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

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 $A\left(t\right), \left(t=1,2,3,\dots ,T\right)$ 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 $L=\left\{ L_1, L_2,\dots , L_g\right\}$. According to the membership function, ${\mu }_L:A\left(t\right)\to \left[0,1\right]$, each element of the time series $A\left(t\right), \left(t=1,2,3,\dots ,T\right)$ can be represented by a fuzzy number $Z\left(t\right)=L_i, \left(t=1,2,3,\dots ,T, i=1,2,\dots ,g\right)$. We called $Z\left(t\right), \left(t=1,2,3,\dots ,T\right)$ a fuzzy time series.

Definition 2.

For a time series $G\left(t\right), \left(t=1, 2, 3, \dots , T\right)$, $Y(t)$ is defined as a fluctuation time series where $Y\left(t\right)=G\left(t\right)-G\left(t-1\right), \left(t=2,3,\cdots ,T\right)$. As described in Definition 1, $t=2, 3, \dots ,T, Y\left(t\right)$ could be represented by a fuzzy time series $H\left(t\right), \left(t=2, 3, \dots ,T\right)$. Thereby, the time series $Y(t)$ is fuzzified into a fuzzy-fluctuation time series (FFTS) $H\left(t\right)$.

Definition 3.

Let $H\left(t\right)$ be a FFTS $\left(t=2, 3,\dots ,T\right)$. If the “next status” of $H\left(t\right)$ is caused by the “current status” of $H\left(t-1\right)$, the first order fuzzy-fluctuation AR(1) is represented by [10,11]:

$$H\left(t-1\right)\to H\left(t\right)$$

(1)

Similarily, let $Q\left(t\right)$ and P(t) be two FFTSs $\left(t=2, 3,\dots ,T\right)$. If the next status of $Q\left(t\right)$ is caused by the current status of $Q\left(t-1\right) \mathrm{and} P\left(t-1\right)$, the two-factor first order fuzzy-fluctuation AR(1) is represented by:

$$Q\left(t-1\right),P\left(t-1\right)\to Q\left(t\right)$$

(2)

This is called the two-factor first order fuzzy-fluctuation logical relationship (FFLR). $\mathrm{Q}\left(\mathrm{t}-1\right),\mathrm{P}\left(\mathrm{t}-1\right)$ is the left-hand side (LHS) and $\mathrm{Q}\left(\mathrm{t}\right)$ 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 $F\left(t\right)$ be a fuzzy time series and $\epsilon \left(t\right)$ be a fuzzy error series obtained from the $F\left(t\right)$. If $F\left(t\right)$ is affected by both the lagged fuzzy time series $\left(F\left(t-1\right),F\left(t-2\right),\dots ,F\left(t-n\right)\right)$ and the lagged fuzzy error series ($\epsilon \left(t-1\right),\epsilon \left(t-2\right),\dots ,\epsilon \left(t-m\right))$, the fuzzy logical relationship can be represented by [40]:

$$F\left(t-1\right),F\left(t-2\right),\dots ,F\left(t-n\right),\epsilon \left(t-1\right),\epsilon \left(t-2\right),\dots ,\epsilon \left(t-m\right)\to F\left(t\right)$$

(3)

Similarly, let $Q\left(t\right)$ and P(t) be two FFTSs $\left(t=2, 3,\dots ,T\right)$ and $\epsilon \left(t\right)$be a fuzzy error series obtained from $Q\left(t\right)$. If $Q\left(t\right)$ is affected by both the lagged fuzzy time series $\left(Q\left(t-1\right),P\left(t-1\right)\right)$ and the lagged fuzzy error series ($\epsilon \left(t-1\right),\epsilon \left(t-2\right),\dots ,\epsilon \left(t-m\right))$, the fuzzy logical relationship can be represented by:

$$Q\left(t-1\right),P\left(t-1\right),\epsilon \left(t-1\right),\epsilon \left(t-2\right),\dots ,\epsilon \left(t-m\right)\to Q\left(t\right)$$

(4)

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.

