MODELING FUZZY TIME SERIES WITH MULTIPLE OBSERVATIONS. Received April 2011; revised August 2011

Transcription

1 International Journal of Innovative Computing, Information and Control ICIC International c 2012 ISSN Volume 8, Number 10(B), October 2012 pp MODELING FUZZY TIME SERIES WITH MULTIPLE OBSERVATIONS Kun-Huang Huarng 1 and Tiffany Hui-Kuang Yu 2 1 Department of International Trade 2 Department of Public Finance Feng Chia University No. 100, Wenhua Road, Seatwen, Taichung 40724, Taiwan { khhuarng; hkyu }@mail.fcu.edu.tw Received April 2011; revised August 2011 Abstract. The application of fuzzy time series models to forecasting has been drawing a great amount of attention. To provide a more sophisticated model to handle real world problems thus becomes important. This study intends to model fuzzy time series with multiple observations at a single time point. The proposed model shows how to fuzzify multiple observations into a fuzzy set. Neural networks are applied for training and then forecasting the consecutive fuzzy sets. The results of the forecasting are defuzzified into forecasts. We use daily observations of the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) as a forecasting target for the period from 2001 to The TAIEX is separated into in-sample and out-of-sample observations. The in-sample observations are used for training and the out-of-sample observations for forecasting. The empirical results demonstrate that the proposed model performs better. Keywords: Financial data processing, Forecasting, Neural networks 1. Introduction. Forecasting has been considered important in various problem domains. How to model multiple observations at a single time point to facilitate forecasting seems even more critical. For example, how can we model multiple stock readings for a single date? Conventional statistical methods have attempted to tackle this type of problem; however, they have suffered from the limitations of these conventional methods [39], such as independent and identically distributed random variables [33,34], stationary autoregression [34], and structural breaks [19]. The application of fuzzy time series models to forecasting has been attracting attention. Recent studies have also been proposed to forecast [9,31]. Advanced models have also been proposed, such as high order models [1,35,38,44], multivariate models [3,10,18,43], and high order and multivariate models [5,6,15,29,40]. The forecasting results from some of the fuzzy time series models have been shown to outperform their conventional counterparts [8,12,30,41]. However, these studies mainly use only one of the multiple observations per time point to represent all the observations for that time point. For example, they often use the closing price of the stock index on a certain date to represent all the index readings of the same date. As a result, the forecasting performances are rather limited. The model proposed in this study aims to handle multiple observations at each single time point. All observations for a particular time point are used to form a fuzzy set. Then, the fuzzy sets for different time points are used to form a fuzzy time series. Neural networks are then applied to establish the fuzzy relationships and then to forecast. For example, the proposed model can model all the index readings on a certain date into a fuzzy set. These fuzzy sets of consecutive dates can form a fuzzy time series. The proposed model can 7415

2 7416 K.-H. HUARNG AND T. H.-K. YU then conduct the fuzzy time series forecasting. As a result, the forecasting results can be greatly improved. Fuzzy relationships play an important role in fuzzy time series models. The proposed model applies neural networks for modeling fuzzy relationships between consecutive observations. Neural networks have been popular for their capability in modeling nonlinear relationships [4,21]. We expect that neural networks can assist in establishing and then forecasting fuzzy relationships among degrees of memberships of consecutive observations. We use the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) as a forecasting target for the years from 2001 to The TAIEX is separated into in-sample and out-of-sample observations. The in-sample observations are used for training and the out-of-sample observations for forecasting. The motivation of this study is twofold. It first seeks to provide a model to handle the problems with multiple observations at a time point. Secondly, it aims to further improve the forecasting results of fuzzy time series models. To that end, the article is organized as follows. Section 2 reviews the relevant studies, including fuzzy time series models, neural networks, and fuzzy time series models with neural networks. Section 3 introduces the algorithm for the proposed model. Section 4 explains the target data, the TAIEX. Section 5 applies the proposed model to forecast the TAIEX. Section 6 provides some discussions about the application of the proposed model and the following section is the conclusion. 2. Literature Review Fuzzy time series model. Conventional time series refer to real values, but fuzzy time series are structured by fuzzy sets [2]. Let U be the universe of discourse, such that U = {u 1, u 2,..., u n }. A fuzzy set A of U is defined as A = f A (u 1 )/u 1 + f A (u 2 )/u f A (u n )/u n, where f A is the membership function of A, and f A : U [0, 1]. f A (u i ) is the degree of membership of u i in A, where f A (u i ) [0, 1] and 1 i n. Definition 2.1. Let Y (t) (t =..., 0, 1, 2,...), a subset of a real number, be the universe of discourse on which fuzzy sets f i (t) (i = 1, 2,...) are defined and F (t) is a collection of f 1 (t), f 2 (t),..., F (t) is referred to as a fuzzy time series on Y (t). Here, F (t) is viewed as a linguistic variable and f i (t) represents possible linguistic values of F (t). If F (t) is caused by F (t 1) only, the relationship can be expressed as F (t 1) F (t) [2]. Various operations have been applied to compute the fuzzy relationship between F (t) and F (t 1) [20,36,37]. Chen [2] suggested that when the maximum degree of membership of F (t) belongs to A i, F (t) is considered to be A i. Hence, F (t 1) F (t) becomes A i A j. The advantage of this proposal is that it simplifies the complicated calculations, but it faces a challenge when the out-of-sample observations do not appear in the in-sample observations. However, this problem can be solved by the proposed model in this study Neural network models. Neural networks consist of an input layer, an output layer, and one or more hidden layers. Each of the layers contains nodes, and these nodes of two consecutive layers are connected with each other. Neural networks have been shown to discover nonlinear relationships among the observations [21]. There have been different applications of neural networks, credit ratings [23], Dow Jones forecasting [22], customer satisfaction analysis [7], stock ranking [32], and tourism demand [24,25,27,28], traffic flow [26], etc.

