AN IMPROVED FUZZY TIME SERIES THEORY WITH APPLICATIONS IN THE SHANGHAI CONTAINERIZED FREIGHT INDEX

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1 Journal of Marine Science and Technology, Vol 5, No, pp (01) 393 DOI: /JMST AN IMPROVED FUZZY TIME SERIES THEORY WITH APPLICATIONS IN THE SHANGHAI CONTAINERIZED FREIGHT INDEX Ming-Tao Chou Key words: fuzzy, Shanghai Containerized Freight Index, significance level, fuzzy tie series ABSTRACT This study presents fuzzy tie series based on the concept of long-ter predictive significance level Fuzzy tie series theory and structural analysis are used to develop a long-ter predictive significance level for evaluating the suitability of historical data New triangular fuzzy nubers by S are subsequently obtained using the graded ean integration representation ethod Finally, S can strengthen fuzzy tie series data for a series and yield additional inforation The Shanghai Containerized Freight Index is used to illustrate the forecasting process The results indicate that the proposed definition can generate forecast levels that provide ore inforation for analysis I INTRODUCTION Since its creation in 1993 by Song and Chisso (1993a, 199), fuzzy tie series has achieved considerable success in both theory developent (Song and Chisso, 1993a, 1993b, 199, 199; Chen, 1996; Liaw, 199; Chen, 00; Lee and Chou, 00; Chou and Lee, 006; Liang et al, 006; Chou, 008, 009, 011, 013; Chou and Chou, 013) and practical applications, ie, education econoics (Song and Chisso, 1993a, 1993b, 199, 199; Chen, 1996; Liaw, 199; Chen, 00; Lee and Chou, 00), business econoics (Chou, 009; Chou and Chou, 013), onetary econoics (Teoh, 008; Lai, 009), etc Each analysis ode has its own advantages and disadvantages and a different procedure or analysis process Currently, fuzzy tie series is increasingly used to ake long-ter predictions, of which long-range forecasting ethods (Chou, 001), are a notable exaple Fuzzy tie series is no longer liited to short-ter predictions but can be used to view and quantify long-ter forecasting The interval settings followed Paper subitted 06/1/16; revised 10/1/16; accepted 03/13/1 Author for correspondence: Ming-Tao Chou (e-ail: tchou@gailco) Departent of Aviation and Maritie Transportation Manageent, Chang Jung Christian University, Tainan, Taiwan, ROC by Chou in the traditional view of triangular fuzzy nubers and the fuzzy techniques developed by Chen and Hsieh (000) to augent the original interval settings can yield accurate predicted values in the long ter These values can be used to deterine long-ter future standards and colun nubers The binding concept (Chou, 001) uses fuzzy intervals to assess the range of an average predicted value, and the interactive use of these two ethods can deterine the scope of its predicted value With the new paradig and definition defined in this paper, we calculate and interpret predictions of the Shanghai Containerized Freight Index (SCFI) (Shanghai Shipping Exchange, 015) The reainder of this paper is organized as follows Section presents the definition and procedure of fuzzy tie series, and Section 3 defines the long-ter predictive significance level A nuerical exaple of SCFI is shown in Section, and concluding rearks are entioned in conclusion II DEFINITION OF FUZZY TIME SERIES Fuzzy sets, introduced by Zadeh (1965), has various applications, such as in fuzzy sets, fuzzy decision analysis, fuzzy regression, and fuzzy tie series (Song and Chisso, 1993a, 1993b, 199, 199; Chen, 1996, 00; Liaw, 199; Lee and Chou, 00; Chou and Lee, 006: Liang et al, 006; Chou, 008, 009, 011, 013; Duru and Yoshida, 01; Chou and Chou, 013; Chou, 016) The theory is also widely applied in social science study and applications Fuzzy tie series is developed rapidly since their introduction by Song and Chisso (Song and Chisso, 1993a, 199) Recent fuzzy tie series ethods have benefited fro both theoretical developents as well as relevant applications in research (Song and Chisso, 1993a, 1993b, 199, 199; Chen, 1996; Liaw, 199; Chen, 00; Lee and Chou 00; Chou and Lee, 006: Liang et al, 006; Chou, 008, 011; Duru and Yoshida, 01; Chou, 016), which has led to ore diverse uses This trend indicates that the developent of fuzzy tie series has arkedly iproved The definitions and analytical procedures of the fuzzy tie series used in this study are described as follows Definition 1 (Song and Chisso, 1993a, 199; Liaw, 199): A fuzzy nuber on the real line is a fuzzy subset of that is noral and convex