Figure 1. Flowchart of the proposed forecasting model.

Step 1:

The first step was to construct a FFTS for the historical main and secondary factor training data. For each element $A_1\left(t\right), \left(t=1,2,\dots ,T\right)$ in the historical training time series of the main factor, its fluctuation trend was determined by $X_1\left(t\right)=A_1\left(t\right)-A_1\left(t-1\right), \mathrm{where}t=2, 3, 4, \dots ,T$. By the values and directions of fluctuation, we fuzzified $X_1\left(t\right)$ 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 $X_1\left(t\right), X_2\left(t\right)\left(t=2, 3, 4, \dots T\right)$, respectively. Then, we had ${\alpha }_1=\left(-\infty ,-\frac{w_1}{2}\right), {\alpha }_2=\left[-\frac{w_1}{2},\frac{w_1}{2}\right)$, and ${\alpha }_3=\left[\frac{w_1}{2},\infty \right)$. Similarly we divided $X_1\left(t\right)$ into 5 intervals such that ${\alpha }_1=\left(-\infty ,-\frac{3w_1}{4}\right)$, ${\alpha }_2=\left[-\frac{3w_1}{4},-\frac{w_1}{4}\right), {\alpha }_3=\left[-\frac{w_1}{4},\frac{w_1}{4}\right), {\alpha }_4=\left[\frac{w_1}{4},\frac{3w_1}{4}\right)$, and ${\alpha }_5=\left[\frac{3w_1}{4},\infty \right)$. We divided $X_1\left(t\right)$ into any $g=2l+1$ intervals, where $l$ is an integer. Next, for each element $A_2\left(t\right), \left(t=1,2,\dots ,T\right)$ in the historical training time series of the secondary factor, its fluctuation trend was determined by $X_2\left(t\right)=A_2\left(t\right)-A_2\left(t-1\right), \mathrm{where} t=2, 3, 4, \dots ,T$. According to Definition 1, $X_2\left(t\right)$ can also be divided into $g$ intervals, namely ${\beta }_i, \left(i=1, 2, \dots , g\right)$. Then we fuzzified $X_1\left(t\right)$ and $X_2\left(t\right)$ into FFTSs $Q_1\left(t\right)$ and $Q_2\left(t\right), \left(t=2,3,\dots ,T\right)$, respectively, where ${\mathrm{Q}}_1\left(\mathrm{t}\right)=L_i$ and ${\mathrm{Q}}_2\left(\mathrm{t}\right)=K_j$ both have the highest membership value of corresponding intervals ${\alpha }_i$ and ${\beta }_j \left(i,j=1, 2, \dots , g\right)$, respectively, and $\left\{L_1,L_2,\dots ,L_g\right\},$ $\left\{K_1,K_2,\dots ,K_g\right\}$ 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 $Q_1\left(t-1\right)=L_i$, $Q_2\left(t-1\right)=K_s$, and $Q_1\left(t\right)=L_j$; the FFLR of this two-factor AR(1) model is $L_i,K_s\to L_j$. 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 $L_1,K_2\to L_2$ and $L_1,K_2\to L_3$, then the FFLRG would be $L_1,K_2\to L_2,L_3$.

Step 3:

The next step was to obtain the fuzzy fluctuation forecast result from AR(n) model. We assumed the lagged variables $Q_1\left(t-1\right)=L_i$, $Q_2\left(t-1\right)=K_s$, and we defined the following conditions:RHS Conditions: If $L_i,K_s\to L_j,\dots ,L_j,L_h,\dots ,L_h,L_l,\dots ,L_l$ 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 $L_j,\dots , L_j,L_h,\dots ,L_h,L_l,\dots ,L_l$. Null RHS Condition: If $L_i,K_s\to empty$ 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:

$$\widehat{X_{AR}}\left(t\right)=\frac{a\times m_j+b\times m_h+c\times m_l}{a+b+c}$$

(5)

Step 5:

Next, we calculated the fluctuation error series $E\left(t\right)$:

$$E\left(t\right)=X_1\left(t\right)-\widehat{X_{AR}}\left(t\right)$$

(6)

where $X_1\left(t\right)$ is the time series of the fluctuation numbers of main factor, and $\widehat{X_1}\left(t\right)$ is calculated result from Step 4.

