Fusion ANFIS models based on multi-stock volatility causality for TAIEX forecasting

oc Ch hno ue a fits. rior mo erie ork proposes a new model based on multi-stock volatility causality, a fusion adaptive-network-based fuzzy inference system (ANFIS) procedure, for forecasting stock price problems in Taiwan. Furthermore, to illustrate the proposed model, three practical, collected stock index datasets from the USA and Taiwan stock markets are used in the empirical experiment. The experimental results indicate that the proposed model is superior to the listing methods in terms of root mean squared error, and further evaluation lacks a n the g gn sec duct. H nomic has been demonstrated. Thus, we utilize volatility causality in a eries tion- th of sting and the length of linguistics intervals. In the process of establishing ARTICLE IN PRESS Contents lists available at ScienceDirect els Neurocom Neurocomputing 72 (2009) 3462–3468 series method to forecast the Taiwan Stock Exchange CapitalizationE-mail address: [email protected] (C.-H. Cheng). multi-stock market in this study. fuzzy relationships and forecasting, Yu [25] argued that recurrent fuzzy relationships should be considered in forecasting and recommended that different weights be assigned to various fuzzy relationships. Therefore, Yu [25] proposed a weighted fuzzy time- 0925-2312/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2008.09.027 � Corresponding author. Tel.: +88655342601x5312; fax: +88655312077. important indicator when forecasting another stock market; in particular, the predictive value of the US stock market’s volatility average-based length to approach this issue. In addition, Chen [5] proposed a new method that used genetic algorithms to tune Stock market investing is an exciting and challenging monetary activity. Market climates can dramatically change in a second, and gain–loss can be realized with a momentary decision; thus, accurate information while planning investments is crucial for investors. Volatility causality in a multi-stock market is an focused on the two major processes of the fuzzy time-s model: (1) fuzzification and (2) establishment of fuzzy rela ships and forecasting. In the fuzzification process, the leng intervals for the universe of discourse can affect foreca accuracy, and Huarng [12] proposed distribution-based circumstances, the impact of world economic fluctuations on Taiwan is significant, especially those originating in the USA. Additionally, Dickinson [10] demonstrated that international stock markets influence the movements of other global stock indexes. more historical data along with some assumptions like normality postulates [16]. Furthermore, fuzzy time-series models have been used for stock-price forecasting. Song and Chissom [22] first proposed the original model of the fuzzy time-series, and the following research 1. Introduction Taiwan is an island country that and thus the degree of dependence o is very high. In particular, the forei 60% of Taiwan’s gross domestic pro member of the international eco reveals that the profits comparison results for the proposed model produce higher profits than the listing models. & 2009 Elsevier B.V. All rights reserved. natural energy source, lobal economic system tor (net export) is over ence, Taiwan is a vital society. Under such For a long time, conventional time-series models have been applied to forecasting application problems of the real world, such as Engle’s [11] autoregressive conditional heteroscedasticity (ARCH) model, Bollerslev’s [2] Generalized ARCH (GARCH) model to refine the ARCH model, Box and Jenkins’ [3] autoregressive moving average (ARMA) model and the autoregressive integrated moving average model. However, traditional time-series requires Fusion ANFIS models based on multi-st TAIEX forecasting Ching-Hsue Cheng �, Liang-Ying Wei, You-Shyang Department of Information Management, National Yunlin University of Science and Tec a r t i c l e i n f o Available online 11 June 2009 Keywords: Multi-stock TAIEX forecasting ANFIS Volatility causality a b s t r a c t Stock market investors val the potential for large pro model, to assemble a supe the time-series model: (1) most conventional time-s from artificial neural netw journal homepage: www. k volatility causality for en logy, 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan, ROC ccurate forecasting of future stock price from trading systems because of Thus, investors use different forecasting models, such as the time-series investment