [2] indicates Song and Chisso s ethod is too coplicated, thus he proposes a siple ethod to copute the fuzzy relation Lee and Chou [7] provide an objective interval ethod to define the universe of discourse Liaw [9] investigates the nonstationary proble and eploys the fuzzy trend technique to define the fuzzy tie series Various definitions and properties of fuzzy tie series forecasting [2, 7, 9, 11, 12] are suarized as follows: Definition 1 [8, 11] A fuzzy nuber on the real line R is a fuzzy subset of R that is noral and convex Definition 2 [11, 12] Let Y(t) (t =, 0, 1, 2, ), a subset of R, be the universe of discourse on which the fuzzy sets f i (t) (t = 1, 2, ) are defined and F(t) is the collection of f i (t) (t = 1, 2, ) Then, F(t) is called a fuzzy tie series on Y(t) (t =, 0, 1, 2, ) Definition 3 [11, 12] Let I and J be the index sets for F(t 1) and F(t) respectively If for any f j (t) F(t) where j J, there exists f i (t 1) F(t 1) where i I such that there exists a fuzzy relation R ij (t, t 1) and f j (t) = f i (t 1) R ij (t, t 1) where is the ax-in coposition, then F(t) is said to be caused by only F(t 1) Denote this as f i (t 1) f j (t), or equivalently, F(t 1) F(t) Definition 4 [11, 12] If for any f j (t 1) F(t) where j J, there exists f i (t 1) F(t 1) where i I and a fuzzy relation R ij (t, t = 1) such that f j (t) = f i (t 1) R ij (t, t 1) Let R(t, t 1) = ij R ij (t, t 1) where is the union operator Then, R(t, t 1) is called the fuzzy relation between F(t) and F(t 1) Thus, we define this as the following fuzzy relational equation: F(t) = F(t 1) R(t, t 1) Definition 5 [11, 12] Suppose that R 1 (t, t 1) = ij R 1 ij(t, t 1) and R 2 (t, t 1) = ij R 2 ij(t, t 1) are two fuzzy relations between

2 510 Journal of Marine Science and Technology, Vol 19, No 5 (2011) F(t) and F(t 1) If for any f j (t) F(t) where j J, there exist f i (t 1) F(t 1) where i I and fuzzy relations R 1 ij(t, t 1) and R 2 ij(t, t 1) such that f i (t) = f i (t 1) R 1 ij(t, t 1) and f i (t) = f i (t 1) R 2 ij(t, t 1), then define R 1 (t, t 1) = R 2 (t, t 1) Definition 6 [11, 12] Suppose that F(t) is only caused by F(t 1) or F(t 1) or F(t 2) or F(t ) ( > 0) This relation can be expressed as the following fuzzy relational equation: F(t) = F(t 1) R 0 (t, t ) This equation is called the first-order odel of F (t) Definition 7 [11, 12] Suppose that F(t) is siultaneously caused by F(t 1), F(t 2),, and F(t ) ( > 0) This relation can be expressed as the following fuzzy relational equation: F(t) = (F(t 1) F(t 2) F(t )) R a (t, t ) This equation is called the th order odel of F(t) Definition 8 [2] F(t) is a fuzzy tie series if F(t) is a fuzzy set The transition is denoted as F(t 1) F(t) Definition 9 [7] The universe of discourse U = [D L, D U ] is defined such that D L = D in st α (n) / n and D U = D ax + st α (n) / n when n 30 or D L = D in σz α / n and D U = D ax σz α / n when n > 30, where t α (n) is the 100(1 α) percentile of the t distribution with n degrees of freedo and z α is the 100(1 α) percentile of the standard noral distribution, that is, if Z is a N(0, 1) distribution, then P(Z z α ) = α Definition 10 [7] Assuing that there are linguistic values under consideration, let A i be the fuzzy nuber that represents the i th linguistic value of the linguistic variable where 1 i The support of A i is defined to be DU DL i( DU DL) DL +, 1 i 1 DU DL i( DU DL) DL +, i = Definition 11 [9] For a test H 0 : nonfuzzy trend against H 1 : fuzzy trend, where the critical region C * = {CC 2 k + C 2 n k > C λ = C 2 n (1 λ)} and the initial value of the significant level α is 02 III THE STATIC LONG-TERM PREDICTIVE VALUE INTERVAL In this section, a useful ethod is proposed to forecast the long-ter predictive value interval using a fuzzy tie series Generally speaking, this ethod is an extended odel of the ethod proposed by Chou and Lee [5, 7] Definition 12 Let d(t) be a set of real nubers: d(t) R An upper interval for d(t) is a nuber b such that x b for all x d(t) d(t) is said to be an interval above if d(t) has an upper interval A nuber, ax, is the axiu of d(t) if ax is an upper interval for d(t) and ax d(t) Definition 13 Let d(t) R The least upper interval of d(t) is a nuber ax satisfying: (1): ax is an upper interval for d(t): x (2): ax for all x d(t) ax is the least upper interval for d(t), that is x b for all x d(t) ax b Definition 14 Let d(t) be a set of real nubers: d(t) R A lower interval for d(t) is a nuber b such that x b for all x d(t) d(t) is said to be an interval below if d(t) has a lower interval A nuber, in, is the iniu of d(t) if in is a lower interval for d(t) and in d(t) Definition 15 Let d(t) R The least lower interval of d(t) is a nuber in satisfying: (1): in (2): in is a lower interval for d(t): x in for all x d(t) is the least lower interval for d(t), that is x b for all b x d(t) in Definition 16 The long-ter predictive value interval, ( in, ax) is called the static long-ter predictive value interval Following Definitions 12-16, ( in, ax) can be obtained In suary, the long-ter predictive value interval for d(t) is given by ( in, ax ) The stepwise procedure of the proposed ethod consists of the following steps and a flow diagra is shown in Fig 1 Step 1 Let d(t) be the data under consideration and F(t) be the fuzzy tie series Following Definition 11, a difference test is used to understand whether or not the inforation is in a stable state Recursion is perfored until the inforation is in a stable state, where the critical region is C * = {CC 2 k + C 2 n k > C λ = C 2 n (1 λ)} Step 2 Deterine the universe of discourse U = [D L, D U ] Step 3 Define A i by letting its ebership function be as follows:

3 M-T Chou: Long-Ter Predictive Value Interval with the Fuzzy Tie Series 511 Is the Data Stable? Define the Universe of Discourse Fuzzy Observed Rules Forecast and Defuzzify Set Up Prediction Value Interval Fig 1 The procedure of the proposed odel DU DL i( DU DL) 1 for x [ DL + ) where 1 i 1; ua ( x) = D ( ) i U DL i DU DL 1 for x [ DL + ] where i = ; 0, otherwise Table 1 Fuzzy historical enrollent data and forecasted enrollent Year Actual Fuzzified enrollents ,055 A1 The forecasted results ,563 A2 14, ,867 A2 14, ,696 A3 14, ,460 A3 15, ,311 A3 15, ,603 A3 15, ,861 A4 15, ,807 A5 16, ,919 A5 17, ,388 A4 17, ,433 A3 16, ,497 A3 15, ,145 A3 15, ,163 A3 15, ,984 A4 15, ,859 A5 16, ,150 A6 17, ,970 A7 18, ,328 A7 19, ,337 A7 19,454 Source: [7] Step 4 Then, F(t) = A i if d(t) supp(a i ), where supp ( ) denotes the support Step 5 Derive the transition rule fro period t 1 to t and denote it as F(t 1) F(t) Aggregate all transition rules Let the set of rules be R = {r i r i : P i Q i } Step 6 The value of d(t) can be predicted using the fuzzy tie series F(t) as follows: Let T(t) = {r j d(t) supp(p j ), where r j R} be the set of rules fired by d(t), where supp(p j ) is the support of P j Let supp( P j ) be the edian of supp(p j ) The predicted value of d(t) is supp( Q j ) r ( 1) ( 1) j T t Tt Step 7 The long-ter