3 MODELING FUZZY TIME SERIES Fuzzy time series with neural networks. There have been studies working on fuzzy time series with neural networks. Huarng and Yu [17] first applied neural networks to train the fuzzy relationships for fuzzy time series. Yu and Huarng [43] proposed the application of neural networks to train the fuzzy relationships for bivariate fuzzy time series. Furthermore, Aladag et al. [1] and Egrioglu et al. [5] applied a similar approach for forecasting high order and high order multivariate fuzzy time series models, respectively. However, these models all take F (t 1) F (t) as A i A j. In other words, only the maximum degree of membership of F (t) is taken into consideration for establishing fuzzy relationships. Yu, Huarng and Rianto [44] started to consider all the degrees of membership of F (t 1) and F (t) in establishing fuzzy relationships for high order fuzzy time series. Many of these models show that they provide better performance in forecasting. 3. Algorithm. The steps for implementing the fuzzy time series model for multiple observations are explained below. Algorithm 1. Define the fuzzy sets for the fuzzy time series as in previous fuzzy time series studies [2,10,15]. We set the beginning of the universe of discourse as s, and the length of interval as l. Suppose there are k intervals. Then, we have the following intervals: u 1 = [s, s + l], u 2 = [s + l, s + 2l],..., u k = [s + (k 1)l, s + kl]. We then define a fuzzy time series as follows [2]: A 1 = 1/u /u /u /u , A 2 = 0.5/u /u /u /u , A 3 = 0.0/u /u /u /u ,... For the calculation, each A j can also be represented by (L j, M1 j, M2 j, R j ), where L j is the left-most value; M1 j, M2 j are the endpoints of the plateau; and R j is the right-most value, respectively. 2. Determine a fuzzy observation, õ t = (l t, m t, r t ), to represent all the (multiple) observations at time point t; l t, m t, r t are the left, middle, and right values of the fuzzy observation, õ t, respectively. There can be various ways to choose from l t, m t, r t. To forecast the stock index, this study considers the lowest and highest values of t as l t and r t, respectively; and the closing price of t as m t. In other words, l t = min(obs z t ), for all z at t. r t = max(obs z t ), for all z at t. m t = close t, where obs z t represents an observation z at a single time point t. 3. Calculate the fuzzy relationship, X t, between each fuzzy observation and the defined fuzzy time series as in Step 1. k X t = (x 1 t, x 2 t,..., x k t ) = max min(õ t, A j ) (1) j=1 where x j t 0. Each x j t can be calculated as follows. The details for deriving the equations are provided in the Appendix. For the x j t for m t : x j t = 1.0. (2) For the x j t to the left of m t : x j t = R j l t m t l t + R j M2 j (3)