portfolio. Unfortunately, there are three major drawbacks to st statistical methods rely on some assumptions about the variables; (2) s models use only one variable in forecasting; and (3) the rules mined s are not easily understandable. To address these shortcomings, this study evier.com/locate/neucom puting 2.2. Fuzzy C-means clustering ARTICLE IN PRESS C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–3468 3463 Weighted Stock Index (TAIEX). TAIEX is an index that reflects the overall market movement and is weighted by the number of outstanding shares. As of 1966, its base year value was set at 100. Further, Cheng et al. [7] proposed a methodology that incorpo- rates trend-weighting into the fuzzy time-series model. To take advantage of neural networks (nonlinear capabilities), Huarng and Yu [13] chose a neural network to establish fuzzy relationships in the fuzzy time-series model, which is also nonlinear, but the process of mining fuzzy logical relationships is not easily under- standable [6]. Moreover, the models mentioned above have been limited to one variable application [26]. Recently, Chen et al. [6] proposed a comprehensive fuzzy time-series, which factors in recent periods of stock prices and fuzzy logical relationships into the forecasting processes. Jilani and Burney [16] proposed a simple time-variant fuzzy time-series method to forecast TAIEX and enrollments at the University of Alabama. Cheng et al. [8] proposed a new fuzzy time-series method, which is based on a weighted-transitional matrix, and also proposed two new fore- casting methods: the expectation method and the grade-selection method. Yu and Huarng [26] proposed a bivariate model, which applies neural networks to fuzzy time-series forecasting. From the literature review, we can conclude the following: (1) fuzzy time-series, where time-series data are represented by fuzzy sets instead of crisp values, can model the qualitative aspects of human knowledge and can be applicable to human recognition; (2) fuzzy time-series are appropriately applied to linguistic values datasets to generate forecasting rules and can get higher accuracy. On the contrary, traditional time-series methods fail to forecast the problems with linguistic value; (3) to improve accuracy, the expert rules validated by expert group should be utilized in the forecasting procedure; and (4) constructing the appropriate fuzzy logical relationships for forecasting is critical. Based on the above information, there are three major drawbacks to the time-series model: (1) most statistical methods rely upon some assumptions about the variables used in the analysis, so they have limited application to all datasets [16]; (2) most conventional time-series models use only one variable in forecasting. However, there is a lot of noise caused by changes in market conditions; therefore, financial analysts should consider many market variables in forecasting. For this reason, forecasting models should use more variables to improve forecasting accuracy [26]; and (3) artificial neural networks (ANN) is a black-box method, and the rules mined from ANN are not easily under- standable [6]. To address these drawbacks, this study considers that the volatility of American stock indexes can significantly affect the volatility of TAIEX. Because this forecasting model uses the relation between the volatility of the American stock index and the volatility of TAIEX, the analytical results can approximate the real world. Furthermore, a fuzzy inference system employing fuzzy if-then rules can model the qualitative aspects of human knowledge and can be applicable for investors. Based on the concept above, this study proposes a new volatility model to forecast the Taiwan stock index. Firstly, this study calculates the volatility of the NASDAQ stock index and the Dow Jones stock index. Then, it uses the fuzzy inference system to forecast the Taiwan stock index; it considers the multi-stock index (NASDAQ stock index (t) and Dow Jones stock index (t) and TAIEX (t)) to forecast the TAIEX (t+1). Secondly, this study optimizes the fuzzy inference system parameters by adaptive network, which can overcome the limitations of statistical methods (data need obey some mathematical distribution). Thus, we expect that this model is viable and useful for investors and will provide higher accuracy in forecasting the TAIEX in the Taiwan stock exchange market or the Taiwan Weighted Average Index stock index in Taiwan’s Futures Market, resulting in huge profits. Clustering has become popular as an efficient data analysis tool to understand and visualize data structures. The prevalent formulation of this task is to use c feature vectors vjðj ¼ 1;2; . . . ; cÞ to represent the c clusters, such that a sample xt is classified into the j-th cluster according to some measure of This rest of the paper is organized as follows: Section 2 describes related studies, Section 3 briefly presents the proposed model, Section 4 describes the experiments and comparisons, and Section 5 presents the findings and discussion. Finally, the conclusions of the study are in Section 6. 