Step 6:

The next step was to construct fuzzy fluctuation time series for the error series $E\left(t\right)$. In the same manner as described in Step 1, we fuzzified $E\left(t\right)$ into FFTSs $R\left(t\right)$. We assumed $h$ is the absolute mean of all elements in the time series $E\left(t\right), \left(t=2, 3, 4, \dots T\right)$, g is the number of intervals of the fuzzy sets, ${\epsilon }_1, {\epsilon }_2,\dots ,{\epsilon }_g$ are corresponding intervals, $\mathrm{R}\left(\mathrm{t}\right)=W_i$ has the highest membership value of corresponding intervals ${\epsilon }_i \left(i=1, 2, \dots , g\right)$, and $\left\{W_1,W_2,\dots ,W_g\right\}$ 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 $Q_1\left(t-1\right)=L_i$, $Q_2\left(t-1\right)=K_s$, $Q_1\left(t\right)=L_j$, $R\left(t-m\right)=W_{i3}$, $R\left(t-\left(m-1\right)\right)=W_{m2}$,…,$R\left(t-1\right)=W_{s2}$, and the FFLR of this two-factor ARMA(1,m) model is $L_i,K_s,W_{i2},W_{m2}\dots ,W_{s2}\to L_j$. 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 $Q_1\left(t-1\right)=L_i$, $Q_2\left(t-1\right)=K_s$, and the lagged error variables $R\left(t-m\right)=W_{i3}$, $R\left(t-\left(m-1\right)\right)=W_{m2}$,…, and$R\left(t-1\right)=W_{s2}$, we defined the following conditions:

RHS Condition: If $L_i,K_s,W_{i3},W_{m3},\dots ,W_{s3}\to L_j,\dots ,L_j,L_h,\dots ,L_h,L_l,\dots ,L_l$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 $L_j,\dots ,L_j,L_h,\dots ,L_h,L_l,\dots ,L_l$.

Null RHS Condition: If $L_i,K_s,W_{i3},W_{m3},\dots ,W_{s3}\to empty$ 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:

$$\widehat{X_{ARMA}}\left(t\right)=\frac{a\times m_j+b\times m_h+c\times m_l}{a+b+c}$$

(7)

Then, we obtained the forecasting value with:

$$\widehat{A_1}\left(t\right)=A_1\left(t-1\right)+\widehat{X_{ARMA}}\left(t\right) \left(6\right)$$

(8)

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, ${\alpha }_1=\left(-\infty ,-50.15\right)$, ${\alpha }_2=\left[-50.15,-16.72\right),{\alpha }_3=\left[-16.72,16.72\right), {\alpha }_4=\left[16.72,50.15\right)$ and ${\alpha }_5=\left[50.15, \infty \right)$, ${\beta }_1=\left(-\infty ,-41.69\right)$, ${\beta }_2=\left[-41.69,-13.90\right), {\beta }_3=\left[-13.90,13.90\right), {\beta }_4=\left[13.90,41.69\right)$, and ${\beta }_5=\left[41.69, \infty \right)$. 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.

Table 1. Fuzzy two-factor first-order autoregressive(AR(1)) solution.