2 39 Journal of Marine Science and Technology, Vol 5, No (01) Definition (Song and Chisso, 1993a, 199): Let Y(t) (t =, 0, 1,, ), a subset of, be the universe of discourse on which the fuzzy sets f i (t) (t = 1,, ) are defined, and let F(t) be the collection of f i (t) (t = 1,, ) Then, F(t) is called fuzzy tie series on Y(t) (t =, 0, 1,, ) Definition 3 (Song and Chisso, 1993a, 199): Let I and J be the index sets for F(t-1) and F(t), respectively If for any fj(t) F(t), where j J, there then exists f i (t-1) F(t-1), where i I, such that there exists a fuzzy relation Rij(t, t-1) and fj(t) = f i (t-1) Rij(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) fj(t), or equivalently, F(t-1) F(t) Definition (Song and Chisso, 1993a, 199): If, for any fj(t-1) F(t), where j J, there exists f i (t-1) F(t-1), where i I, and a fuzzy relation Rij(t, t-1), such that fj(t) = f i (t-1) Rij(t, t-1) Let R(t, t-1) ijrij(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 (Song and Chisso, 1993a, 199): Suppose that R 1 = ijr 1 ij(t, t-1) and R (t, t-1) = ijr ij(t, t-1) are two fuzzy relations between F(t) and F(t-1) If, for any fj(t) F(t), where j J, there exists f i (t-1) F(t-1), where i I, and fuzzy relations R 1 ij(t, t-1) and R 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 ij(t, t-1), then define R 1 (t, t-1) = R (t, t-1) Definition 6 (Song and Chisso, 1993a, 199): Suppose that F(t) is only caused by F(t-1), F(t-),, or F(t-) ( > 0) This relation can be expressed as the following fuzzy relational equation: F(t) = F(t-1) R 0 (t, t-), which is called the first-order odel of F(t) Definition (Song and Chisso, 1993a, 199): Suppose that F(t) is siultaneously caused by F(t-1), F(t-),, and F(t-) ( > 0) This relation can be expressed as the following fuzzy relational equation: F(t) = (F(t-1) F(t-) F(t-)) R a (t, t-)), which is called the th -order odel of F(t) Definition 8 (Chen, 1996): F(t) is fuzzy tie series if F(t) is a fuzzy set The transition is denoted as F(t-1) F(t) Definition 9 (Chou, 009): Let d(t) be a set of real nubers: d(t) R We define an exponential function where (1) y = exp d(t) ln y = d(t) and () exp(ln d(t)) = d(t), ln(exp x) = d(t) Definition 10 (Lee and Chou, 00): The universe of discourse U = [D L, D U ] is defined such that D Din st ( n) n and DU Dax st ( n) n when n 30 or DL Din Z n and D D Z n when n > 30, where t (n) is the 100(1-) U ax L percentile of the t distribution with n degrees of freedo z is the 100(1-) percentile of the standard noral distribution Briefly, if Z is an N (0, 1) distribution, then P(Z z ) = Definition 11 (Lee and Chou, 00): 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 as follows: DU DL i( DU DL) DL ( i1), DL, 1i 1 DU DL i( DU DL) DL ( i1), DL, i Definition 1 (Liaw, 199): For a test H 0 : nonfuzzy trend against * k n k H 1 : fuzzy trend, where the critical region C CC C n C C (1 ), the initial value of the significance level is 0 Definition 13 (Chou, 011): 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) The set d(t) is said to be an interval higher 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 1 (Chou, 011): 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) such that x ax for all x d(t) and () ax is the least upper interval for d(t), that is, x b for all x dt () ax b Definition 15 (Chou, 011): 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) The set 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 16 (Chou, 011): Let d(t) R The least lower interval of d(t) is a nuber in satisfying: (1) in is a lower interval for d(t) such that x in for all x d(t) and () in is the least lower interval for d(t), that is, x b for all x dt () in b

3 M-T Chou: An Iproved Fuzzy Tie Series Theory with Applications in the Shanghai Containerized Freight Index 395 Definition 1 (Chou, 011): The long-ter predictive value interval ( in, ax ) is called the static long-ter predictive value interval III DEFINITION OF LONG-TERM PREDICTIVE SIGNIFICANCE LEVEL AND PROCEDURE This section proposes a ethod to forecast the long-ter predictive significance level by using fuzzy tie series, extending the ethod proposed by Chou (011) Based on Chou s predictive value interval dt ˆ( ), any ethods to construct triangular fuzzy nubers are developed using Chen s technique (1996) These perit the prediction of long-ter profit levels and the prediction of phase discriination values for discriinating future trends Definition 18 (Chen and Hsieh, 