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Article

A Two-Factor Autoregressive Moving Average Model Based on Fuzzy Fluctuation Logical Relationships

1
Rensselaer Polytechnic Institute, Troy, NY 12180, USA
2
School of management, Jiangsu University, Zhenjiang 212013, China
*
Author to whom correspondence should be addressed.
Symmetry 2017, 9(10), 207; https://doi.org/10.3390/sym9100207
Submission received: 26 August 2017 / Revised: 25 September 2017 / Accepted: 28 September 2017 / Published: 1 October 2017
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)

Abstract

:
Many of the existing autoregressive moving average (ARMA) forecast models are based on one main factor. In this paper, we proposed a new two-factor first-order ARMA forecast model based on fuzzy fluctuation logical relationships of both a main factor and a secondary factor of a historical training time series. Firstly, we generated a fluctuation time series (FTS) for two factors by calculating the difference of each data point with its previous day, then finding the absolute means of the two FTSs. We then constructed a fuzzy fluctuation time series (FFTS) according to the defined linguistic sets. The next step was establishing fuzzy fluctuation logical relation groups (FFLRGs) for a two-factor first-order autoregressive (AR(1)) model and forecasting the training data with the AR(1) model. Then we built FFLRGs for a two-factor first-order autoregressive moving average (ARMA(1,m)) model. Lastly, we forecasted test data with the ARMA(1,m) model. To illustrate the performance of our model, we used real Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and Dow Jones datasets as a secondary factor to forecast TAIEX. The experiment results indicate that the proposed two-factor fluctuation ARMA method outperformed the one-factor method based on real historic data. The secondary factor may have some effects on the main factor and thereby impact the forecasting results. Using fuzzified fluctuations rather than fuzzified real data could avoid the influence of extreme values in historic data, which performs negatively while forecasting. To verify the accuracy and effectiveness of the model, we also employed our method to forecast the Shanghai Stock Exchange Composite Index (SHSECI) from 2001 to 2015 and the international gold price from 2000 to 2010.

1. Introduction

A historic time series can show the rules and patterns of some phenomena and can be applied to forecast the same event in the future [1]. Many researchers have described time series models to predict the future of a given system, including regression analysis [2], artificial neural networks (ANN) [3], evolutionary computation [4], support vector machines (SVM) [5], and immune systems [6]. However, although these models satisfy the constraints, they might overemphasize the randomness of the dataset and distort the internal evolutionary rules, and may not perform optimally. To solve this problem, Song and Chissom proposed the fuzzy time series forecasting model [7] which introduced the fuzzy set theory by Zadeh [8] into a time series. Chen [9] developed a first order fuzzy time series to simplify the fuzzy relationships in Song and Chissom’s model [7,10,11], described by complex matrix operations. Chen’s method [9] has been the basis for the future research of fuzzy logic groups because of its universality and level of performance. For the selection of the length of the intervals, Huarng [12] proposed two methods: based on averages and on distribution. Since then, the fuzzy time series model has been widely used for forecasting in many nonlinear and complex forecasting problems. In order to forecast the fluctuation of the stock market, Chen [13] proposed a hybrid first order fuzzy time series model using granular computing as the partitioning method. Many studies [14,15,16] used a second-order fuzzy time series model to create the rules for the forecasting of future trends. The biggest differences between these fuzzy time series models are the detailed partitioning method and the trend rules. Efendi et al. [17] used a fuzzy time series model to forecast daily electricity load demand. Sadaei et al. [18] proposed a short-term load forecasting model based on the seasonality memory process and fuzzy time series model. These fuzzy time series models are all autoregressive (AR) models. With fuzzy lagged variables of a time series, these models can be represented as AR(n). Such models are also used for project cost forecasting [19] and the enrollment forecasting at Alabama University [20,21].
In order to improve the accuracy of fuzzy time series models, many researchers have proposed other models on the basis of Chen’s model. For example, an unequal interval length method was proposed by Huarng and Yu [22] based on the ratios of data in which the length of interval was exponentially variable. In addition to determining the intervals, the definition of the universe of discourse also plays an effective role in the forecasting accuracy. To establish a suitable universe of discourse, in addition to the maximum and minimum values of the historical data of the main factor, the models need two proper real numbers to cover the noise.
Another essential step when creating fuzzy time series models is the establishment of fuzzy logical relationships (FLR). In this realm, the research by Egrioglu et al. [23] is regarded as a basic high-order method for forecasting based on artificial neural networks. Moreover, Egrioglu [24] employed generic algorithms to establish fuzzy relations. Some other soft computing techniques have been used to forecast in many studies [25,26,27]. In fact, fuzzy time series forecasting studies are frequently based on fuzzy autoregressive (AR) structures [28,29,30,31,32]. To further improve the performance of fuzzy AR models, an adaptive fuzzy inference system (ANFIS) [33] has been used in time series prediction [34,35,36,37]. However, only using an AR structure for some of the time series may lead to unsatisfactory and flawed results. To address this, we combined moving average (MA) structures and produced an ARMA-type fuzzy time series forecasting model that includes both AR and MA structures. Because of the excellent performance of the ARMA model, it has been widely mentioned in the. For example, Kocak [38] and Kocak el al. [39] researched first-order ARMA fuzzy time series models based on fuzzy logical relation tables and an artificial neural network, respectively. Kocak [40] also studied a high-order ARMA fuzzy time series model.
Most of the existing fuzzy time series models first fuzzify the exact values of the time series, then use AR models of the dataset itself to forecast its future. Such methods usually improve the performance by using extra solution steps, such as the use of artificial neural networks. In this paper, we propose a new first-order ARMA model based on two-factor fuzzy logical relationships. The advantages of this model are that it uses the fluctuation values rather than the exact values of the time series, and a secondary factor is used to help forecast the main factor with ARMA fuzzy time series models. Since the fluctuation orientations, including up, equal, and down, and the extent to which the trends would be realized, are the crucial ingredients for financial forecasting. Because of this, using a fluctuation time series for further rules generation would be more reasonable. Although internal rules determine future changes, we could not ignore the effects of relative external changes. Therefore, we chose an external element as the secondary factor to generate the logical rules. The experiment results indicate that the proposed two-factor fluctuation method outperforms the one-factor method, based on real historic data, because the secondary factor may have some effects on the main factor and thereby impact the forecasting results. Using fuzzified fluctuations, rather than fuzzified real data, could avoid the influence of extreme values in the historic data which negatively affects forecasting.
The remainder of the paper is organized as follows. The next section presents the basic preliminaries of fuzzy-fluctuation time series. The third section introduces the procedure used to build the ARMA(1,m) model. Next, the proposed model is used to forecast the stock market using TAIEX datasets from 1997 to 2005, SHSECI from 2001 to 2015, and internal gold prices from 2000 to 2010. Finally, we discuss the conclusions and potential future research.