predictive value interval for d(t) is given as ( in, ax) IV ENROLLMENT FORECASTING Enrollent forecasting will be used to deonstrate how the proposed fuzzy tie series odel can be applied to forecast long-ter data The walk through of enrollent forecasting is presented as follows 1 Forecasting of Enrollent: Step 1 Let d(t) be the historical data under consideration and F(t) be the fuzzy tie series Following Definition 11, a difference test is used to understand whether or not the inforation is in a stable state Recursion is perfored until the inforation is in a stable Since = C" = {CC = C C 4 2 } = 142 < {CC 21 2 (1 02)} = 168, that the inforation is in a stable state is not rejected Step 2 Fro Definition 9, the discourse U = {D L, D U } Fro Table 1, D in = 13,055, D ax = 19,337, s = 1,757, and n = 21 can be obtained Let α = 005 Since n is less than 30, the Student-t distribution with 21 degrees of freedo is used as a substitute for the noral distribution Thus, t α (n) = t 005 (21) = 1721, D L = D in st α / n 12,396, and D U = D ax + st α / n 19,996 That is, U = [12,396, 19,996] Step 3 Assue that the following linguistic values are under consideration: extreely few, very few, few, soe, any, very any, and extreely any According to Definition 10, the supports of fuzzy nubers that represent the linguistic values are given by

4 512 Journal of Marine Science and Technology, Vol 19, No 5 (2011) Table 2 Fuzzy transitions derived fro Table 1 r 1 : A 1 A 2 r 2 : A 2 A 2 r 3 : A 2 A 3 r 4 : A 3 A 3 r 5 : A 3 A 4 Source: [7] r 6 : A 4 A 3 r 7 : A 4 A 5 r 8 : A 5 A 4 r 9 : A 5 A 6 r 10 : A 5 A 5 r 11 : A 6 A 7 r 12 : A 7 A 7 1 for x [12,396 + ( i 1)(1,086),12,396 + i(1,086)) where 1 i 1; ua i ( x) = 1 for x [12,396 + ( i 1)(1, 086), 12,396 + i(1, 086)] where i = ; 0 otherwise where A 1 = extreely few, A 2 = very few, A 3 = few, A 4 = soe, A 5 = any, A 6 = very any, and A 7 = extreely any Thus, the supports are supp(a 1 ) = [12,396, 13482], supp(a 2 ) = [13,482, 14,568], supp(a 3 ) = [14,568, 15,654], supp(a 4 ) = [15,654, 16,740], supp(a 5 ) = [16,740, 17,826], supp(a 6 ) = [17,826, 18,912], and supp(a 7 ) = [18,912, 19,996] Step 4 The fuzzy tie series F(t) is given by F(t) = A i when d(t) supp(a i ) Therefore, F(1971) = A 1, F(1972) = A 2, F(1974) = A 3, F(1979) = A 5,, and F(1989) = A 7 A coparison between actual enrollent data and the fuzzy enrollent data is shown in Table 1 Step 5 The transition rules are derived fro Table 1 For exaple, F(1977) F(1978) is A 3 A 4 All transition rules obtained fro Table 1 are shown in Table 2 Step 6 The forecasting results fro 1972 to 1991 are shown in Table 1 Step 7 The long-ter predictive value interval for d(t) is given as (14,025, 19,454) 2 Discussion One of the ajor liitations of the existing fuzzy tie series forecasting odels [2-5, 7, 11-14] is that they can only provide a single-point forecast value The current paper proposes a ethod which is a coposite of the fuzzy support [7] and the stability concept of fuzzy tie series [9] This ethod not only provides a ore objective interval setup technique, [7] but also increases the forecasting applicability [2-5, 7, 11-14] It is deonstrated that the proposed tie series odel provides an adequate fit to the data and values for the derived fuzzy transitions can be forecasted by using Definitions To forecast the long-ter predictive value interval for d(t), the in-ax interval, is given by ( in, ax) = (14,025, 19,454) According to the ( in, ax), it is clear that the proposed