000): Let A i = ( i, i, i ), i = 1,,, n, be n triangular fuzzy nubers By using the graded ean integration representation (GMIR) ethod, the GMIR value P(A i ) of A i is P(A i ) = ( i i i )/6 P(A i ) and P(A j ) are the GMIR values of the triangular fuzzy nubers A i and A j, respectively Definition 19 Set up new triangular fuzzy nubers by S = ( in, dt (), ax ) After GMIR transforation, S becoes a real nuber S This is called the long-ter significance level with fuzzy tie series The S is a real nuber satisfying the following: (1) S is called a long-ter significance level up, only if: S > dt (); () S is called a long-ter significance level down, only if: S < dt (); and (3) S is called a long-ter significance level stable, only if: S = dt () The stepwise procedure of the proposed ethod consists the following steps (Chou, 016), illustrated as a flowchart in Fig 1 (Chou, 008; Chou, 016) Step 1 Let d(t) be the data under consideration and let F(t) be fuzzy tie series Following Definition 11, a difference test is perfored to deterine whether stability of the inforation Recursion is perfored until the inforation is in a stable state, where the critical region is * k nk n C C C C C C (1 ) Step Deterine the universe of discourse U = [D L, D U ] Step 3 Define A i by letting its ebership function be as follows: Is the Data Stable? Define the Universe of Discourse Fuzzy Observed Rules Forecast and Defuzzify Set Up Prediction Value Interval Set Up New Triangular Fuzzy Defuzzify by GMIR Fig 1 Procedure of the proposed odel DU DL i( DU DL) 1 for x[ DL ( i1), DL ) where 1 i 1; DU DL i( DU DL) ua ( x) 1 for x[ D ( 1), ] i L i DL where i ; 0 otherwise Step 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 R r r : P Q Let the set of rules be i i i 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 ) Tt ( 1) rjt( t1) Step The long-ter predictive value interval for dt () is given as ( in, ax ) Step 8 Set up new triangular fuzzy nubers by S = ( in, dt () ax ) Step 9 Defuzzify S to be S

4 396 Journal of Marine Science and Technology, Vol 5, No (01) Q 011Q 01Q 013Q Fig Rate of return of the SCFI The SCFI return rate 01Q 015Q IV NUMERICAL EXAMPLE OF SCFI DATA In this study, the SCFI is used for a nuerical exaple The SCFI reflects the spot rates of the Shanghai export container transport arket, including both freight rates (indices) of 15 individual shipping routes and a coposite index (Shanghai Shipping Exchange, 015) The SCFI data are sourced fro the Shanghai Shipping Exchange (015), the historical data for which is defined here as the SCFI, and season-averaged data for the period between Quarter 1, 010, and Quarter, 015, was collected Over these data points, the analysis produces an average of 110, with a standard deviation of 1883, axiu value of 15151, and iniu value of 6965 These descriptive statistics show that the SCFI has largely reained at the 110 level Copared with figures fro Quarter 1, 010, this value represents the peak level for recent years while China s doestic deand has reained weak, because of the global financial crisis and European debt crisis Therefore, the weak doestic deand has allowed China s econoic growth to reain crucial despite these negative influences We discovered that although China is considerably affected by the SCFI, the consequent adjustents result in a synergetic change in the rate of return of the SCFI As shown in Fig, the SCFI has recovered fro the effect of the financial crisis, although its current rate of return is negative The following steps in the procedure are perfored when using fuzzy tie series to analyze visitor arrivals Step 1 First, we take the logarith of the SCFI data to reduce variation and iprove the forecast accuracy, letting ~ SCFI( t) lnscfi( t) Step Maintaining stationary data while forecasting helps to iprove the forecast quality; therefore, we conduct a stationary test on the SCFI data For fuzzy tie series, a fuzzy trend test can easure whether the SCFI s fuzzy trend oves upward or downward Using this fuzzy trend test, the SCFI data can be converted into a stationary series If the original SCFI data exhibited a fuzzy trend, it can be eliinated by taking the difference We then repeat the test after taking the first difference to Table 1 Fuzzy historical SCFI data and the forecasted results Year ln(actual) Fuzzified The forecast value 010Q1 18 A Q 0 A Q3 33 A Q 0 A Q1 696 A Q 6950 A Q3 693 A Q 6801 A Q A 68 01Q 68 A Q3 190 A Q 05 A Q1 03 A Q 690 A Q A Q 6935 