2. Preliminaries

In this section, the general definitions of a fuzzy fluctuation time series in ARMA(1,m) models are outlined.
Definition 1.
Let A ( t ) ,   ( t = 1 , 2 , 3 , , T ) be a time series of real numbers, where T is the number of the time series, and can be defined as the universe of discourse of the fuzzy sets L = {   L 1 ,   L 2 , ,   L g } . According to the membership function, μ L : A ( t ) [ 0 , 1 ] , each element of the time series A ( t ) ,   ( t = 1 , 2 , 3 , , T ) can be represented by a fuzzy number Z ( t ) = L i ,   ( t = 1 , 2 , 3 , , T ,   i = 1 , 2 , , g ) . We called Z ( t ) ,   ( t = 1 , 2 , 3 , , T ) a fuzzy time series.
Definition 2.
For a time series G ( t ) ,   ( t = 1 ,   2 ,   3 ,   ,   T ) , Y ( t ) is defined as a fluctuation time series where   Y ( t ) = G ( t ) G ( t 1 ) ,   ( t = 2 , 3 , , T ) . As described in Definition 1, t = 2 ,   3 ,   , T ,   Y ( t ) could be represented by a fuzzy time series   H ( t ) ,   ( t = 2 ,   3 ,   , T ) . Thereby, the time series Y ( t ) is fuzzified into a fuzzy-fluctuation time series (FFTS) H ( t ) .
Definition 3.
Let H ( t ) be a FFTS ( t = 2 ,   3 , , T ) . If the “next status” of H ( t )   is caused by the “current status” of H ( t 1 ) , the first order fuzzy-fluctuation AR(1) is represented by [10,11]:
H ( t 1 ) H ( t )
Similarily, let Q ( t ) and P(t) be two FFTSs ( t = 2 ,   3 , , T ) . If the next status of Q ( t )   is caused by the current status of Q ( t 1 )   and   P ( t 1 ) , the two-factor first order fuzzy-fluctuation AR(1) is represented by:
Q ( t 1 ) , P ( t 1 ) Q ( t )
This is called the two-factor first order fuzzy-fluctuation logical relationship (FFLR). Q ( t 1 ) ,   P ( t 1 ) is the left-hand side (LHS) and Q ( t ) is the right-hand side (RHS) of the FFLR. A forecasting model based on these relationships is called a two-factor first order time series forecasting model.
Definition 4.
Let F ( t ) be a fuzzy time series and ε ( t )   be a fuzzy error series obtained from the F ( t ) . If F ( t ) is affected by both the lagged fuzzy time series ( F ( t 1 ) , F ( t 2 ) , , F ( t n ) ) and the lagged fuzzy error series ( ε ( t 1 ) , ε ( t 2 ) , , ε ( t m ) ) , the fuzzy logical relationship can be represented by [40]:
F ( t 1 ) , F ( t 2 ) , , F ( t n ) , ε ( t 1 ) , ε ( t 2 ) , , ε ( t m ) F ( t )
Similarly, let Q ( t ) and P(t) be two FFTSs ( t = 2 ,   3 , , T ) and ε ( t )   be a fuzzy error series obtained from Q ( t ) . If Q ( t ) is affected by both the lagged fuzzy time series ( Q ( t 1 ) , P ( t 1 ) ) and the lagged fuzzy error series ( ε ( t 1 ) , ε ( t 2 ) , , ε ( t m ) ) , the fuzzy logical relationship can be represented by:
Q ( t 1 ) , P ( t 1 ) , ε ( t 1 ) , ε ( t 2 ) , , ε ( t m ) Q ( t )
This is called a two-factor ARMA(1,m) fuzzy-fluctuation time series forecasting model, where m ≤ T. In this expression, m gives the order of the MA model.

3. New Forecasting Model Based on Two-Factor ARMA(1,m) FFLRs

In this paper, we propose a new forecasting model with two-factor first-order fuzzy fluctuation logical relationships ARMA model. To make a comparison with the forecasting results of other researchers’ work [29,30,41,42], we used the real Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) to show the forecasting procedure. We used the data from January to October of the given year as a training time series and the data from November to December of the same year as the testing dataset. The basic steps of the proposed model are shown in Figure 1.
Step 1:
The first step was to construct a FFTS for the historical main and secondary factor training data. For each element A 1 ( t ) ,   ( t = 1 , 2 , , T ) in the historical training time series of the main factor, its fluctuation trend was determined by   X 1 ( t ) = A 1 ( t ) A 1 ( t 1 ) ,   where   t = 2 ,   3 ,   4 ,   , T . By the values and directions of fluctuation, we fuzzified X 1 ( t ) into a linguistic set {down, equal, up}. We assumed w 1 and w 2 are the absolute means of all elements in the fluctuation time series   X 1 ( t ) ,   X 2 ( t ) ( t = 2 ,   3 ,   4 ,   T ) , respectively. Then, we had α 1 = ( , w 1 2 ) ,   α 2 = [ w 1 2 , w 1 2 ) , and   α 3 = [ w 1 2 , ) . Similarly we divided X 1 ( t ) into 5 intervals such that   α 1 = ( , 3 w 1 4 ) ,   α 2 = [ 3 w 1 4 , w 1 4 ) ,   α 3 = [ w 1 4 , w 1 4 ) ,   α 4 = [ w 1 4 , 3 w 1 4 ) , and α 5 = [ 3 w 1 4 , ) . We divided X 1 ( t ) into any   g = 2 l + 1   intervals, where l is an integer. Next, for each element A 2 ( t ) ,   ( t = 1 , 2 , , T ) in the historical training time series of the secondary factor, its fluctuation trend was determined by X 2 ( t ) = A 2 ( t ) A 2 ( t 1 ) ,   where   t = 2 ,   3 ,   4 ,   , T . According to Definition 1, X 2 ( t ) can also be divided into g intervals, namely β i ,   ( i = 1 ,   2 ,   ,   g ) . Then we fuzzified X 1 ( t )   and   X 2 ( t ) into FFTSs   Q 1 ( t )   and   Q 2 ( t ) ,   ( t = 2 , 3 , , T )   , respectively, where Q 1 ( t ) = L i and Q 2 ( t ) = K j both have the highest membership value of corresponding intervals α i and β j   ( i , j = 1 ,   2 ,   ,   g ) , respectively, and { L 1 , L 2 , , L g } , { K 1 , K 2 , , K g } are fuzzy sets.
Step 2:
The second step was to determine the two-factors fuzzy-fluctuation logic relationships for the AR(1) model. In this step, we determined the two-factor fuzzy-fluctuation logical relationships for the AR(1) model as outlined in Definition 3. Let the lagged variables Q 1 ( t 1 ) = L i , Q 2 ( t 1 ) = K s , and Q 1 ( t ) = L j   ; the FFLR of this two-factor AR(1) model is   L i , K s L j . Then, the FFLRs with the same LHS were grouped into a fuzzy-fluctuation logical relationship group (FFLRG) by putting all their RHSs together, as on the RHS of the FFLRG. For example, when the FFLRs for a two-factor AR(1) model are L 1 , K 2 L 2 and L 1 , K 2 L 3 , then the FFLRG would be   L 1 , K 2 L 2 , L 3 .
Step 3:
The next step was to obtain the fuzzy fluctuation forecast result from AR(n) model. We assumed the lagged variables   Q 1 ( t 1 ) = L i , Q 2 ( t 1 ) = K s   , and we defined the following conditions:RHS Conditions: If L i , K s L j , , L j , L h , , L h , L l , , L l   exists and assuming the numbers of   L j ,   L h   ,   and L l from the previous equation are a ,   b , and c   respectively, then the fuzzy fluctuation forecast result would be L j , ,   L j , L h , , L h , L l , , L l . Null RHS Condition: If   L i , K s e m p t y exists on the FFLRG, then the fuzzy forecast is   L i , K s .
Step 4:
Next, we defuzzified the fluctuation forecast result for the AR(1) model. We used the centralization method to defuzzify the forecast results. For example, assuming m j ,   m h   , and m l are the middle points of corresponding sub-intervals of   L j ,   L h   ,   and L l respectively, the defuzzified fluctuation forecast result is represented by:
X A R ^ ( t ) = a ×   m j + b ×   m h + c ×   m l   a + b + c
Step 5:
Next, we calculated the fluctuation error series E ( t ) :
E ( t ) = X 1 ( t ) X A R ^ ( t )
where X 1 ( t )   is the time series of the fluctuation numbers of main factor, and X 1 ^ ( t ) is calculated result from Step 4.
Step 6:
The next step was to construct fuzzy fluctuation time series for the error series E ( t ) . In the same manner as described in Step 1, we fuzzified E ( t ) into FFTSs R ( t ) . We assumed h is the absolute mean of all elements in the time series   E ( t ) ,   ( t = 2 ,   3 ,   4 ,   T ) , g is the number of intervals of the fuzzy sets, ε 1 ,   ε 2 , , ε g are corresponding intervals, R ( t ) = W i has the highest membership value of corresponding intervals ε i   ( i = 1 ,   2 ,   ,   g ) , and { W 1 , W 2 , , W g } are the corresponding fuzzy sets.
Step 7:
Next, we determined the two-factor fuzzy logical relationships for the ARMA(n,m) model. In this step, we determined the fuzzy logical relationships for ARMA(n,m) model as outlined in Definition 4. Let the lagged variables Q 1 ( t 1 ) = L i , Q 2 ( t 1 ) = K s , Q 1 ( t ) = L j   , R ( t m ) = W i 3   , R ( t ( m 1 ) ) = W m 2 ,…,   R ( t 1 ) = W s 2 , and the FFLR of this two-factor ARMA(1,m) model is   L i , K s , W i 2 , W m 2 , W s 2 L j . Then, as described in Step 2, the FFLRs with the same LHS were grouped into a FFLRG for the ARMA(1,m) model.
Step 8:
Next, we obtained the fuzzy fluctuation forecast result from the ARMA(1,m) model. In the same manner as described in Step 3, we forecasted the future based on the two-factor FFLRG and the lagged variables. Assuming the lagged variables Q 1 ( t 1 ) = L i , Q 2 ( t 1 ) = K s , and the lagged error variables R ( t m ) = W i 3   , R ( t ( m 1 ) ) = W m 2 ,…, and   R ( t 1 ) = W s 2 , we defined the following conditions:
RHS Condition: If L i , K s , W i 3 , W m 3 , , W s 3 L j , , L j , L h , , L h , L l , , L l   exists and assume the number of   L j   L h   and L l from the previous equation is a ,   b , and c , respectively, then the fuzzy fluctuation forecast result would be L j , ,   L j , L h , , L h , L l , , L l .
Null RHS Condition: If   L i , K s , W i 3 , W m 3 , , W s 3 e m p t y exists on the FFLRG, then it was replaced with the FFLRG of its corresponding AR(1) model of L i , K s .
Step 9:
In the final step, we defuzzified the forecast fluctuation and obtained forecast results. As described in Step 4, we defuzzified the obtained new forecast fluctuation:
X A R M A ^ ( t ) = a ×   m j + b ×   m h + c ×   m l   a + b + c
Then, we obtained the forecasting value with:
A 1 ^ ( t ) = A 1 ( t 1 ) + X A R M A ^ ( t )   ( 6 )

