Modeling cayenne production data in Central Java using adaptive neuro fuzzy inference system (ANFIS)

Abstract

The aim of this research is to develop the novel procedure of Adaptive Neuro-Fuzzy Inference System (ANFIS) modeling for forecasting time series data. The procedure development applies statistical inference based on Lagrange Multiplier (LM) test for selecting input variables, determining the number of clusters, and generating the rule-bases. For selecting inputs, several lags which are indicated significantly different to zero are divided into 2 clusters (minimum number of clusters), and then the lags are selected as optimal inputs of ANFIS based on LM test procedure. The cluster numbers of optimal inputs are added using LM-test procedure such optimal clusters are obtained. Based on those results, a number of rule-bases are generated. The developed model is applied for forecasting cayenne production data in Central Java. The result of proposed procedure is that the optimal inputs consist of 2 lags (lag-1 and lag-3) which are divided into 2 clusters. In this case, the two rules are selected as optimal rules. Finally, the model can work well, and generates very satisfying result in forecasting cayenne production data. Based on the Root Mean Squares Error (RMSE) value, the ANFIS performance is better than performance of Autoregressive Integrated Moving Average (ARIMA) for forecasting cayenne production data in Central Java.

Keywords

Time series cayenne production forecasting ARIMA ANFIS LM-test

1. Introduction

Time series data which characterized by uncertainty, autocorrelation persistence and leptokurtic behavior are usually non-stationary and non-linear (Abiyev et al., 2005; Cheng & Wei, 2010; Maciel, 2012; Maciel et al., 2012; Makridakis et al., 1998; Tsay, 2005; Talebizadeh & Moridnejad, 2011; Samanta, 2011). Autoregressive Integrated Moving Average (ARIMA) is one of the most popular models which used for time series forecasting (Box et al., 1994; Brockwell & Davis, 1991; Engle, 1982; Haykin, 1999; Wei, 2006). Autoregressive Conditional Heteroscedasticity (ARCH) proposed by (Engle, 1982) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) developed by (Bollerslev, 1986) are the most popular variance models. ARIMA-GARCH has been applied in a lot of research for forecasting nonlinear time series data (Fahimifard et al., 2009; Tsay, 2005). The model still has some disadvantages when applied for forecasting nonlinear time series data, because the series is not appropriate with standardized theoretical phenomenon and also the model can not capture the uncertainty of data. Using linear models for non-linear data would cause cointegration errors in time series (Chen et al., 2009).

In recent years, alternative models have been developed to analyze non-linear time series such as neural networks (NN), fuzzy system and its hybrid (Fausset, 2009; Haykin, 1999; Jang, 1993; Jang et al., 1997). ANFIS that combines NN and fuzzy system has been implemented in many fields of time series research such as application of ANFIS based on singular spectrum analysis for forecasting chaotic time series (Abdollahzade et al., 2015); chaotic time series prediction using improved ANFIS (Behmanesh et al., 2014); fuzzy time series forecasting (Cheng et al., 2016); developing a new approach for forecasting the trends of oil price (Mombeini & Yazdani-Chamzini, 2015); computational intelligence for prediction of chaotic time series; modeling minimum temperature (Daneshmand et al., 2015); financial trading (Kablan, 2009); forecasting of mobile sales (Wang et al., 2011); forecasting of stock return (Boyacioglu & Avci, 2010; Hung, 2009; Wei et al., 2011); prediction of government bond yield (Chen et al., 2009); prediction for EPS of leading industries (Wei et al., 2011); forecasting of financial volatility (Luna & Ballini, 2012; Maciel, 2012; Maciel et al., 2012) and prediction of exchange rates (Alakhras, 2005, Alizadeh et al., 2009; Fahimifard et al., 2009). Conclusively ANFIS is better than the other methods.

All ANFIS models that have been proposed by the previous researchers did not apply LM-test for selecting the best model. To date, there is no research which using statistical inference based on LM test for selecting optimal model in ANFIS. Therefore, this research focused on procedure development of ANFIS modeling which uses LM-test for selecting the input variables, determining the number of membership function (cluster) and determining the number of rules. The procedure is extended based on LM-test for selecting NN model proposed by White in 1989 (Anders & Korn, 1999). The new procedure for determining input variables, determining number of clusters and generating rule-bases of ANFIS applies statistical inference based on LM-test. Finally, the model was implemented for forecasting paddy production in Central Java Province. This paper is organized as follows; Section 2 discusses an architecture of ANFIS and estimation of consequent parameters; Section 3 discusses about proposed procedure for selecting model by using statistical inference based on LM-test and application of developed model; and conclusion is discussed in Section 4.