A 68 01Q1 698 A 68 01Q 6996 A 68 01Q3 699 A 68 01Q 6933 A Q1 689 A Q 658 A 1 63 easure if the SCFI data exhibits a fuzzy trend If a fuzzy trend is again observed, then we take the second difference, and so on Letting SCFI(t) be the historical data under consideration and fuzzy tie series, a difference test is used (following Definition 11) to understand whether the stability of the inforation Recursion is perfored until the inforation is deterined to be stable Once the region '' C C C C C, 106 CC (1 0) the SCFI data are considered in a stable state and are not rejected Step 3 According to the interval setting of the SCFI data, we define the upper and lower bounds, which facilitate dividing the linguistic value intervals later Fro Definition 10, the discourse U = [D L, D U ] Fro Table 1, D in = 658, D ax = 33, s = 010, and n = can be obtained Letting = 055, since n is less than 30, a Student t distribution with degrees of freedo was used as a substitute for the noral distribution Thus, t (n) =

5 M-T Chou: An Iproved Fuzzy Tie Series Theory with Applications in the Shanghai Containerized Freight Index 39 t 005 () = 11, DL Din st n 685, and D U = D ax st n 386 That is, U = [685, 386] Step After defining the upper and lower bounds of the SCFI data in Step 3, we can define the SCFI range by deterining the ebership function as well as the linguistic values We can also define the range of the subinterval for each linguistic value, assuing that the following linguistic values are under consideration: extreely few, very few, few, soe, any, very any, and extreely any According to Definition 11, the supports of fuzzy nubers that represent these linguistic values are given as follows: 1 for x[685 ( i1)(019), 685 i(019)) where 1 i 1; ua i ( x) 1 for x[685 ( i1)(019), 685 i(019)] where i ; 0 otherwise where A 1 = extreely few, A = very few, A 3 = few, A = soe, A 5 = any, A 6 = very any, and A = extreely any Thus, the supports are supp(a 1 ) = [685, 661), supp(a ) = [661, 63), supp(a 3 ) = [63, 68), supp(a ) = [68, 001), supp(a 5 ) = [001, 130), supp(a 6 ) = [130, 59), and supp(a ) = [59, 386) Step 5 According to the subinterval setting of each linguistic value, we classified each historical dataset of the SCFI into its corresponding interval to easure the value corresponding to the linguistic value for each interval The fuzzy tie series F(t) was given by F(t) = A i when d(t) supp(a i ) Therefore, F(010Q1) = A 6, F(010Q) = A 6, F(010Q3) = A, F(010Q) = A 5,, and F(015) = A 1 Table 1 shows the coparison between the actual SCFI data and the fuzzy enrollent data Step 6 We apply fuzzy theory to define the corresponding value for the intervals of the SCFI data, arrange the corresponding ethod for the SCFI data, and integrate the changes fro all the rules to deterine the rules for the SCFI quantity The transition rules are derived fro Table 1 For exaple, F(010Q1) F(010Q) is A 6 A 6 Table shows all transition rules obtained fro Table 1 Step We calculate each rule by deterining all the rules of the SCFI, and the calculation results can be used to forecast future values Table 1 shows the forecasting results fro 010Q1 to 015Q Step 8 The calculated SCFI rules can define the intervals of the SCFI data; using these intervals, we can deterine the variation in future long-ter intervals The longter predictive value interval for the SCFI is given as (63, 19) Thus, the long-ter predictive interval Table Fuzzy transitions derived fro Table 1 1: 3 r5 : A A6 r9 : A6 A6 : 1 r6 : A5 A r10 : A6 A 3 : 3 r : A5 A5 r11 : A A5 : r8 : A6 A Actual Proposed value 010Q1 010Q 011Q3 01Q 013Q1 013Q 01Q3 015Q Fig 3 Forecast SCFI and actual SCFI for the SCFI is given as (88005, ) Therefore, the current long-ter SCFI is bounded by this interval According to Step 8, the fuzzy SCFI of 015Q shown in Table 1 is A 1, and fro Table, we can see that the rules are the fuzzy logical relationships in Rule 11 of Table, in which the current state of fuzzy logical relationships is A 6 Thus, the 015Q3 SCFI predictive value is Step 9 Letting defuzzified S be S, the SCFI 015Q3 forecast value based on our investigation is 88005, and its trading range is between and Thus, the new triangular fuzzy nubers by S = (88005, , ) Thus, the defuzzified S is S = 9853, and S = 9853 > dt () = The result shows that based on the long-ter significance level, the SCFI is currently oversold This result and the riskreward ratio are both related within the group We used Table 1 data in our analysis according to the root ean square percentage error ethod, with an average prediction error of 08% Fig 3 shows the forecast visitor arrivals deterined through fuzzy tie series analysis and the actual SCFI values Based on the fuzzy tie series results, the average SCFI is estiated to be in 015Q3 (Fig 3) V CONCLUSION In this paper, a long-ter predictive value interval odel is developed for forecasting the SCFI This odel facilitates ini-