4. Applications

4.1. Forecasting TAIEX 2004

We used the 2004 TAIEX data as an example to illustrate our method. As the secondary factor, the 2004 Dow Jones data was used.
Step 1: Construct FFTS for historical main and secondary factor training data.
Firstly, the absolute mean of the fluctuation historical dataset of TAIEX 2004 from January to October was 66.87and the absolute mean of the fluctuation of the Dow Jones was 55.58. Then we divided both TAIEX 2004 and Dow Jones 2004 from January to October into 5 intervals according to their absolute means. Therefore, α 1 = ( , 50.15 ) ,   α 2 = [ 50.15 , 16.72 ) ,   α 3 = [ 16.72 , 16.72 ) ,   α 4 = [ 16.72 , 50.15 ) and α 5 = [ 50.15 ,   ) , β 1 = ( , 41.69 ) ,   β 2 = [ 41.69 , 13.90 ) ,     β 3 = [ 13.90 , 13.90 ) ,     β 4 = [ 13.90 , 41.69 ) , and β 5 = [ 41.69 ,   )   . In this way, the historical training dataset was represented by a fuzzified fluctuation dataset (Appendix A).
Step 2: Determine the fuzzy logical relationships (FFLRs) for two-factor AR(1) model.
Step 3: Obtain fuzzy fluctuation forecast result for time series.
Based on the results obtained from Step 2, the two-factor AR(1) FFLRs are shown in Table 1.
Step 4: Defuzzify the fluctuation forecast result.
The fluctuation forecast result was defuzzified according to Equation (3); the results are shown in Table 1.
Step 5: Calculate the fluctuation error series   E ( t ) of the historic training data.
We first added the forecast fluctuation to the previous day and obtained our forecast results. Then we calculated the difference between our forecast values and actual values.
Step 6: Fuzzify the fluctuation error series.
Based on the results of Step 5, we fuzzified the fluctuation error series   E ( t ) . as we did in Step 1. The absolute mean of the fluctuation error series was 64.32. Then we divided the fluctuation error series   E ( t ) into 5 intervals according to their absolute mean. The results are shown in Appendix B.
Step 7: Determine the fuzzy logical relationships for the ARMA(1,m) model.
In this case, to obtain optimal results, we used m = 3 to build our model.
Step 8: Obtain fuzzy fluctuation forecast result for the time series based on the FFLRGs of the ARMA(1,m) model.
Based on the results obtained in Step 2, the two-factor ARMA(1,3) fuzzy logic relationships are shown in Appendix C.
Step 9: Defuzzify the fluctuation forecast result.
We defuzzified the fluctuation forecast result according to Equation (3). The results are shown in Appendix C.
Then we used the fuzzy two-factor ARMA(1,3) solution to forecast the test dataset, which is the TAIEX 2004 from November to December. The forecast result is shown in Table 2. The forecast values were obtained by adding the fluctuation values to the current values. The forecast results are shown in Table 2.
We assessed the forecast performance by comparing the difference between the forecast values and the actual values. The widely used indicators in time series model comparisons are the mean squared error (MSE), root of the mean squared error (RMSE), mean absolute error (MAE), and mean percentage error (MPE),. To compare the performance of different forecasting methods, the Diebold-Mariano test statistic (S) is also used. These formulas are defined by Equations (9)–(13):
MSE = t = 1 n ( forecast ( t ) actual ( t ) ) 2 n
RMSE = t = 1 n ( forecast ( t ) actual ( t ) ) 2 n
MAE = t = 1 n | ( forecast ( t ) actual ( t ) ) | n
MPE = t = 1 n | ( forecast ( t ) actual ( t ) ) | / actual ( t ) n
S = d ¯ ( Variance ( d ¯ ) ) 1 / 2 , d ¯ = t = 1 n ( error o f F o r e c a s t 1 ) t 2 t = 1 n ( error o f F o r e c a s t 2 ) t 2 n
where n denotes the number of values forecasted, and forecast(t) and actual(t) denote the predicted value and actual value at time t, respectively. S is a test statistic of the Diebold method, that is used to compare the predictive accuracy of two forecasts obtained by different methods. Forecast1 represents the dataset obtained by Method 1, and Forecast2 represents another dataset from Method 2. If S > 0 and | S | > Z = 1 . 64 , at the 0.05 significance level, then Forecast2 has better predictive accuracy than Forecast1. With respect to the proposed method for two-factor ARMA(1,3), the MSE, RMSE, MAE, and MPE were 2814.65, 53.05, 42.09, and 0.0071, respectively.
To compare the forecasting results with different parameters, such as the number m of the two-factor ARMA(1,m) model and the element number g of linguistic sets, used in the fluctuation fuzzifying process, we completed different experiments and calculated the results. The forecasting errors of the averages for the experiments are shown in Table 3 and Table 4.
In Table 4, g = 3 means the linguistic set is {down, equal, up}, g = 5 means {greatly down, slightly down, equal, slightly up, greatly up}, g = 7 means {very greatly down, greatly down, slightly down, equal, slightly up, greatly up, very greatly up}, etc. “None” means that the model only used the AR(1) method to forecast.
We employed the proposed method to forecast the TAIEX from 1997 to 2005. The forecast results and errors are shown in Figure 2 and Table 5.
Table 6 shows a comparison of the RMSEs for the different methods when forecasting the TAIEX 2004. From this table, the performance of the proposed method is excellent. Though some of the other methods have better RMSEs results, they often need to build complex discretization partitioning rules or employ adaptive expectation models to modify the final forecast results. The method proposed in this paper is easily achieved by a computer program.