4 7418 K.-H. HUARNG AND T. H.-K. YU For the x j t to the right of m t : x j r t L j t = (4) r t m t L j + M1 j 4. Randomly draw a certain portion from all the observations as the out of sample, and treat the rest as the in sample. 5. Calculate the overall relationships from all of the in-sample observations. We apply back-propagation neural networks to train the relationships between consecutive fuzzy relationships, X t and X t To forecast the out-of-sample observations at t + 1, we use X t as the input to the trained neural networks. The output from the trained neural networks is the fuzzy forecast for t + 1, fr t,t+1 = (µ 1 t,t+1, µ 2 t,t+1,..., µ k t,t+1). 7. Defuzzify the fuzzy relationships to obtain the forecasts. k µ g t,t+1 m g forecast t+1 = g=1 k µ g t,t+1 g=1 where forecast t+1 is the forecast for the out of sample at t + 1; µ g t,t+1 is the forecasted degrees of membership and m g represents the corresponding midpoints of the interval, µ g t,t We use two indicators to demonstrate the performance of the proposed model: root mean squared errors (RMSEs) and the hit percentage. RMSEs have been used to measure the performance of fuzzy time series models [14,17,42]: (forecastt+1 close t+1 ) 2 (5) RMSE = where there are a total of q out-of-sample observations. Meanwhile, we are interested in knowing the percentage of the defuzzified forecast falling into the range of its forecasting fuzzy observation, the hit percentage. Suppose the out-of-sample forecast for t + 1 is forecast t+1 and the actual fuzzy observation to forecast is õ t+1 = (l t+1, m t+1, r t+1 ). Hence, count = 0 Loop for all the out-of-sample forecasts If l t+1 forecast t+1 r t+1 then count = count + 1 End Loop hit = count (7) q 4. Data. This study uses the TAIEX as the forecasting target, ranging from 2001 to Each year, 20% of the observations are randomly drawn as the out of sample and the remaining 80% as the in sample. The in-sample observations are used to establish relationships and the out-of-sample observations are used to test the proposed model. 5. Stock Forecasting. This study uses TAIEX as the forecasting target to demonstrate how the proposed model can be used to model the time series with multiple observations. Step 1. Define the fuzzy sets for the fuzzy time series We set the universe of discourse as [0, 12000]. The length of interval is set as Hence, we have the following intervals: u 1 = [0, 1000], u 2 = [1000, 2000],..., u k = [1000 (k 1), 1000 k], where k = 12 in this example. We define a fuzzy time series as follows [2]: A 1 = 1/u /u /u /u , q (6)

5 MODELING FUZZY TIME SERIES 7419 A 2 = 0.5/u /u /u /u , A 3 = 0.0/u /u /u /u ,... The fuzzy time series is defined as shown in Figure 1. Figure 1. The fuzzy time series consisting of A i Step 2. Determine fuzzy observations For each day, there are multiple observations for the stock index. We choose low, close, and high to determine a fuzzy observation for that day. For example, on January 4, 2001, the low is , the close is , and the high is Hence, the fuzzy observation for January 4, 2001 is õ 01/04/2001 = (l 01/04/2001, m 01/04/2001, r 01/04/2001 ) = ( , , ). The fuzzy observation, õ 01/04/2001, is depicted in Figure 2. Similarly, we can obtain the Figure 2. The fuzzy observation obtained from the low, close and high of the multiple observations on 01/04/2001. Hence, õ 01/04/2001 = (5023, 5136, 5169).

6 7420 K.-H. HUARNG AND T. H.-K. YU Figure 3. max min(õ 01/04/2001, A 5 ). First, the bold dotted lines represent the min(õ 01/04/2001, A 5 ). Then, the maximal of the min(õ 01/04/2001, A 5 ) is In other words, max min(õ 01/04/2001, A 5 ) = fuzzy observations for the indices for the other days. Step 3. Calculate the fuzzy relationships Following Equations (2)-(4), the fuzzy relationship for January 4, 2001 is calculated by X 01/04/2001 = 12 max min(õ 01/04/2001, A j ). j=1 x 6 01/04/2001 = 1.0 x 5 R 5 l 01/04/ /04/2001 = m 01/04/2001 l 01/04/ R 5 M = = 0.88 x 7 r 01/04/2001 L 7 01/04/2001 = r 01/04/2001 m 01/04/2001 L 7 + M = = 0.16 The remaining x j t are all zeros. Conceptually, we can also obtain the same results from the dashed bold lines in Figure 3 and then Figure 4. Hence, X 01/04/2001 = (x 1 01/04/2001, x 2 01/04/2001,..., x 12 01/04/2001) = 12 j=1 = (0.0, 0.0, 0.0, 0.0, 0.88, 1.0, 0.16, 0.0, 0.0, 0.0, 0.0, 0.0). max min(õ 01/04/2001, A j ) The fuzzy relationships for the other fuzzy observations can be calculated similarly. Step 4. Sampling As stated in the previous section, we randomly draw 20% of the observations as the out of sample and the remaining 80% as the in sample for every year. Step 5. Neural network training