2. Related studies This section reviews related studies of different forecasting models for the stock market, including the adaptive-network- based fuzzy inference system (ANFIS), fuzzy C-means clustering (FCM) and subtractive clustering (Subclust). 2.1. Different forecasting models for the stock market Researchers have presented many different methods to deal with forecasting stock price problems. For instance, a study by Dickinson [10] shows that the stock price indexes in different countries influence each other. Huarng et al. [14] have used the volatility of the NASDAQ (the largest USA electronic stock market) stock index and the Dow Jones (Dow Jones Industrial Average) stock index to forecast the Taiwan stock index. Time-series models have been applied to handle economic forecasting, such as stock index forecasting, and various models have been proposed. Engle [11] proposed the ARCH model, which has been used by many financial analysts, and the GARCH model [2]. While Box and Jenkins [3] proposed the ARMA model, which combines a moving average process with a linear difference equation. During the past years, many researchers have applied data mining techniques to financial analysis. Huarng and Yu [13] applied the backpropagation neural network to establish fuzzy relationships in fuzzy time-series for forecasting stock price. Kinoto et al. [19] developed a prediction system for the stock market by using neural network. Nikolopoulos and Fellrath [20] combined genetic algorithms and neural network to develop a hybrid expert system for investment advising. Kim and Han [17] proposed a genetic algorithms approach to feature discretization and the determination of connection weights for ANN to predict the stock price index. Roh [21] integrated neural network and time-series models for forecasting the volatility of the stock price index. Thawornwong and Enke [24] proposed redeveloped neural network models for predicting the directions of future excess stock return. Kim [18] applied support vector machine to predict the stock price index. Summarily, this provides the following arguments: (1) in the fuzzy time-series process, the concept of fuzzy logic was introduced to cope with the ambiguity and uncertainly of most of the real-world problems; (2) traditional, crisp time-series forecasting methods cannot deal with the historical data repre- sented by linguistic values, but a fuzzy time-series model can overcome the drawback of traditional forecasting methods; (3) the fuzzy time-series are appropriately applied to linguistic values datasets to generate high accuracy forecasting rules; and (4) the fuzzy time-series models are easily understandable for the researchers. similarity and its corresponding objective function. FCM, proposed by Bezdek [1], is the most famous and basic fuzzy clustering algorithm. FCM attempts to find a fuzzy partition of the dataset by minimizing the following within group least-squares error objective function with respect to fuzzy memberships uit and center vi: Jm0 ðX;U;VÞ ¼ Xc i¼1 Xn t¼1 um 0 it d 2ðxt;viÞ, (1) where m041 is the fuzziness index used to tune out the noise in the data, n is the number of feature vectors xt , c42 is the number of clusters in the set and dðxt;viÞ is the similarity measure between a datum and a center. Minimization of Jm0 under the following constraints: ð1Þ 0 � uit � 1; 8i; t; ð2Þ 0oPn t¼1 uit � n; 8i; Pc (2) potential Pi is revised of each data point Wi by Eq. (5) pi ¼ pi � p�1 expð�bjjWi �W�1jj2Þ, (5) where b is a positive constant defining the neighborhood which will have measurable reductions in potential. W�1 is the first cluster center and P�1 is its potential value. From Eq. (5), the method selects the data point with the highest remaining potential as the second cluster center. For the general equation, we can rewrite Eq. (5) as Eq. (6). pi ¼ pi � p�k expð�bkWi �W�kk2Þ, (6) whereW�k ¼ ðx�k; y�kÞ is the location of the k’th