4.2. Forecasting SHSECI

The SHSECI (Shanghai Stock Exchange Composite Index) is the most influential stock market index in China. We chose Dow Jones as a secondary factor to build our model. For each year, the authentic datasets of historical daily SHSECI closing prices from January to October were used as the training data, and the datasets from November to December were employed as the testing data. The RMSEs of forecast errors are shown in Table 7. The proposed model accurately forecasted the SHSECI stock market.

4.3. Forecasting Gold Price

We also applied the proposed method to forecast the international gold price in USD from 2000 to 2010. We chose the COMEX gold price as a secondary factor. For each year, the authentic datasets of the historical daily closing prices from January to October were used as the training data, and the datasets from November to December were the testing data. The RMSEs of the forecast errors are shown in Table 8. Taking the 2010 gold price as an example, the forecast results are shown in Figure 3.
We can see that the proposed model can accurately forecast the international gold price.

5. Conclusions

In this paper, a new forecasting model is proposed based on a first-order two-factor ARMA(1,m) model. The proposed method is based on the fluctuations of two time series. The secondary factor was used to modify the forecast performance of the main factor. The experiments showed that the fuzzy logic relations of the main and secondary factors obtained from the two training datasets can successfully predict the testing dataset of the main factor. To compare the performance with other methods, we employed TAIEX 2004 as an example to illustrate our process. We also forecasted TAIEX 1997–2005, SHSECI 2001–2015, and the international gold price 2000–2010 to show its accuracy and versatility. For future research, we may consider additional aspects of the stock markets such as volumes, ending prices, opening prices, etc. A third factor, or more, could be used to modify the forecasting process.

Acknowledgments

The authors are indebted to anonymous reviewers for their very insightful comments and constructive suggestions, which help ameliorate the quality of this paper. This work supported by the National Research Foundation of Korea Grant funded by the Korean Government(NRF-2014S1A2A2027622) and the Foundation Program of Jiangsu University (16JDG005).

Author Contributions

Shuang Guan designed the experiments and wrote the paper; Aiwu Zhao conceived the main idea of the method.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