7 MODELING FUZZY TIME SERIES 7421 Figure 4. X 01/04/2001 = (x 1 01/04/2001, x2 01/04/2001,..., x12 01/04/2001 ) = 12 max j=1 min(õ 01/04/2001, A j ) = (0.0, 0.0, 0.0, 0.0, 0.88, 1.0, 0.16, 0.0, 0.0, 0.0, 0.0, 0.0) We pair X t and X t+1 as the input layer and output layer of the neural network. In the year 2001, X t = (x 1 t, x 2 t,..., x k t ), where k = 12. Hence, there are 12 input and output nodes in the input and output layers, respectively. For example, X 01/04/2001 = (x 1 01/04/2001, x 2 01/04/2001,..., x 12 01/04/2001) = (0.0, 0.0, 0.0, 0.0, 0.88, 1.0, 0.16, 0.0, 0.0, 0.0, 0.0, 0.0), X 01/05/2001 = (x 1 01/05/2001, x 2 01/05/2001,..., x 12 01/05/2001) = (0.0, 0.0, 0.0, 0.0, 0.75, 1.0, 0.30, 0.0, 0.0, 0.0, 0.0, 0.0). The former can serve as the input nodes and the latter as the output nodes of the neural network. The relationships can then be established via the neural network training. Step 6. Neural network forecasting After training, we can proceed to forecast the out-of-sample observations by using the trained neural networks. For example, õ 01/10/2001 = ( , , ) whose X 01/10/2001 = (0.0, 0.0, 0.0, 0.0, 0.60, 1.0, 0.46, 0.0, 0.0, 0.0, 0.0, 0.0). Hence, if we use that fuzzy relationship as an input to the trained neural network, the output of the neural network is fr 01/10/2001,01/11/2001 = (µ 1 01/10/2001,01/11/2001, µ 2 01/10/2001,01/11/2001,..., µ k 01/10/2001,01/11/2001) = (0.0, 0.0, , , , , , , 0.0, 0.0, 0.0, 0.0). And this is the forecast for the fuzzy relationship on January 11, Step 7. Defuzzification Following Equation (5), the weighted average of the fuzzy relationships and the midpoints of the corresponding intervals are used to defuzzify the fuzzy relationships. A forecast can be obtained as follows: ( )/( ) = Step 8. Performance evaluation

8 7422 K.-H. HUARNG AND T. H.-K. YU We calculate all of the out-of-sample forecasts and then the performance indicators, including the RMSE using Equation (6) and the hit percentage using Equation (7). The RMSEs for the years from 2001 to 2006 are listed in Table 1. To demonstrate the performance, we also apply Chen s model [2] to conduct similar forecasts. The RMSEs are compared in Table 1. From the results, we can tell that the proposed model performs better in all the years. Table 1. Comparisons of RMSEs The proposed model Chen s model Table 2. Analysis of hit percentages Year Range Range average /24=38% 15/27=56% 24/41=59% 23/36=64% 30/48=63% 21/41=51% 56% /22=64% 15/20=75% 3/8=38% 7/11=64% 0/1=0% 5/7=71% 64% /2=50% 2/2=100% 2/2=100% 1/1=100% 86% Year average 50% 65% 55% 65% 61% 55% 59% The hit percentages for all the years are also listed in Table 2. By taking the difference between the highest and lowest stock index readings, we categorize the range for each forecast into 0-100, and For example, the number in the second row and second column indicates that there are 9 hits over 24 forecasts falling in the range of in the year The result is equal to 38%. Similarly, we calculate all the hit percentages for the years 2001 to 2006 in the ranges 0-100, and , respectively. Meanwhile, the range average in the rightmost column divides the sum of the hits by that of the forecasts for each range. For example, the range average for range is equal to ( )/( ), which is 56%. Table 2 shows that the range averages for different ranges are 56%, 64% and 86%, respectively. The year average divides the sum of the hits by that of the forecasts by year. For example, that for the year 2001 is equal to ( )/( ), which is 50%. Table 2 shows that the year averages are between 50% and 65%. The overall average of the hit percentage is 59%. 6. Discussions. Fuzzy time series models have been studied for years and different models have been proposed. However, most models are suitable for utilizing only one among multiple observations at a particular time point. In this case, the information from the rest of the observations is missing, which may greatly affect forecasting performance. The model proposed in this study handles multiple observations at a time point. All observations for a particular time point are used to form a fuzzy set. Then, the fuzzy sets for different time points are used to form a fuzzy time series. Neural networks are then applied to establish the fuzzy relationships and then to forecast. As a result, the forecasting results are greatly improved by the proposed model as shown in the above example. In the proposed model, the number of input and output nodes increases as the number of intervals increases. When the number of intervals becomes too large, more input and output nodes may need more computations, which may turn to a concern. However,