ethod can provide better forecasting inforation than other ethods [2-5, 7, 11-14] V CONCLUSIONS In this paper, a long-ter predictive value interval odel has been developed for the fuzzy tie series This odel helps to iniize the uncertainties of the fuzzy nubers and handle fuzzy-trend/non-fuzzy-trend [9] The ethod is exained by forecasting the enrollent of a university fro its enrollent data fro which the in-ax interval, (in, ax), is obtained In suary, the proposed ethod is as accurate as the ethods used in the past [2, 3, 7, 11-14], and is ore robust and systeatic in forecasting ACKNOWLEDGMENTS The author gratefully acknowledges the helpful coents and suggestions of the reviewers, which have iproved the presentation This research work was partially supported by the National Science Council of the Republic of China under grant No NSC E REFERENCES 1 Chen, S-H, Wang, C-C, and Chang, S-M, Fuzzy econoic production quantity odel for ites with iperfect quality, International Journal of Innovative Coputing, Inforation and Control, Vol 3, No 1, pp (2007) 2 Chen, S-M, Forecasting enrollents based on fuzzy tie series, Fuzzy Sets and Systes, Vol 81, No 3, pp (1996) 3 Chen, S-M, Forecasting enrollents based on high order fuzzy tie series, Cybernetics and Systes: An International Journal, Vol 33, No 1, pp 1-16 (2002) 4 Chou, M-T, A fuzzy tie series odel to forecast the BDI, IEEE Proceeding of the Fourth International Conference on Networked Coputing and Advanced Inforation Manageent, Gyeongju, Korea, pp (2008) 5 Chou, M-T and Lee, H-S, Increasing and decreasing with fuzzy tie series, Joint Conference on Inforation Sciences, Kaohsiung, Taiwan, pp (2006) 6 Han, T-C, Chung, C-C, and Liang, G-S, Application of fuzzy critical path ethod to operation systes, Journal of Marine Science and Technology, Vol 14, No 3, pp (2006) 7 Lee, H-S and Chou, M-T, Fuzzy forecast based on fuzzy tie series, International Journal of Coputer Matheatic, Vol 81, No 7, pp (2004) 8 Liang, M-T, Wu, J-H, and Liang, G-S, Applying fuzzy atheatics to evaluating the ebership of existing reinforced concrete bridges in Taipei, Journal of Marine Science and Technology, Vol 8, No 1, pp (2006) 9 Liaw, M-C, The Order Identification of Fuzzy Tie Series, Models Construction and Forecasting, PhD Dissertation, Departent of Statistics, National Chengchi University, Taiwan (1997) 10 Lin, S-C, Liang, G-S, and Lee, K-L, Applying fuzzy analytic hierarchy process in location ode of international logistics on airports copetition evaluation, Journal of Marine Science and Technology, Vol 14, No 1, pp (2006) 11 Song, Q and Chisso, B S, Fuzzy tie series and its odels, Fuzzy Sets and Systes, Vol 54, No 3, pp (1993)

5 M-T Chou: Long-Ter Predictive Value Interval with the Fuzzy Tie Series Song, Q and Chisso, B S, Forecasting enrollent with fuzzy tie series-part I, Fuzzy Sets and Systes, Vol 54, No 1, pp 1-9 (1993) 13 Song, Q and Chisso, B S, Forecasting enrollent with fuzzy tie series-part II, Fuzzy Sets and Systes, Vol 62, No 1, pp 1-8 (1994) 14 Song, Q, Leland, R P, and Chisso, B S, Fuzzy stochastic tie series and its odels, Fuzzy Sets and Systes, Vol 88, No 3, pp (1997) 15 Zadeh, L A, Fuzzy set, Inforation and Control, Vol 8, No 3, pp (1965) 16 Zadeh, L A, Siilarity relations and fuzzy orderings, Inforation Sciences, Vol 3, No 2, pp (1971) 17 Zadeh, L A, Outline of a new approach to the analysis of coplex syste and decision process, IEEE Transactions on Syste, Man and Cybernetics, Vol 3, No 1, pp (1973)