The historical training dataset can be represented by a fuzzified fluctuation dataset as shown in Table A1 and Table A2.
Table A1. Historical training data and fuzzified fluctuation data of TAIEX2004.
Table A1. Historical training data and fuzzified fluctuation data of TAIEX2004.
Date (MM/DD/YYYY)TAIEXFluctuationFuzzifiedDate (MM/DD/YYYY)TAIEXFluctuationFuzzifiedDate (MM/DD/YYYY)TAIEXFluctuationFuzzified
01/02/20046041.56--04/16/20046818.2081.41507/23/20045373.85−14.113
01/05/20046125.4283.86504/19/20046779.18−39.02207/26/20045331.71−42.142
01/06/20046144.0118.59404/20/20046799.9720.79407/27/20045398.6166.905
01/07/20046141.25−2.76304/21/20046810.2510.28307/28/20045383.57−15.043
01/08/20046169.1727.92404/22/20046732.09−78.16107/29/20045349.66−33.912
01/09/20046226.9857.81504/23/20046748.1016.01307/30/20045420.5770.915
01/12/20046219.71−7.27304/26/20046710.70−37.40208/02/20045350.40−70.171
01/13/20046210.22−9.49304/27/20046646.80−63.90108/03/20045367.2216.824
01/14/20046274.9764.75504/28/20046574.75−72.05108/04/20045316.87−50.351
01/15/20046264.37−10.60304/29/20046402.21−172.54108/05/20045427.61110.745
01/16/20046269.715.34304/30/20046117.81−284.40108/06/20045399.16−28.452
01/27/20046384.63114.92505/03/20046029.77−88.04108/09/20045399.450.293
01/28/20046386.251.62305/04/20046188.15158.38508/10/20045393.73−5.723
01/29/20046312.65−73.60105/05/20045854.23−333.92108/11/20045367.34−26.392
01/30/20046375.3862.73505/06/20045909.7955.56508/12/20045368.020.683
02/02/20046319.96−55.42105/07/20046040.26130.47508/13/20045389.9321.914
02/03/20046252.23−67.73105/10/20045825.05−215.21108/16/20045352.01−37.922
02/04/20046241.39−10.84305/11/20045886.3661.31508/17/20045342.49−9.523
02/05/20046268.1426.75405/12/20045958.7972.43508/18/20045427.7585.265
02/06/20046353.3585.21505/13/20045918.09−40.70208/19/20045602.99175.245
02/09/20046463.09109.74505/14/20045777.32−140.77108/20/20045622.8619.874
02/10/20046488.3425.25405/17/20045482.96−294.36108/23/20045660.9738.114
02/11/20046454.39−33.95205/18/20045557.6874.72508/26/20045813.39152.425
02/12/20046436.95−17.44205/19/20045860.58302.90508/27/20045797.71−15.683
02/13/20046549.18112.23505/20/20045815.33−45.25208/30/20045788.94−8.773
02/16/20046565.3716.19305/21/20045964.94149.61508/31/20045765.54−23.402
02/17/20046600.4735.10405/24/20045942.08−22.86209/01/20045858.1492.605
02/18/20046605.855.38305/25/20045958.3816.30309/02/20045852.85−5.293
02/19/20046681.5275.67505/26/20046027.2768.89509/03/20045761.14−91.711
02/20/20046665.54−15.98305/27/20046033.055.78309/06/20045775.9914.853
02/23/20046665.890.35305/28/20046137.26104.21509/07/20045846.8370.845
02/24/20046589.23−76.66105/31/20045977.84−159.42109/08/20045846.02−0.813
02/25/20046644.2855.05506/01/20045986.208.36309/09/20045842.93−3.093
02/26/20046693.2548.97406/02/20045875.67−110.53109/10/20045846.193.263
02/27/20046750.5457.29506/03/20045671.45−204.22109/13/20045928.2282.035
03/01/20046888.43137.89506/04/20045724.8953.44509/14/20045919.77−8.453
03/02/20046975.2686.83506/07/20045935.82210.93509/15/20045871.07−48.702
03/03/20046932.17−43.09206/08/20045986.7650.94509/16/20045891.0519.984
03/04/20047034.10101.93506/09/20045965.70−21.06209/17/20045818.39−72.661
03/05/20046943.68−90.42106/10/20045867.51−98.19109/20/20045864.5446.154
03/08/20046901.48−42.20206/11/20045735.07−132.44109/21/20045949.2684.725
03/09/20046973.9072.42506/14/20045574.08−160.99109/22/20045970.1820.924
03/10/20046874.91−98.99106/15/20045646.4972.41509/23/20045937.25−32.932
03/11/20046879.114.20306/16/20045560.16−86.33109/24/20045892.21−45.042
03/12/20046800.24−78.87106/17/20045664.35104.19509/27/20045849.22−42.992
03/15/20046635.98−164.26106/18/20045569.29−95.06109/29/20045809.75−39.472
03/16/20046589.72−46.26206/21/20045556.54−12.75309/30/20045845.6935.944
03/17/20046577.98−11.74306/23/20045729.30172.76510/01/20045945.3599.665
03/18/20046787.03209.05506/24/20045779.0949.79410/04/20046077.96132.615
03/19/20046815.0928.06406/25/20045802.5523.46410/05/20046081.013.053
03/22/20046359.92−455.17106/28/20045709.84−92.71110/06/20046060.61−20.402
03/23/20046172.89−187.03106/29/20045741.5231.68410/07/20046103.0042.394
03/24/20046213.5640.67406/30/20045839.4497.92510/08/20046102.16−0.843
03/25/20046156.73−56.83107/01/20045836.91−2.53310/11/20046089.28−12.883
03/26/20046132.62−24.11207/02/20045746.70−90.21110/12/20045979.56−109.721
03/29/20046474.11341.49507/05/20045659.78−86.92110/13/20045963.07−16.493
03/30/20046494.7120.60407/06/20045733.5773.79510/14/20045831.07−132.001
03/31/20046522.1927.48407/07/20045727.78−5.79310/15/20045820.82−10.253
04/01/20046523.491.30307/08/20045713.39−14.39310/18/20045772.12−48.702
04/02/20046545.5422.05407/09/20045777.7264.33510/19/20045807.7935.674
04/05/20046682.73137.19507/12/20045758.74−18.98210/20/20045788.34−19.452
04/06/20046635.54−47.19207/13/20045685.57−73.17110/21/20045797.248.903
04/07/20046646.7411.20307/14/20045623.65−61.92110/22/20045774.67−22.572
04/08/20046672.8626.12407/15/20045542.80−80.85110/26/20045662.88−111.791
04/09/20046620.36−52.50107/16/20045502.14−40.66210/27/20045650.97−11.913
04/12/20046777.78157.42507/19/20045489.10−13.04310/28/20045695.5644.594
04/13/20046794.3316.55307/20/20045325.68−163.42110/29/20045705.9310.373
04/14/20046880.1885.85507/21/20045409.1383.455
04/15/20046736.79−143.39107/22/20045387.96−21.172
Table A2. Historical training data and fuzzified fluctuation data of Dow Jones 2004.
Table A2. Historical training data and fuzzified fluctuation data of Dow Jones 2004.
Date (MM/DD/YYYY)TAIEXFluctuationFuzzifiedDate (MM/DD/YYYY)TAIEXFluctuationFuzzifiedDate (MM/DD/YYYY)TAIEXFluctuationFuzzified
01/02/200410409.85--04/14/200410377.95−3.33307/26/20049961.92−0.303
01/05/200410544.07134.22504/15/200410397.4619.51407/27/200410085.14123.225
01/06/200410538.66−5.41304/16/200410451.9754.51507/28/200410117.0731.934
01/07/200410529.03−9.63304/19/200410437.85−14.12207/29/200410129.2412.173
01/08/200410592.4463.41504/20/200410314.50−123.35107/30/200410139.7110.473
01/09/200410458.89−133.55104/21/200410317.272.77308/02/200410179.1639.454
01/12/200410485.1826.29404/22/200410461.20143.93508/03/200410120.24−58.921
01/13/200410427.18−58.00104/23/200410472.8411.64308/04/200410126.516.273
01/14/200410538.37111.19504/26/200410444.73−28.11208/05/20049963.03−163.481
01/15/200410553.8515.48404/27/200410478.1633.43408/06/20049815.33−147.701
01/16/200410600.5146.66504/28/200410342.60−135.56108/09/20049814.66−0.673
01/20/200410528.66−71.85104/29/200410272.27−70.33108/10/20049944.67130.015
01/21/200410623.6294.96504/30/200410225.57−46.70108/11/20049938.32−6.353
01/22/200410623.18−0.44305/03/200410314.0088.43508/12/20049814.59−123.731
01/23/200410568.29−54.89105/04/200410317.203.20308/13/20049825.3510.763
01/26/200410702.51134.22505/05/200410310.95−6.25308/16/20049954.55129.205
01/27/200410609.92−92.59105/06/200410241.26−69.69108/17/20049972.8318.284
01/28/200410468.37−141.55105/07/200410117.34−123.92108/18/200410083.15110.325
01/29/200410510.2941.92505/10/20049990.02−127.32108/19/200410040.82−42.331
01/30/200410488.07−22.22205/11/200410019.4729.45408/20/200410110.1469.325
02/02/200410499.1811.11305/12/200410045.1625.69408/23/200410073.05−37.092
02/03/200410505.186.00305/13/200410010.74−34.42208/24/200410098.6325.584
02/04/200410470.74−34.44205/14/200410012.872.13308/25/200410181.7483.115
02/05/200410495.5524.81405/17/20049906.91−105.96108/26/200410173.41−8.333
02/06/200410593.0397.48505/18/20049968.5161.60508/27/200410195.0121.604
02/09/200410579.03−14.00205/19/20049937.71−30.80208/30/200410122.52−72.491
02/10/200410613.8534.82405/20/20049937.64−0.07308/31/200410173.9251.405
02/11/200410737.70123.85505/21/20049966.7429.10409/01/200410168.46−5.463
02/12/200410694.07−43.63105/24/20049958.43−8.31309/02/200410290.28121.825
02/13/200410627.85−66.22105/25/200410117.62159.19509/03/200410260.20−30.082
02/17/200410714.8887.03505/26/200410109.89−7.73309/07/200410341.1680.965
02/18/200410671.99−42.89105/27/200410205.2095.31509/08/200410313.36−27.802
02/19/200410664.73−7.26305/28/200410188.45−16.75209/09/200410289.10−24.262
02/20/200410619.03−45.70106/01/200410202.6514.20409/10/200410313.0723.974
02/23/200410609.62−9.41306/02/200410262.9760.32509/13/200410314.761.693
02/24/200410566.37−43.25106/03/200410195.91−67.06109/14/200410318.163.403
02/25/200410601.6235.25406/04/200410242.8246.91509/15/200410231.36−86.801
02/26/200410580.14−21.48206/07/200410391.08148.26509/16/200410244.4913.133
02/27/200410583.923.78306/08/200410432.5241.44509/17/200410284.4639.974
03/01/200410678.1494.22506/09/200410368.44−64.08109/20/200410204.89−79.571
03/02/200410591.48−86.66106/10/200410410.1041.66509/21/200410244.9340.044
03/03/200410593.111.63306/14/200410334.73−75.37109/22/200410109.18−135.751
03/04/200410588.00−5.11306/15/200410380.4345.70509/23/200410038.90−70.281
03/05/200410595.557.55306/16/200410379.58−0.85309/24/200410047.248.343
03/08/200410529.48−66.07106/17/200410377.52−2.06309/27/20049988.54−58.701
03/09/200410456.96−72.52106/18/200410416.4138.89409/28/200410077.4088.865
03/10/200410296.89−160.07106/21/200410371.47−44.94109/29/200410136.2458.845
03/11/200410128.38−168.51106/22/200410395.0723.60409/30/200410080.27−55.971
03/12/200410240.08111.70506/23/200410479.5784.50510/01/200410192.65112.385
03/15/200410102.89−137.19106/24/200410443.81−35.76210/04/200410216.5423.894
03/16/200410184.6781.78506/25/200410371.84−71.97110/05/200410177.68−38.862
03/17/200410300.30115.63506/28/200410357.09−14.75210/06/200410239.9262.245
03/18/200410295.78−4.52306/29/200410413.4356.34510/07/200410125.40−114.521
03/19/200410186.60−109.18106/30/200410435.4822.05410/08/200410055.20−70.201
03/22/200410064.75−121.85107/01/200410334.16−101.32110/11/200410081.9726.774
03/23/200410063.64−1.11307/02/200410282.83−51.33110/12/200410077.18−4.793
03/24/200410048.23−15.41207/06/200410219.34−63.49110/13/200410002.33−74.851
03/25/200410218.82170.59507/07/200410240.2920.95410/14/20049894.45−107.881
03/26/200410212.97−5.85307/08/200410171.56−68.73110/15/20049933.3838.934
03/29/200410329.63116.66507/09/200410213.2241.66510/18/20049956.3222.944
03/30/200410381.7052.07507/12/200410238.2225.00410/19/20049897.62−58.701
03/31/200410357.70−24.00207/13/200410247.599.37310/20/20049886.93−10.693
04/01/200410373.3315.63407/14/200410208.80−38.79210/21/20049865.76−21.172
04/02/200410470.5997.26507/15/200410163.16−45.64110/22/20049757.81−107.951
04/05/200410558.3787.78507/16/200410139.78−23.38210/25/20049749.99−7.823
04/06/200410570.8112.44307/19/200410094.06−45.72110/26/20049888.48138.495
04/07/200410480.15−90.66107/20/200410149.0755.01510/27/200410002.03113.555
04/08/200410442.03−38.12207/21/200410046.13−102.94110/28/200410004.542.513
04/12/200410515.5673.53507/22/200410050.334.20310/29/200410027.4722.934
04/13/200410381.28−134.28107/23/20049962.22−88.111