9 MODELING FUZZY TIME SERIES 7423 following Yu and Huarng s model [45], we can always take the differences in the observations and then conduct the subsequent fuzzy time series forecasting. In this case, the universe of discourse becomes smaller and hence the number of intervals becomes smaller. Therefore, this concern can be resolved. Due to the same reason, the proposed model can be easily extended into a high order model [15] or a multivariate model [8,18]. Other research results for fuzzy time series studies can also be integrated with the proposed model, such as lengths of intervals [11,16] and defuzzification [36], etc. Hence, the model can greatly assist the subsequent fuzzy time series studies. 7. Conclusion. This study proposes a novel fuzzy time series model to handle multiple observations for forecasting. A systematic method is proposed to model the multiple observations into a fuzzy set. Neural networks are then applied to establish the fuzzy relationships between consecutive fuzzy sets. The Taiwan Stock Exchange Capitalization Weighted Stock Index is taken as the forecasting target to demonstrate the forecasting process and performance. We compare the forecasting RMSEs with those of the previous study. The comparison shows that the proposed model performs better. In addition, in the empirical analysis, the range average can reach as high as 86% and the year average 65%. The overall hit percentage is nearly 60%, which implies that the forecasted results are satisfactory. There are many domain problems with multiple observations at a single time point. For instance, there are many temperature readings in a day; and different numbers of arriving and departing tourists in a month, etc. This study proposes a fuzzy time series model that can handle the problems of multiple observations and then forecast. The length of intervals is always an interesting topic to explore in relation to fuzzy time series models [11,13,16]. Future studies could examine the relationships between the lengths of intervals and hit percentages to see how the length of intervals may affect forecasting performance. The proposed model can also be extended to high-order models as well as multivariate models. Acknowledgments. The authors acknowledge and are grateful for the financial support by the National Science Council, Taiwan, under grant NSC H and NSC H MY2. Ms. Pei-Chin Jocelyn Shen, Librarian of Kreative Commons, at Feng Chia University Library helped to prepare this paper. Mr. Bruce Stewart helped with the editing. Special thanks are also extended to Yan Shi, the Executive Editor of IJICIC and anonymous reviewers for their valuable comments in further revising this paper. REFERENCES [1] C. H. Aladag, M. A. Basaran, E. Egrioglu, U. Yolcu and V. R. Uslu, Forecasting in high order fuzzy times series by using neural networks to define fuzzy relations, Expert Systems with Applications, vol.36, no.3, pp , [2] S.-M. Chen, Forecasting enrollments based on fuzzy time series, Fuzzy Sets and Systems, vol.81, no.3, pp , [3] C.-H. Cheng, G.-W. Cheng and J. W. Wang, Multi-attribute fuzzy time series method based on fuzzy clustering, Expert Systems with Applications, vol.34, no.2, pp , [4] R. G. Donaldson and M. Kamstra, Forecast combining with neural networks, Journal of Forecasting, vol.15, no.1, pp.49-61, [5] E. Egrioglu, C. H. Aladag, U. Yolcu, M. A. Basaran and V. R. Uslu, A new hybrid approach based on SARIMA and partial high order bivariate fuzzy time series forecasting model, Expert Systems with Applications, vol.36, no.4, pp , 2009.