cluster center and P�k is its potential value. At the end of the clustering process, the method obtains q cluster centers and D corresponding spreads Si, i ¼ (1,y,D). Then we define their membership functions. The spread is calculated according to b. ARTICLE IN PRESS aye C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–34683464 ð3Þ i¼1 uit ¼ 1; 8t; yields an iterative minimization pseudo-algorithm well known as the FCM algorithm. The components vij of each center vi and the membership degrees uit are updated according to the expressions vij ¼ Pn t¼1u m0 it xkjPn i¼1u m0 it and uit ¼ 1 Pc j¼1 dðxt ;viÞ dðxt;vjÞ � �2=m0�1, (3) where j is a variable on the feature space, i.e., j ¼ 1,2,y,m. 2.3. Subtractive clustering Chiu [9] developed subtractive clustering, a type of fuzzy clustering, to estimate both the number and initial locations of cluster centers. Consider a set T of N data points in a D- dimensional hyperspace, where each data point Wiði ¼ 1;2; . . . ;NÞ. Wi ¼ ðxi; yiÞ, where xi denotes the p input variables and yi denotes the output variable. The potential value Pi of data point is calculated by Eq. (4) Pi ¼ XN j¼1 e�akWi�Wjk 2 , (4) where a ¼ 4=r2, r is the radius defining a Wi neighborhood and k � k denotes the Euclidean distance. The data point with many neighboring data points is chosen as the first cluster center. To generate the other cluster centers, the x y A1 A2 Π B1 B2 Π Layer 1 Layer 2 L W1 W2 Fig. 1. The architecture 2.4. ANFIS: adaptive-network-based fuzzy inference system Jang [15] proposed ANFIS, which is a fuzzy inference system implemented in the framework of adaptive networks. For illustrating the system, we assume the fuzzy inference system consists of five layers of adaptive network with two inputs x and y and one output z. The architecture of ANFIS is shown in Fig. 1. Then, we suppose that the system consists of two fuzzy if–then rules based on Takagi and Sugeno’s type [23]: Rule 1. If x is A1 and y is B1, then f 1 ¼ p1xþ q1yþ r1. Rule 2. If x is A2 and y is B2, then f 2 ¼ p2xþ q2yþ r2. The node in the i-th position of the k-th layer is denoted as Ok;i, and the node functions in the same layer are of the same function family as described below: Layer 1: This layer is the input layer, and every node i in this layer is a square node with a node function (see Eq. (7)). O1;i is the membership function of Ai, and it specifies the degree to which the given x satisfies the quantifier Ai. Usually, we select the bell- shaped membership function as the input membership function (see Eq. (8)), with maximum equal to 1 and minimum equal to 0. O1;i ¼ mAiðxÞ for i ¼ 1;2, (7) mAiðxÞ ¼ 1 1þ x� ci ai � �2" #bi, (8) where ai, bi, and ci are the parameters, b is a positive value and c denotes the center of the curve. w2 f2 w1 f1 w2 w1 N N r 3 Layer 4 Layer 5 f X Y X Y ∑ of ANFIS network. some data under the NASDAQ and Dow Jones are empty because there were no transactions on those days. For this reason, this study fills in the last volatility as the differences. diffðNðtÞÞ ¼ NðtÞ � Nðt � 1Þ. (13) diffðDðtÞÞ ¼ DðtÞ � Dðt � 1Þ. (14) Step 3: Define and partition the universe of discourse for input variables (see B1 block of Fig. 2) Firstly, we define each universe of discourse for three variables (TAIEX (t), diff(N(t)), diff(D(t))) according to the minimum and maximum value in each variable. Secondly, we partition the universe of discourse into three linguistic intervals by using FCM clustering [1] (triangular membership function) and subtractive clustering [9] (Gaussian membership function), respectively. Step 4: Set the type of membership function for output variables (see B2 block of Fig. 2) There are two types of membership functions for output variables as follows: (1) Lineal type: a typical rule in a Sugeno fuzzy model has the form as follows: If x (TAIEX (t)) ¼ Ai, y (diff(N(t))) ¼ Bj and z ARTICLE IN PRESS Generate fuzzy inference system Train fuzzy inference system Forecast the testing TAIEX (t+1) by the four types of forecasting models and calculate RMSE Select the best model based on minimal RMSE and compare the forecasting performances of the different models Fig. 2. Flowchart of proposed procedure. C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–3468 3465 Layer 2: Every node in this layer is a square node labeled P which multiplies the incoming signals and sends the product out by Eq. (9). O2;i ¼ wi ¼ mAiðxÞ � mBiðyÞ for i ¼ 1;2. (9) Layer 3: Every node in this layer is a square node labeled N. The i-th node calculates the ratio of the