6 398 Journal of Marine Science and Technology, Vol 5, No (01) izing the uncertainties associated with fuzzy nubers The ethod is exained by forecasting the SCFI by using data fro which S = 9853 and S > dt () is obtained For index returns, the current rate of return is negative and its volatility is increasing The long-ter predictive significance level of the SCFI is at the S level; the SCFI should thus exhibit extree volatility The current odel for the SCFI 015Q3 forecast level deviates insignificantly fro the actual values for an average of and is within the group; the prediction error does not exceed 08% of the significance level By constructing a fuzzy tie series forecasting odel for the SCFI with an error of less than 08%, with the traditional fuzzy tie excluded fro the singlepoint forecast coparison, this odel provides a long-ter predictive significance level Furtherore, the proposed ethod can be coputerized Thus, by iproving fuzzy linguistic assessents as well as the evaluation of fuzzy tie series, decision akers can autoatically obtain the final long-ter predictive significance level REFERENCES Chen, S-H and C-H Hsieh (000) Representation, ranking, distance, and siilarity of L-R type fuzzy nuber and application Australian Journal of Intelligent Inforation Processing Systes 6, 1-9 Chen, S-M (1996) Forecasting enrollents based on fuzzy tie series Fuzzy Sets and Systes 81, Chen, S-M (00) Forecasting enrollents based on high order fuzzy tie series Cybernetics and Systes: An International Journal 33, 1-16 Chou, M-T and H-S Lee (006) Increasing and decreasing with fuzzy tie series Joint Conference on Inforation Sciences, Chou, M-T (008) A fuzzy tie series odel to forecast the BDI IEEE Proceeding of the Fourth International Conference on Networked Coputing and Advanced Inforation Manageent, Chou, M-T (009) The logarith function with a fuzzy tie series Journal of Convergence Inforation Technology, -51 Chou, M-T (011) Long-ter predictive value interval with the fuzzy tie series Journal of Marine Science and Technology 19, Chou, M-T (013) An application of fuzzy tie series: a long range forecasting ethod in the global steel price index forecast Review of Econoics and Finance 3, Chou, M-T and C-C Chou (013) The iplication of Taiwan s ore trap carrier cargo on the blast furnace plant Advanced Materials Research 69-69, Chou, M T (016) Fuzzy tie series theory application for the China containerized freight index Applied Econoics and Finance 3, -53 Duru, O and S Yoshida (01) Modeling principles in fuzzy tie series forecasting 01 IEEE Conference on Coputational Intelligence for Financial Engineering and Econoics, 1- Lee, H-S and M-T Chou (00) Fuzzy forecast based on fuzzy tie series International Journal of Coputer Matheatics 81, Lai, R K, C Y Fan, W H Huang and P C Chang (009) Evolving and clustering fuzzy decision tree for financial tie series data forecasting Expert Systes with Applications 36, Liang M-T, J-H Wu and G-S Liang (006) Applying fuzzy atheatics to evaluating the ebership of existing reinforced concrete bridges in Taipei Journal of Marine Science and Technology 8, 16-9 Liaw, M-C (199) The order identification of fuzzy tie series, odels construction and forecasting, PhD Thesis, National Chengchi University, Taiwan Shanghai Shipping Exchange (015) Retrieved August 10, 015, website: Song, Q and B S Chisso (1993a) Forecasting enrollent with fuzzy tie series-part I Fuzzy Sets and Systes 5, 1-9 Song, Q and B S Chisso (1993b) Fuzzy tie series and its odels Fuzzy Sets and Systes 5, 69- Song, Q and B S Chisso (199) Forecasting enrollent with fuzzy tie series-part II Fuzzy Sets and Systes 6, 1-8 Song, Q, R P Leland and B S Chisso (199) Fuzzy stochastic tie series and its odels Fuzzy Sets and Systes 88, Teoh, H J, C H Chen, H H Chu and J S Chen (008) Fuzzy tie series odel based on probabilistic approach and rough set rule induction for epirical research in stock arkets Data and Knowledge Engineering 6, Zadeh, L A (1965) Fuzzy set Inforation and Control 8,