Appendix B

The fluctuation error series of training data is shown in Table A3.
Table A3. The Fluctuation Error Series.
Table A3. The Fluctuation Error Series.
DateTAIEX GroupDow Jones GroupActualForecastFluctuationFuzzifiedDateTAIEX GroupDow Jones GroupActualForecastFluctuationFuzzified
TAIEXTAIEXGroup of FluctuationTAIEXTAIEXGroup of Fluctuation
01/05/2004336125.426041.5683.86506/03/2004155671.455871.95−200.501
01/06/2004556144.016144.81−0.80306/04/2004115724.895671.4553.445
01/07/2004436141.256118.8822.37406/07/2004555935.825744.28191.545
01/08/2004336169.176141.2527.92406/08/2004555986.765955.2131.554
01/09/2004456226.986213.8413.14306/09/2004555965.706006.15−40.452
01/12/2004516219.716214.804.91306/10/2004215867.515961.51−94.001
01/13/2004346210.226216.66−6.44306/11/2004155735.075863.79−128.721
01/14/2004316274.976210.2264.75506/14/2004115574.085735.07−160.991
01/15/2004556264.376294.36−29.99206/15/2004115646.495574.0872.415
01/16/2004346269.716261.328.39306/16/2004555560.165665.88−105.721
01/27/2004356384.636298.4286.21506/17/2004135664.355560.16104.195
01/28/2004516386.256372.4513.80306/18/2004535569.295644.81−75.521
01/29/2004316312.656386.25−73.60106/21/2004145556.545582.69−26.152
01/30/2004156375.386308.9366.45506/23/2004315729.305556.54172.765
02/02/2004526319.966341.88−21.92206/24/2004555779.095748.6930.404
02/03/2004136252.236319.96−67.73106/25/2004425802.555784.6717.884
02/04/2004136241.396252.23−10.84306/28/2004415709.845787.66−77.821
02/05/2004326268.146234.6933.45406/29/2004125741.525698.6742.854
02/06/2004446353.356284.8968.46506/30/2004455839.445786.1953.255
02/09/2004556463.096372.7490.35507/01/2004545836.915849.01−12.103
02/10/2004526488.346429.5958.75507/02/2004315746.705836.91−90.211
02/11/2004446454.396505.09−50.70107/05/2004115659.785746.70−86.921
02/12/2004256436.956471.14−34.19207/06/2004115733.575659.7873.795
02/13/2004216549.186432.76116.42507/07/2004515727.785721.396.393
02/16/2004516565.376537.0028.37407/08/2004345713.395724.73−11.343
02/17/2004336600.476565.3735.10407/09/2004315777.725713.3964.335
02/18/2004456605.856645.14−39.29207/12/2004555758.745797.11−38.372
02/19/2004316681.526605.8575.67507/13/2004245685.575741.99−56.421
02/20/2004536665.546661.983.56307/14/2004135623.655685.57−61.921
02/23/2004316665.896665.540.35307/15/2004125542.805612.48−69.681
02/24/2004336589.236665.89−76.66107/16/2004115502.145542.80−40.662
02/25/2004116644.286589.2355.05507/19/2004225489.105477.0112.093
02/26/2004546693.256653.8539.40407/20/2004315325.685489.10−163.421
02/27/2004426750.546698.8351.71507/21/2004155409.135321.9687.175
03/01/2004536888.436731.00157.43507/22/2004515387.965396.95−8.993
03/02/2004556975.266907.8267.44507/23/2004235373.855415.37−41.522
03/03/2004516932.176963.08−30.91207/26/2004315331.715373.85−42.142
03/04/2004237034.106959.5874.52507/27/2004235398.615359.1239.494
03/05/2004536943.687014.56−70.88107/28/2004555383.575418.00−34.432
03/08/2004136901.486943.68−42.20207/29/2004345349.665380.52−30.862
03/09/2004216973.906897.2976.61507/30/2004235420.575377.0743.504
03/10/2004516874.916961.72−86.81108/02/2004535350.405401.03−50.631
03/11/2004116879.116874.914.20308/03/2004145367.225363.803.423
03/12/2004316800.246879.11−78.87108/04/2004415316.875352.33−35.462
03/15/2004156635.986796.52−160.54108/05/2004135427.615316.87110.745
03/16/2004116589.726635.98−46.26208/06/2004515399.165415.43−16.272
03/17/2004256577.986606.47−28.49208/09/2004215399.455394.974.483
03/18/2004356787.036606.69180.34508/10/2004335393.735399.45−5.723
03/19/2004536815.096767.4947.60408/11/2004355367.345422.44−55.101
03/22/2004416359.926800.20−440.28108/12/2004235368.025394.75−26.732
03/23/2004116172.896359.92−187.03108/13/2004315389.935368.0221.914
03/24/2004136213.566172.8940.67408/16/2004435352.015364.80−12.793
03/25/2004426156.736219.14−62.41108/17/2004255342.495368.76−26.272
03/26/2004156132.626153.01−20.39208/18/2004345427.755339.4488.315
03/29/2004236474.116160.03314.08508/19/2004555602.995447.14155.855
03/30/2004556494.716493.501.21308/20/2004515622.865590.8132.054
03/31/2004456522.196539.38−17.19208/23/2004455660.975667.53−6.563
04/01/2004426523.496527.77−4.28308/26/2004425813.395666.55146.845
04/02/2004346545.546520.4425.10408/27/2004535797.715793.853.863
04/05/2004456682.736590.2192.52508/30/2004345788.945794.66−5.723
04/06/2004556635.546702.12−66.58108/31/2004315765.545788.94−23.402
04/07/2004236646.746662.95−16.21209/01/2004255858.145782.2975.855
04/08/2004316672.866646.7426.12409/02/2004535852.855838.6014.253
04/09/2004426620.366678.44−58.08109/03/2004355761.145881.56−120.421
04/12/2004116777.786620.36157.42509/06/2004125775.995749.9726.024
04/13/2004556794.336797.17−2.84309/07/2004335846.835775.9970.845
04/14/2004316880.186794.3385.85509/08/2004555846.025866.22−20.202
04/15/2004536736.796860.64−123.85109/09/2004325842.935839.323.613
04/16/2004146818.206750.1968.01509/10/2004325846.195836.239.963
04/19/2004556779.186837.59−58.41109/13/2004345928.225843.1485.085
04/20/2004226799.976754.0545.92409/14/2004535919.775908.6811.093
04/21/2004416810.256785.0825.17409/15/2004335871.075919.77−48.701
04/22/2004336732.096810.25−78.16109/16/2004215891.055866.8824.174
04/23/2004156748.106728.3719.73409/17/2004435818.395865.92−47.532
04/26/2004336710.706748.10−37.40209/20/2004145864.545831.7932.754
04/27/2004226646.806685.57−38.77209/21/2004415949.265849.6599.615
04/28/2004146574.756660.20−85.45109/22/2004545970.185958.8311.353
04/29/2004116402.216574.75−172.54109/23/2004415937.255955.29−18.042
04/30/2004116117.816402.21−284.40109/24/2004215892.215933.06−40.852
05/03/2004116029.776117.81−88.04109/27/2004235849.225919.62−70.401
05/04/2004156188.156026.05162.10509/29/2004215809.755845.03−35.282
05/05/2004535854.236168.61−314.38109/30/2004255845.695826.5019.194
05/06/2004135909.795854.2355.56510/01/2004415945.355830.80114.555
05/07/2004516040.265897.61142.65510/04/2004556077.965964.74113.225
05/10/2004515825.056028.08−203.03110/05/2004546081.016087.53−6.523
05/11/2004115886.365825.0561.31510/06/2004326060.616074.31−13.703
05/12/2004545958.795895.9362.86510/07/2004256103.006077.3625.644
05/13/2004545918.095968.36−50.27110/08/2004416102.166088.1114.053
05/14/2004225777.325892.96−115.64110/11/2004316089.286102.16−12.883
05/17/2004135482.965777.32−294.36110/12/2004345979.566086.23−106.671
05/18/2004115557.685482.9674.72510/13/2004135963.075979.56−16.492
05/19/2004555860.585577.07283.51510/14/2004315831.075963.07−132.001
05/20/2004525815.335827.08−11.75310/15/2004115820.825831.07−10.253
05/21/2004235964.945842.74122.20510/18/2004345772.125817.77−45.652
05/24/2004545942.085974.51−32.43210/19/2004245807.795755.3752.425
05/25/2004235958.385969.49−11.11310/20/2004415788.345792.90−4.563
05/26/2004356027.275987.0940.18410/21/2004235797.245815.75−18.512
05/27/2004536033.056007.7325.32410/22/2004325774.675790.54−15.873
05/28/2004356137.266061.7675.50510/26/2004215662.885770.48−107.601
05/31/2004525977.846103.76−125.92110/27/2004155650.975659.16−8.193
06/01/2004115986.205977.848.36310/28/2004355695.565679.6815.883
06/02/2004345875.675983.15−107.48110/29/2004435705.935670.4335.504