i-th rule’s firing strength to the sum of all rules’ firing strengths by Eq. (10). Output of this layer can be called normalized firing strengths. O3;i ¼ w¯i ¼ wi w1 þw2 for i ¼ 1;2. (10) Layer 4: Every node i in this layer is a square node with a node function (see Eq. (11)). Parameters in this layer will be referred to as consequent parameters. O4;i ¼ w¯if i ¼ w¯iðpi þ qi þ riÞ, (11) where pi, qi and ri are the parameters. Layer 5: The single node in this layer is a circle node labeled P that computes the overall output as the summation of all incoming signals (see Eq. (12)) O5;i ¼ X i w¯if i ¼ P i¼1wifP i¼1wi ¼ overall output. (12) 3. Proposed model As stated in Section 1, there are three major drawbacks to the time-series model: (1) statistical methods rely upon some assumptions [16]; (2) time-series models use only one variable [26]; and (3) ANN is a black-box method, and the rules mined from ANN are not easily understandable [6]. Nevertheless, the forecasting rules are useful for investors buying and selling stocks. To reconcile these drawbacks, this study considers that the volatility of American stock indexes can significantly affect the volatility of TAIEX. Because this forecasting model uses the relation between the volatility of the American stock index and the volatility of TAIEX, the analytical results can approximate the real world. Furthermore, a fuzzy inference system employing fuzzy if-then rules can model the qualitative aspects of human knowledge and can be applicable for investors. Based on the concept above, this study proposes a new volatility model to forecast the Taiwan stock index. Firstly, this study calculates the volatility of the NASDAQ stock index and the Dow Jones stock index by Eqs. (1) and (2). Then, using the fuzzy inference system to forecast the Taiwan stock index, it considers multi-stock indexes (NASDAQ stock index (t) and Dow Jones stock index (t) and TAIEX (t)) to forecast TAIEX (t+1). Secondly, this study optimizes the fuzzy inference system parameters by adaptive network, which can overcome the limitations of statistical methods (data need obey some mathematical distribution). The overall flowchart of the proposed model is shown in Fig. 2. This section uses some numerical data as an example, and the core concept of the proposed algorithm is shown step by step. Step 1: Collect datasets In this section, we choose TAIEX data from 1997 to 2003 (7 sub-datasets) to illustrate the proposed model (such as year-2000 sub-datasets, which contains 271 transaction days). Training data are from January to October, and the remaining data (from November and December) are used for testing. Step 2: Calculate multivariate volatility (NASDAQ stock index and Dow Jones stock index) In this section, we define two variables, namely (1) the NASDAQ (N) and (2) the Dow Jones (D), and calculate the volatility of the two variables by Eqs. (13) and (14). Table 1 lists the differences in the variables NASDAQ and Dow Jones. In Table 1, Define and partition the universe of discourse (B1) Calculate multivariate volatility (NASDAQ and Dow Jones) FCM Subtractive Clustering Set output MF (B2) Linear ConstantConstant Linear Collect datasets (diff(D(t))) ¼ Ck, then output is fl ¼ pl x+ql y+rl z+sl, where x (2) S F mo mea line con diff fore T des inte ship if–t mem out F ing, as (dif W vari fl de (i ¼ T pl ¼ S tive des (dif n ARTICLE IN PRESS C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–34683466 Bj and Ck are the linguistic labels (high, middle, low), fl denotes the l-th output value, pl, ql, rl and sl are the parameters (i ¼ 1, 2, 3; j ¼ 1, 2, 3; k ¼ 1, 2, 3; and l ¼ 1,2,y,27). Constant type: a zero-order Sugeno model, the output level f l is a constant (p ¼ q ¼ s ¼ 0). (TAIEX (t)), y (diff(N(t))), z (diff(D(t))) are linguistic variables, Ai, Table 1 Differences in variables. Date NASDAQ diff(N(t)) Dow Jones diff(D(t)) 2000/1/3 4131.15 11357.51 2000/1/4 3901.69 �229.46 10997.93 �359.58 2000/1/5 3877.54 �24.15 11122.65 124.72 2000/1/6 3727.13 �150.41 11253.26 130.61 2000/1/7 3882.62 155.49 11522.56 269.3 2000/1/8 155.49 269.3 2000/1/9 155.49 269.3 2000/1/10 4049.67 167.05 11572.2 49.64 2000/1/11 3921.19 �128.48 11511.08 �61.12 2000/1/12 3850.02 �71.17 11551.1 40.02 2000/1/13 3957.21 107.19 11582.43 31.33 2000/1/14 4064.27 107.06 11722.98 140.55 2000/1/15 107.06 140.55 2000/1/16 107.06 140.55 2000/1/17 107.06 140.55 2000/1/18 4130.81 66.54 11560.72 �162.26 2000/1/19 4151.29 20.48 11489.36 �71.36 2000/1/20 4189.51 38.22 11351.3 �138.06 2000/1/21 4235.4 