10 7424 K.-H. HUARNG AND T. H.-K. YU [6] E. Egrioglu, C. H. Aladag, U. Yolcu, V. R. Uslu and M. A. Basaran, A new approach based on artificial neural networks for high order multivariate fuzzy time series, Expert Systems with Applications, vol.36, no.7, pp , [7] L. Gronholdt and A. Martensen, Analysing customer satisfaction data: A comparison of regression and artificial neural networks, International Journal of Market Research, vol.47, no.2, pp , [8] Y.-Y. Hsu, S.-M. Tse and B. Wu, A new approach of bivariate fuzzy time series analysis to the forecasting of a stock index, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol.11, no.6, pp , [9] H.-L. Hsu and B. Wu, Evaluating forecasting performance for interval data, Computers & Mathematics with Applications, vol.56, no.9, pp , [10] K. Huarng, Heuristic models of fuzzy time series for forecasting, Fuzzy Sets and Systems, vol.123, no.3, pp , [11] K. Huarng, Effective lengths of intervals to improve forecasting in fuzzy time series, Fuzzy Sets and Systems, vol.123, no.3, pp , [12] K. Huarng, L. Moutinho and T. H.-K. Yu, An advanced approach to forecasting tourism demand in Taiwan, Journal of Travel and Tourism Marketing, vol.21, no.4, pp.15-24, [13] K. Huarng and H.-K. Yu, A dynamic approach to adjusting lengths of intervals in fuzzy time series forecasting, Intelligent Data Analysis, vol.8, no.1, pp.3-27, [14] K. Huarng and H.-K. Yu, A type 2 fuzzy time series model for stock index forecasting, Physica A, vol.353, pp , [15] K. Huarng and H.-K. Yu, An n-th order heuristic fuzzy time series model for TAIEX forecasting, International Journal of Fuzzy Systems, vol.5, no.4, pp , [16] K. Huarng and T. H.-K. Yu, Ratio-based lengths of intervals to improve fuzzy time series forecasting, IEEE Transactions on Systems, Man and Cybernetics Part B, vol.36, no.2, pp , [17] K. Huarng and H.-K. Yu, The application of neural networks to forecast fuzzy time series, Physica A, vol.363, no.2, pp , [18] K. Huarng, T. H.-K. Yu and Y. W. Hsu, A multivariate heuristic model for fuzzy time series forecasting, IEEE Transactions on Systems, Man and Cybernetics Part B, vol.37, no.4, pp , [19] K.-H. Huarng, T. H.-K. Yu and T.-T. Kao, Analyzing structural changes using clustering techniques, International Journal of Innovative Computing, Information and Control, vol.4, no.5, pp , [20] J. R. Hwang, S.-M. Chen and C.-H. Lee, Handling forecasting problems using fuzzy time series, Fuzzy Sets and Systems, vol.100, no.1-3, pp , [21] D. C. Indro, C. X. Jiang, B. E. Patuwo and G. P. Zhang, Predicting mutual fund performance using artificial neural networks, Omega, vol.27, pp , [22] A. Kanas, Neural network linear forecasts for stock returns, International Journal of Finance and Economics, vol.6, no.3, pp , [23] K. Kumar and S. Bhattacharya, Artificial neural network vs. linear discriminant analysis in credit ratings forecast: A comparative study of prediction performances, Review of Accounting & Finance, vol.5, no.3, pp , [24] R. Law, Back-propagation learning in improving the accuracy of neural network-based tourism demand forecasting, Tourism Management, vol.21, pp , [25] R. Law and N. Au, A neural network model to forecast Japanese demand for travel to Hong Kong, Tourism Management, vol.20, pp.89-97, [26] W. Ma, X. Lu and G. Han, Research of predictive model of short-time traffic flow based on fuzzy neural networks, ICIC Express Letters, vol.5, no.5, pp , [27] C. A. Martin and S. F. Witt, Accuracy of econometric forecasts of tourism, Annals of Tourism Research, vol.16, pp , [28] A. Palmer, J. J. Montaño and A. Sesé, Designing an artificial neural network for forecasting tourism time series, Tourism Management, vol.27, pp , [29] J.-I. Park, D.-J. Lee, C.-K. Song and M.-G. Chun, TAIFEX and KOSPI 200 forecasting based on two-factors high-order fuzzy time series and particle swarm optimization, Expert Systems with Applications, vol.37, no.2, pp , [30] P. Qin and D. Huang, Comparison of fuzzy time series and grey model to passenger volume forecasting, Journal of Wuhan University of Technology, vol.31, no.4, pp , 2007.