Fuzzy Value of Main Factor	Fuzzy Value of Secondary Factor	Fuzzy Forecast	Defuzzified Forecast
1	1	2,1,5,5,1,1,5,5,3,1,5,3,1,5,3,5,2,1,	0
1	2	4,3,1,	−11.17
1	3	1,1,5,4,1,3,2,3,5,5,	0
1	4	5,1,4,4,3,	13.4
1	5	3,2,1,5,1,1,5,5,3,	−3.72
2	1	1,4,5,1,5,2,3,2,	−4.19
2	2	4,3,1,1,	−25.13
2	3	5,5,3,3,5,3,3,3,2,5,5,	27.41
2	4	1,4,	−16.75
2	5	4,4,3,2,5,3,	16.75
3	1	2,3,2,1,1,5,4,1,5,4,5,1,5,5,1,3,	0
3	2	3,4,3,2,2,	−6.7
3	3	2,4,3,5,5,1,4,2,1,	0
3	4	3,5,1,5,2,3,3,2,3,1,4,	−3.05
3	5	1,5,5,2,5,5,4,	28.71
4	1	3,1,1,3,5,2,5,1,2,	−14.89
4	2	5,3,1,1,5,4,	5.58
4	3	3,1,2,3,	−25.13
4	4	2,5,	16.75
4	5	5,3,5,5,4,4,	44.67
5	1	1,1,3,2,2,4,5,2,3,3,3,	−12.18
5	2	4,2,1,1,	−33.5
5	3	3,3,3,1,1,3,1,1,1,3,5,4,	−19.54
5	4	2,3,3,4,5,2,4,	9.57
5	5	2,4,2,2,3,5,5,3,3,3,2,5,4,4,1,5,5,5,5,	19.39

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$E\left(t\right)$ 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$E\left(t\right)$. as we did in Step 1. The absolute mean of the fluctuation error series was 64.32. Then we divided the fluctuation error series$E\left(t\right)$ 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.

Table 2. Forecasting results from 1 November 2004 to 31 December 2004.

Date (MM/DD/YYYY)	Actual	Forecast	(Forecast–Actual)2	Date (MM/DD/YYYY)	Actual	Forecast	(Forecast–Actual)2
11/05/2004	5931.31	5889.44	1753.10	12/06/2004	5919.17	5868.14	2604.06
11/08/2004	5937.46	5950.70	175.30	12/07/2004	5925.28	5904.28	441.00
11/09/2004	5945.20	5937.46	59.91	12/08/2004	5892.51	5925.28	1073.87
11/10/2004	5948.49	5945.20	10.82	12/09/2004	5913.97	5909.26	22.18
11/11/2004	5874.52	5948.49	5471.56	12/10/2004	5911.63	5958.64	2209.94
11/12/2004	5917.16	5870.80	2149.25	12/13/2004	5878.89	5911.63	1071.91
11/15/2004	5906.69	5961.83	3040.42	12/14/2004	5909.65	5895.64	196.28
11/16/2004	5910.85	5906.69	17.31	12/15/2004	6002.58	5926.40	5803.39
11/17/2004	6028.68	5910.85	13883.91	12/16/2004	6019.23	6012.15	50.13
11/18/2004	6049.49	6048.07	2.02	12/17/2004	6009.32	6019.23	98.21
11/19/2004	6026.55	6066.24	1575.30	12/20/2004	5985.94	6026.07	1610.42
11/22/2004	5838.42	5993.05	23910.44	12/21/2004	5987.85	6013.35	650.25
11/23/2004	5851.10	5851.82	0.52	12/22/2004	6001.52	6016.56	226.20
11/24/2004	5911.31	5851.10	3625.24	12/23/2004	5997.67	6030.23	1060.15
11/25/2004	5855.24	5920.88	4308.61	12/24/2004	6019.42	5997.67	473.06
11/26/2004	5778.65	5855.24	5866.03	12/27/2004	5985.94	6036.17	2523.05
11/29/2004	5785.26	5711.65	5418.43	12/28/2004	6000.57	5981.75	354.19
11/30/2004	5844.76	5785.26	3540.25	12/29/2004	6088.49	6029.28	3505.82
12/01/2004	5798.62	5832.58	1153.28	12/30/2004	6100.86	6054.99	2104.06
12/02/2004	5867.95	5815.37	2764.66	12/31/2004	6139.69	6094.16	2072.98
12/03/2004	5893.27	5800.95	8522.98		RMSE	53.05