45.89 11251.71 �99.59 2000/1/22 45.89 �99.59 2000/1/23 45.89 �99.59 2000/1/24 4096.08 �139.32 11008.17 �243.54 2000/1/25 4167.41 71.33 11029.89 21.72 2000/1/26 4069.91 �97.5 11032.99 3.1 2000/1/27 4039.56 �30.35 11028.02 �4.97 2000/1/28 3887.07 �152.49 10738.87 �289.15 2000/1/29 �152.49 �289.15 2000/1/30 �152.49 �289.15 2000/1/31 3940.35 53.28 10940.53 201.66 i i i tep 5: Generate fuzzy inference system rom steps 3 and 4, we can obtain four types of forecasting dels: (1) fuzzy C-means with linear type (FCM_L), (2) fuzzy C- ns with constant type (FCM_C), (3) subtractive clustering with ar type (Subclust_L) and (4) subtractive clustering with stant type (Subclust_C). Then, we can generate the four erent fuzzy inference systems according to the four types of casting models, respectively. he detailed steps of generating fuzzy inference system are cribed as follows: Firstly, from step 3, we can get the linguistic rvals as input membership functions, and the output member- functions are set by step 4. Secondly, we can generate fuzzy hen rules, where the linguistic values (Ai,Bi and Ci) from input bership functions are used as the if–condition part and the put membership functions (f i) as the then part. CM case: the input membership partitioned by FCM cluster- we generate 27 rules (3�3�3). The general rule is described follows: If x(TAIEX (t)) ¼ Ai, y (diff(N(t))) ¼ Bj and z f(D(t))) ¼ Ck, then output is fl ¼ pl x+ql y+rl z+sl. here x (TAIEX (t)), y (diff(N(t))) and z (diff(D(t))) are linguistic ables, Ai, Bj and Ck are the linguistic labels (high, middle, low), notes the l-th output value, pl, ql, rl and sl are the parameters 1, 2, 3; j ¼ 1, 2, 3; k ¼ 1, 2, 3, and l ¼ 1,2,y,27). he output membership function is constant when ql ¼ rl ¼ 0. ubclust case: the input membership partitioned by subtrac- clustering, we generate three rules. The general rule is cribed as follows: If x (TAIEX (t)) ¼ Ai, y (diff(N(t))) ¼ Bi and z f(D(t))) ¼ Ci, then fi ¼ pi x+qi y+ri z+si. Where x (TAIEX (t)), y Where actualðtÞ denotes the real TAIEX value, forecastðtÞ denotes the predicting TAIEX value, and n is the number of data. Step 8: Select the best model based on minimal RMSE and compare the forecasting performances of the different models Based on minimal RMSE for the target testing datasets from step 7, the best forecasting model among the four models can be obtained. Then the minimal RMSE is taken as evaluation criterion to compare with different models. 4. Experiments and comparisons This section provides an evaluation of the model’s accuracy and comparison to other models, as well as profit evaluations and comparisons. To verify the proposed model, experimentation, using the TAIEX from 1997 to 2003 (7 sub-datasets), was implemented. The sub-datasets for the first 10-month period are used for training, and those from November to December are selected for testing. 4.1. Accuracy evaluations and comparisons After the experiments, we generate 7 forecasting performances for the 7 testing sub-datasets. Then, this study compares the performances of the proposed model with the conventional fuzzy time-series model, Chen’s [4] model. Furthermore, to examine whether the proposed model surpasses the latest fuzzy time- series model, the performance of Yu’s [25] model is compared with the proposed model. The forecasting performances of Chen’s model, Yu’s model and the proposed model are listed in Table 2. From Table 2, we can see that the proposed model outperforms the performances of the listing model. 4.2. Profit evaluations and comparisons For making simulation trades and showing the profits, we set (diff(N(t))), z (diff(D(t))) are linguistic variables, Ai, Bi, Ci are the linguistic labels (high, middle, low), fi denotes the i-th output value, pi, qi, ri, si are the parameters (i ¼ 1,2,3). The output membership function is constant when pi ¼ qi ¼ ri ¼ 0. Step 6: Train fuzzy inference system parameters from training datasets In this section, we employ a combination of the least-squares method and the backpropagation gradient-descent method for training four types of forecasting models and use fuzzy inference system membership function parameters to emulate a given training dataset. This study sets epoch as 50 (the process is executed for the predetermined fixed number (50) of iterations unless it terminates while the training error converges) for the training stopping criterion and then obtains the parameters for the selected output membership function. Step 7: Forecast the testing TAIEX (t+1) by