Appendix C

The fuzzy two-factor ARMA (1,3) solution is shown in Table A4.
Table A4. Fuzzy two-factor AR (1,3) solution.
Table A4. Fuzzy two-factor AR (1,3) solution.
Fuzzy Value of Main FactorFuzzy Value of Secondary FactorFuzzy Value of Lagged ErrorsFuzzy ForecastDefuzzified ForecastFuzzy Value of Main FactorFuzzy Value of Secondary FactorFuzzy Value of Lagged ErrorsFuzzy ForecastDefuzzified Forecast
123123
111112,1,5,5,8.38335331,−67
111213,0341533,0
112111,1,−67342132,−33.5
112211,−67342335,67
112415,67342422,−33.5
112513,0343234,33.5
113115,2,5,33.5343533,0
113315,67343523,0
114513,0344333,0
115311,−67344325,67
115411,−67344331,−67
115515,67345131,−67
122111,−67351225,67
124414,33.5352332,−33.5
125313,0352531,−67
131325,67353134,33.5
131515,67353445,67
131521,−67355235,5,67
131515,67411245,67
132512,−33.5412541,−67
133313,0413252,−33.5
134114,33.5413343,0
135111,−67414131,−67
135211,3,−33.5414245,67
141424,33.5414532,−33.5
141513,0415143,0
142414,33.5415441,−67
143515,67421141,−67
144221,−67421241,−67
151115,67421545,67
151311,1,−67422544,33.5
151412,−33.5425323,0
152313,5,33.5425435,67
154211,−67431242,−33.5
154413,0431333,0
155315,67433141,−67
212212,−33.5441345,67
212523,0445552,−33.5
213231,−67452345,67
215125,5,67452534,33.5
215322,−33.5453445,67
215314,33.5454145,67
215421,−67455443,0
221123,0455544,33.5
221421,−67511153,0
221514,33.5511253,1,−33.5
225511,−67511551,−67
231533,0512353,0
232533,0512554,33.5
233225,2,16.75513152,−33.5
233313,0513252,−33.5
233523,0514433,0
234125,67515155,67
234225,67515552,−33.5
234513,0521552,−33.5
235525,67523151,−67
235535,67524451,−67
241324,33.5524554,33.5
243521,−67531151,−67
251123,0532241,−67
252124,33.5532254,33.5
252433,0532343,0
253325,67533253,0
255334,33.5533353,0
255512,−33.5534253,0
311231,−67534353,0
311535,67535151,−67
312533,0535251,−67
313124,1,−16.75535351,−67
313335,67535455,67
313433,0541453,0
313531,−67541552,−33.5
314425,67542454,33.5
314531,−67543154,33.5
315125,67544553,0
315131,−67545155,67
315124,33.5545352,−33.5
315332,5,16.75551151,5,5,22.33
315322,−33.5551254,4,33.5
322134,33.5551453,0
324523,0551555,67
325233,0552243,0
325322,−33.5552455,67
325532,−33.5553255,67
331441,−67553352,3,−16.75
332544,33.5553452,5,16.75
333145,67554153,0
333532,−33.5554555,67
334142,−33.5555152,−33.5
335233,0555542,−33.5
335344,33.5