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):

$$\mathrm{MSE}=\frac{{\displaystyle \sum _{t=1}^n{(\mathrm{forecast}(t)-\mathrm{actual}(t))}^2}}{n}$$

(9)

$$\mathrm{RMSE}=\sqrt{\frac{{\displaystyle \sum _{t=1}^n{(\mathrm{forecast}(t)-\mathrm{actual}(t))}^2}}{n}}$$

(10)

$$\mathrm{MAE}=\frac{{\displaystyle \sum _{t=1}^n\left|(\mathrm{forecast}(t)-\mathrm{actual}(t))\right|}}{n}$$

(11)

$$\mathrm{MPE}=\frac{{\displaystyle \sum _{t=1}^n\left|(\mathrm{forecast}(t)-\mathrm{actual}(t))\right|}/\mathrm{actual}(t)}{n}$$

(12)

$$S=\frac{\overline{d}}{{(\mathrm{Variance}(\overline{d}))}^{1/2}},\hspace{0.17em}\overline{d}=\frac{{\displaystyle \sum _{t=1}^n{(\mathrm{error}\begin{array}{c}\end{array}of\begin{array}{c}\end{array}Forecast1)}_t^2}-{\displaystyle \sum _{t=1}^n{(\mathrm{error}\begin{array}{c}\end{array}of\begin{array}{c}\end{array}Forecast2)}_t^2}}{n}$$

(13)

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 $\left|S\right|>Z=1.64$, 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.

Table 3. Comparison of forecasting errors for different two-factor first-order autoregressive moving average (ARMA(1,m)) model (g = 5).

m	None	1	2	3	4	5
RMSE	57.59	59.32	61.74	53.05	60.84	63.22

Table 4. Comparison of forecasting errors for different linguistic sets (m = 3).

g	3	5	7	9
RMSE	57.25	53.05	58.99	65.8

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.

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)”.)

Table 5. RMSEs of forecast errors for TAIEX 1997 to 2005.

Year	1997	1998	1999	2000	2001	2002	2003	2004	2005
RMSE	130.9	111.95	101.11	127.47	114.19	61.92	53.05	53.07	52.27

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.

Table 6. A comparison of RMSEs for different methods for forecasting the TAIEX 2004.

Methods	RMSE
	1999	2000	2001	2002	2003	2004	Average
Huarng et al.’s Method [41]	N/A	158.7 **	136.49 **	95.15 **	65.51 **	73.57 **	105.88
Chen and Chang’s Method [29]	123.64 **	131.1	115.08	73.06 **	66.36 **	60.48 **	94.95
Chen and Chen’s Method [30]	119.32 **	129.87	123.12	71.01	65.14 **	61.94 **	95.07
Chen et al.’s Method [42]	102.34	131.25	113.62	65.77	52.23	56.16	86.89
Cheng et al.’s method [43]	100.74	125.62	113.04	62.94	51.46	54.24	84.68
Chen and Kao’s method [44]	87.63	125.34	114.57	76.86 **	54.29	58.17	86.14
Yu and Huarng’s method [45]	N/A	149.59 **	98.91	78.71 **	58.78	55.91	88.38
The Proposed Method	101.11	127.47	114.19	61.92	53.05	53.07	85.14

** Use Diebold-Mariano test statistic (S), the proposed method has better accuracy than other methods at 5% significance level at least.

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.

Table 7. Root of the mean squared error (RMSE)s of forecast errors for Shanghai Stock Exchange Composite Index (SHSECI) from 2007 to 2015.

Year	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010	2011	2012	2013	2014	2015
RMSE	24.86	21.75	26.57	19.07	9.63	28.98	129.22	79.77	59.96	49.48	29.7	23.14	22.13	44.11	58.89

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.

Table 8. RMSEs of forecast errors for gold price from 2000 to 2010.

Year	2000	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010
RMSE	1.27	1.52	2.33	2.81	3.34	6.65	5.44	11.48	19.81	14.61	14.33

Figure 3. Comparison of actual and forecast results for gold prices in 2010.

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.