the four types of forecasting models and calculate root mean squared error (RMSE) Firstly, the fuzzy inference system parameters of the four types of forecasting models are determined when the stopping criterion is reached from step 6, then the four training forecasting models are used to forecast T (t+1) for the target testing datasets, respectively. Secondly, the four RMSE values are calculated in testing datasets by Eq. (15). RMSE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn t¼1 jactualðtÞ � forecastðtÞj2 vuuut . (15) two trade rules by the Taiwan Futures Exchange (TAIFEX) [27], use ARTICLE IN PRESS 1997 1998 1999 2000 2001 2002 2003 C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–3468 3467 the two trade rules to calculate profits, and assume that the profits unit is equal to one. Therefore, the profit formula is defined as Eq. (16). Rule 1: sell rule IF jforecastðtÞ � actualðtÞj actualðtÞ � a And forecastðt þ 1Þ � actualðtÞ40 Then sell Rule 2: buy rule IF jforecastðtÞ � actualðtÞj � a And Yu’s model [25] 165 164 145 191 167 75 66 Chen’s model [4] 154 134 120 176 148 101 74 Proposed model 130a 113a 103a 138a 118a 67a 50a a The best performance among three models. Table 3 The profits comparisons of different models (TAIEX). Year a Models Yu’s model [25] Chen’s model [4] Proposed model 1997 0.02 �107 �127 935a 1998 0.03 �864 661 878a 1999 0.03 �756 �702 124a 2000 0.01 �200 �106a �154 2001 0.02 450a �340 91 2002 0.005 �96 �17 466a 2003 0.005 �190 �238 481a Cumulated profits �1763 �869 2821a a The best profits among three models. Table 2 The performance comparisons of different models (TAIEX). Models Year actualðtÞ forecastðt þ 1Þ � actualðtÞo0 Then buy where a denotes threshold parameter (0oa � 0:07, the threshold parameter depends on daily fluctuation of TAIEX). Definition of profit: Profit ¼ Xp ts¼1 ðactualðt þ 1Þ � actualðtÞÞ þ Xq tb¼1 ðactualðtÞ � actualðt þ 1ÞÞ (16) where p represents the total number of days for selling, q represents the total number of days for buying, ts represents the t-th day for selling and tb represents the t-th day for buying. The optimal threshold parameter a is obtained when the forecasting performance reaches best profits in the training dataset. From the optimal threshold parameter a and Eq. (16), the profits for different models are calculated, and the profits results are shown in Table 3. Based on Table 3, we can see that the proposed model has higher profits than the listing models in five testing periods (excluding 2000 and 2001). 5. Findings and discussion Based on the verification and comparison results, the proposed model outperforms the listing methods. Also of interest is the conditions and environments, to improve forecasting perfor- mance; and (3) the study generates the decision rules regarded as references of investment for stock investors. For subsequent research, we can use datasets of other countries, such as China, Japan and Hong Kong, to further validate the proposed model. Moreover, there are two methods suitable for integration into the proposed model that will improve the forecasting accuracy: (1) employ data discretization in the preprocessing step, which granulates (partitions) attributes to enhance performance of the proposed model (can generate fewer rules by data discretization) and (2) validate the generated rules by expert group to improve accuracy. References [1] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, NY, 1981. [2] T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, From the experimental results, there are two findings: (1) According to Table 2, it is evident that the proposed model is superior to the listing methods in terms of RMSE. The main reason is that the proposed model takes into account multi-stock volatility causality with ANFIS learning for TAIEX forecasting. (2) From the empirical results, the best performance in four types of forecasting models is subtractive clustering with linear type (Subclust_L). That is, in Subclust_L, the parameters of the model are not equal to 0 (output function). This means that the American stock indexes significantly affect the volatility of TAIEX. These results confirm the study of Dickinson [10]. 