11 MODELING FUZZY TIME SERIES 7425 [31] W. Qiu, X. Liu and L. Wang, Forecasting in time series based on generalized fuzzy logical relationship, ICIC Express Letters, vol.4, no.5(a), pp , [32] A. N. Refenes, M. Azema-Barac and A. D. Zapranis, Stock ranking: Neural networks vs. multiple linear regression, IEEE International Conference on Neural Networks, pp , [33] S. Sethuraman and I. V. Basawa, Large sample estimation in nonstationary autoregressive processes with multiple observations, Stochastic Processes and Their Applications, vol.54, no.2, pp , [34] S. Sethuraman and I. V. Basawa, Parameter estimation in a stationary autoregressive process with correlated multiple observations, Journal of Statistical Planning and Inference, vol.39, pp , [35] S. R. Singh, A computational method of forecasting based on high-order fuzzy time series, Expert Systems with Applications, vol.36, no.7, pp , [36] Q. Song and B. S. Chissom, Forecasting enrollments with fuzzy time series Part 2, Fuzzy Sets and Systems, vol.62, no.1, pp.1-8, [37] J. Sullivan and W. H. Woodall, A comparison of fuzzy forecasting and Markov modeling, Fuzzy Sets and Systems, vol.64, no.3, pp , [38] H. J. Teoh, T.-L. Chen, C.-H. Cheng and H.-H. Chu, A hybrid multi-order fuzzy time series for forecasting stock markets, Expert Systems with Applications, vol.36, no.4, pp , [39] F.-M. Tseng, G.-H. Tzeng, H.-C. Yu and B. J. C. Yuan, Fuzzy ARIMA model for forecasting the foreign exchange market, Fuzzy Sets and Systems, vol.118, pp.9-19, [40] N.-Y. Wang and S.-M. Chen, Temperature prediction and TAIFEX forecasting based on automatic clustering techniques and two-factor high-order fuzzy time series, Expert Systems with Applications, vol.36, no.2, pp , [41] H.-L. Wong, Y.-H. Tu and C.-C. Wang, Application of fuzzy time series models for forecasting the amount of Taiwan export, Expert Systems with Applications, vol.37, no.2, pp , [42] H.-K. Yu, Weighted fuzzy time series models for TAIEX forecasting, Physica A, vol.349, pp , [43] T. H.-K. Yu and K.-H. Huarng, A bivariate fuzzy time series model to forecast the TAIEX, Expert Systems with Applications, vol.34, no.4, pp , [44] T. H.-K. Yu, K.-H. Huarng and R. Rianto, Neural network-based fuzzy auto-regressive models of different orders to forecast Taiwan stock index, International Journal of Economics & Business Research, vol.1, no.3, pp , [45] T. H.-K. Yu and K.-H. Huarng, A neural network-based fuzzy time series model to improve forecasting, Expert Systems with Applications, vol.37, no.4, pp , Appendix. The fuzzy relationship between each fuzzy observation and the defined fuzzy time series is denoted by X t and X t = (x 1 t, x 2 t,..., x k t ) = k max min(õ t, A j ). A j = (L j, M1 j, M2 j, R j ) and õ t = (l t, m t, r t ). To facilitate the derivation, z j t represents a stock reading. The calculation of x j t is derived as follows: (i) For the x j t for m t : (ii) For the x j t (x j t 0) to the left of m t : x j t = 1.0. j=1 From Equations (A1) and (A2), we have 1.0 m t l t = xj t, z j t l t i.e., z j t = l t + x j t(m t l t ) 1.0 R j M2 j = xj t, R j z j t i.e., z j t = R j x j t(r j M2 j ) (A1) (A2) l t + x j t(m t l t ) = R j x j t(r j M2 j )

12 7426 K.-H. HUARNG AND T. H.-K. YU Hence, x j t = R j l t m t l t+r j M2 j. (iii) For the x j t (x j t 0) to the right of m t : 1.0 r t m t = xj t, r t z j t i.e., z j t = r t x j t(r t m t ) 1.0 M1 j L j = xj t, z j t L j i.e., z j t = L j + x j t(m1 j L j ) From Equations (A3) and (A4), we have Hence, x j t = r t L j r t m t L j +M1 j. r t x j t(r t m t ) = L j + x j t(m1 j L j ) (A3) (A4)