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Figure 1. Flowchart of the proposed forecasting model.
Figure 1. Flowchart of the proposed forecasting model.
Symmetry 09 00207 g001
Figure 2. Comparison of actual and forecast results for Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) test dataset (1997–2005). (X coordinate is the TAIEX and Y coordinate is the time series number remarked by “time(s)”.)
Figure 2. Comparison of actual and forecast results for Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) test dataset (1997–2005). (X coordinate is the TAIEX and Y coordinate is the time series number remarked by “time(s)”.)
Symmetry 09 00207 g002
Figure 3. Comparison of actual and forecast results for gold prices in 2010.
Figure 3. Comparison of actual and forecast results for gold prices in 2010.
Symmetry 09 00207 g003
Table 1. Fuzzy two-factor first-order autoregressive(AR(1)) solution.
Table 1. Fuzzy two-factor first-order autoregressive(AR(1)) solution.
Fuzzy Value of Main FactorFuzzy Value of Secondary FactorFuzzy ForecastDefuzzified Forecast
112,1,5,5,1,1,5,5,3,1,5,3,1,5,3,5,2,1,0
124,3,1,−11.17
131,1,5,4,1,3,2,3,5,5,0
145,1,4,4,3,13.4
153,2,1,5,1,1,5,5,3,−3.72
211,4,5,1,5,2,3,2,−4.19
224,3,1,1,−25.13
235,5,3,3,5,3,3,3,2,5,5,27.41
241,4,−16.75
254,4,3,2,5,3,16.75
312,3,2,1,1,5,4,1,5,4,5,1,5,5,1,3,0
323,4,3,2,2,−6.7
332,4,3,5,5,1,4,2,1,0
343,5,1,5,2,3,3,2,3,1,4,−3.05
351,5,5,2,5,5,4,28.71
413,1,1,3,5,2,5,1,2,−14.89
425,3,1,1,5,4,5.58
433,1,2,3,−25.13
442,5,16.75
455,3,5,5,4,4,44.67
511,1,3,2,2,4,5,2,3,3,3,−12.18
524,2,1,1,−33.5
533,3,3,1,1,3,1,1,1,3,5,4,−19.54
542,3,3,4,5,2,4,9.57
552,4,2,2,3,5,5,3,3,3,2,5,4,4,1,5,5,5,5,19.39
Table 2. Forecasting results from 1 November 2004 to 31 December 2004.
Table 2. Forecasting results from 1 November 2004 to 31 December 2004.
Date (MM/DD/YYYY)ActualForecast(Forecast–Actual)2Date (MM/DD/YYYY)ActualForecast(Forecast–Actual)2
11/05/20045931.315889.441753.1012/06/20045919.175868.142604.06
11/08/20045937.465950.70175.3012/07/20045925.285904.28441.00
11/09/20045945.205937.4659.9112/08/20045892.515925.281073.87
11/10/20045948.495945.2010.8212/09/20045913.975909.2622.18
11/11/20045874.525948.495471.5612/10/20045911.635958.642209.94
11/12/20045917.165870.802149.2512/13/20045878.895911.631071.91
11/15/20045906.695961.833040.4212/14/20045909.655895.64196.28
11/16/20045910.855906.6917.3112/15/20046002.585926.405803.39
11/17/20046028.685910.8513883.9112/16/20046019.236012.1550.13
11/18/20046049.496048.072.0212/17/20046009.326019.2398.21
11/19/20046026.556066.241575.3012/20/20045985.946026.071610.42
11/22/20045838.425993.0523910.4412/21/20045987.856013.35650.25
11/23/20045851.105851.820.5212/22/20046001.526016.56226.20
11/24/20045911.315851.103625.2412/23/20045997.676030.231060.15
11/25/20045855.245920.884308.6112/24/20046019.425997.67473.06
11/26/20045778.655855.245866.0312/27/20045985.946036.172523.05
11/29/20045785.265711.655418.4312/28/20046000.575981.75354.19
11/30/20045844.765785.263540.2512/29/20046088.496029.283505.82
12/01/20045798.625832.581153.2812/30/20046100.866054.992104.06
12/02/20045867.955815.372764.6612/31/20046139.696094.162072.98
12/03/20045893.275800.958522.98 RMSE53.05
Table 3. Comparison of forecasting errors for different two-factor first-order autoregressive moving average (ARMA(1,m)) model (g = 5).
Table 3. Comparison of forecasting errors for different two-factor first-order autoregressive moving average (ARMA(1,m)) model (g = 5).
mNone12345
RMSE57.5959.3261.7453.0560.8463.22
Table 4. Comparison of forecasting errors for different linguistic sets (m = 3).
Table 4. Comparison of forecasting errors for different linguistic sets (m = 3).
g3579
RMSE57.2553.0558.9965.8
Table 5. RMSEs of forecast errors for TAIEX 1997 to 2005.
Table 5. RMSEs of forecast errors for TAIEX 1997 to 2005.
Year199719981999200020012002200320042005
RMSE130.9111.95101.11127.47114.1961.9253.0553.0752.27
Table 6. A comparison of RMSEs for different methods for forecasting the TAIEX 2004.
Table 6. A comparison of RMSEs for different methods for forecasting the TAIEX 2004.
MethodsRMSE
199920002001200220032004Average
Huarng et al.’s Method [41]N/A158.7 **136.49 **95.15 **65.51 **73.57 **105.88
Chen and Chang’s Method [29]123.64 **131.1115.0873.06 **66.36 **60.48 **94.95
Chen and Chen’s Method [30]119.32 **129.87123.1271.0165.14 **61.94 **95.07
Chen et al.’s Method [42]102.34131.25113.6265.7752.2356.1686.89
Cheng et al.’s method [43]100.74125.62113.0462.9451.4654.2484.68
Chen and Kao’s method [44]87.63125.34114.5776.86 **54.2958.1786.14
Yu and Huarng’s method [45]N/A149.59 **98.9178.71 **58.7855.9188.38
The Proposed Method101.11127.47114.1961.9253.0553.0785.14
** Use Diebold-Mariano test statistic (S), the proposed method has better accuracy than other methods at 5% significance level at least.
Table 7. Root of the mean squared error (RMSE)s of forecast errors for Shanghai Stock Exchange Composite Index (SHSECI) from 2007 to 2015.
Table 7. Root of the mean squared error (RMSE)s of forecast errors for Shanghai Stock Exchange Composite Index (SHSECI) from 2007 to 2015.
Year200120022003200420052006200720082009201020112012201320142015
RMSE24.8621.7526.5719.079.6328.98129.2279.7759.9649.4829.723.1422.1344.1158.89
Table 8. RMSEs of forecast errors for gold price from 2000 to 2010.
Table 8. RMSEs of forecast errors for gold price from 2000 to 2010.
Year20002001200220032004200520062007200820092010
RMSE1.271.522.332.813.346.655.4411.4819.8114.6114.33

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Guan, S.; Zhao, A. A Two-Factor Autoregressive Moving Average Model Based on Fuzzy Fluctuation Logical Relationships. Symmetry 2017, 9, 207. https://doi.org/10.3390/sym9100207

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Guan S, Zhao A. A Two-Factor Autoregressive Moving Average Model Based on Fuzzy Fluctuation Logical Relationships. Symmetry. 2017; 9(10):207. https://doi.org/10.3390/sym9100207

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Guan, Shuang, and Aiwu Zhao. 2017. "A Two-Factor Autoregressive Moving Average Model Based on Fuzzy Fluctuation Logical Relationships" Symmetry 9, no. 10: 207. https://doi.org/10.3390/sym9100207

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