6. Conclusions A new model, based on multi-stock volatility causality joined to the fusion ANFIS procedure, was proposed to forecast stock index problems in Taiwan; furthermore, the proposed model was compared with two different models, Chen’s model and Yu’s model, to evaluate the results. This proposed model mainly uses input variables of stock index (e.g., Dow Jones, NASDAQ and TAIEX) to forecast the TAIEX in the next trading day for investors. To illustrate the proposed model, three practical, collected stock index datasets from the USA and the Taiwan stock market, the Dow Jones, NASDAQ and TAIEX, were employed in this empirical experiment, all of which consist of datum from 1997 to 2003 (7 years in total). Each stock index dataset is split into 7 sub- datasets, based on year. The first 10 months of the sub-dataset for each year, January to October, were used for training data, and the last 2 months, November to December, were used for testing data. From Table 2, the experimental results of three datasets indicate that the proposed model outperforms the listing models in terms of RMSE. Moreover, the results of this study should be useful and viable for stock investors, decision makers and future research. Investors can utilize this forecasting model to uncover superior target investments in the stock market. Generally, the proposed model is based on the following, which overcome the drawbacks mentioned in Section 1: (1) ANFIS combines the advantages of ANN and fuzzy logic system; thus, it does not pre-assume the data distribution; (2) the study uses more variables, which addresses the issue of only one variable in forecasting accuracy and better approximates actual market degree of influence of other countries’, particularly neighboring countries’ (e.g., China, Japan and Hong Kong), stock indexes; thus, in the future, we will further verify the influence of the stock indexes (e.g., A-share index, Nikkei 225 and Hang Seng index) of different countries on the proposed model. Journal of Econometrics 31 (1986) 307–327. [3] G. Box, G. Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, 1976. [4] S.M. Chen, Forecasting enrollments based on fuzzy time-series, Fuzzy Sets Systems 81 (1996) 311–319. [5] S.M. Chen, N.Y. 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His research is mainly in the field of fuzzy logic, fuzzy time series, soft computing, reliability, and data mining. He has published more than 200 papers (include 97 significant journal papers). LIANG-YING WEI received the Bachelor’s degree in Environmental Engineering and Science from Feng Chia University in 1998 and the Master’s degree in Technology in 1988, and the Master’s degree from National Yunlin University of Science and Technology ARTICLE IN PRESS C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–34683468 [15] J.S. Jang, ANFIS: adaptive-network-based fuzzy inference systems, IEEE Transactions on Systems, Man, and Cybernetics 23 (3) (1993) 665–685. [16] T.A. Jilani, S.M.A. Burney, A refined fuzzy time series model for stock market forecasting, Physica A 387 (2008) 2857–2862. [17] K. Kim, I. Han, Genetic algorithms approach to feature discretization in artificial neural networks for prediction of stock index, Expert System with Applications 19 (2000) 125–132. [18] K.J. 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Enke, The adaptive selection of financial and economic variables for use with artificial neural networks, Neurocomputing 56 (2004) 205–232. University of Science and Technology and majors in information management. His research interests in- clude financial analysis and bankruptcy prediction. in 2006. He is now a Ph.D. student in National Yunlin Management Department in National Yunlin Univer- sity of Science and Technology and works in national genotyping center at academia sinica for research. His research is mainly in the field of fuzzy time series, soft computing, machine learning, data mining and bioin- formatics. YOU-SHYANG CHEN received the Bachelor’s degree from National Taiwan University of Sciences and (2007) 836–846. Information Management from Huafan University in 2005. He is now a Ph.D. student of Information [13] K.H. Huarng, H.K. Yu, The application of neural networks to forecast fuzzy time series, Physica A 336 (2006) 481–491. [14] K.H. Huarng, H.K. Yu, Y.W. Hsu, A multivariate heuristic model for fuzzy, IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics 37 (4) Fusion ANFIS models based on multi-stock volatility causality for TAIEX forecasting Introduction Related studies Different forecasting models for the stock market Fuzzy C-means clustering Subtractive clustering ANFIS: adaptive-network-based fuzzy inference system Proposed model Experiments and comparisons Accuracy evaluations and comparisons Profit evaluations and comparisons Findings and discussion Conclusions References