2. Theoretical framework

2.1 ANFIS architecture

The ANFIS architecture consists of fuzzyfication (layer-1), fuzzy inference system (layer-2 and layer-3), defuzzyfication (layer-4) and aggregation (layer-5). The NN architecture which used in ANFIS architecture has 5 fixed-layers (Jang et al., 1997). Generally, the architecture of ANFIS for time series modeling with p input variables $Z_{t-1}$ , $Z_{t-2},\ldots,Z_{t-p}$ and one output variable $Z_{t}$ by assuming rule-base of Sugeno order-one with m rules is as follow.

$\displaystyle\text{If }Z_{t-1}\text{ is }A_{11}\text{ and }Z_{t-2}\text{ is }A% _{21}\ldots\text{ and }Z_{t-p}\text{ is }A_{p1}\text{ then }Z_{t}^{(1)}{=}% \theta_{11}Z_{t-1}+\theta_{12}Z_{t-2}+\ldots+\theta_{1p}Z_{t-p}+\theta_{10};$ $\displaystyle\text{If }Z_{t-1}\text{ is }A_{12}\text{ and }Z_{t-2}\text{ is }A% _{22}\ldots\text{ and }Z_{t-p}\text{ is }A_{p2}\text{ then }Z_{t}^{(2)}{=}% \theta_{21}Z_{t-1}+\theta_{22}Z_{t-2}+\ldots+\theta_{2p}Z_{t-p}+\theta_{20}$ $\displaystyle\vdots$ $\displaystyle\text{If }Z_{t-1}\text{ is }A_{1m}\text{ and }Z_{t-2}\text{ is }A% _{2m}\ldots\text{ and }Z_{t-p}\text{ is }A_{pm}\text{ then }Z_{t}^{(m)}=\theta% _{m1}Z_{t-1}+\theta_{m2}Z_{t-2}+\ldots+\theta_{mp}Z_{t-p}$ $\displaystyle\quad∼{}+\theta_{m0};$

where $Z_{t-k}$ is $A_{kj}$ as premise (nonlinear) section, whereas $Z_{t}^{(j)}=\theta_{j0}+\sum\nolimits_{k=1}^{p}{\theta_{jk}Z_{t-k}}$ as consequent (linear) section; $\theta_{jk}$ , $\theta_{j0}$ as linear parameters; $A_{kj}$ as nonlinear parameters; $j=1,2,\ldots,m$ ; $k=1,2,\ldots,p$ . If the firing strength for m values $Z_{t}^{(1)}$ , $Z_{t}^{(2)}$ , $\ldots$ , $Z_{t}^{(m)}$ are $w_{1},w_{2},\ldots,w_{m}$ respectively then the output $Z_{t}$ can be determined as:

$Z_{t}=\frac{w_{1}Z_{t}^{(1)}+w_{2}Z_{t}^{(2)}+\ldots+w_{m}Z_{t}^{(m)}}{w_{1}+w% _{2}+\ldots+w_{m}}$ (1)

The architecture of ANFIS (see Fig. 1) consist of 5 layers that can be described as follows (Jang et al., 1997).

Figure 1.

ANFIS architecture for time series modeling.

Layer-1: Layer-1:

Every node in the first layer is adaptive with one parametric activation function. The output is membership degree of given inputs which satisfy membership function $\mu_{A_{11}}(Z_{t-1}),\mu_{A_{12}}(Z_{t-1}),\ldots,\mu_{A_{1m}}$ $(Z_{t-1}),\mu_{A_{21}}(Z_{t-2}),\mu_{A_{22}}(Z_{t-2}),\ldots,\mu_{A_{2m}}(Z_{t% -2}),\ldots,\mu_{A_{p1}}(Z_{t-p}),\mu_{A_{p2}}(Z_{t-p}),\ldots,\mu_{A_{pm}}(Z_% {t-p})$ .One example of membership function is Gaussian membership function (gaussmf) which can be written as:

$\mu_{A_{kj}}(Z_{t-k})=\exp\left({-\frac{1}{2}\left({\frac{Z_{t-k}-c_{jk}}{a_{% jk}}}\right)^{2}}\right),∼{}∼{}j=1,2,\ldots,m,∼{}∼{}k=1,2,\ldots,p;$

$c_{jk}:$ location parameters and $a_{jk}$ : scaling parameters. The parameters are called as premise parameters.

Layer-2:

Every node in the second layer is fixed node which the output of this layer is the product of incoming signal. Generally, it uses fuzzy operation AND. The output of each node represents firing strength $w_{j}$ of the $j$ -th rule.

$w_{j}=\prod\limits_{k=1}^{p}{\mu_{A_{kj}}(Z_{t-k})},∼{}∼{}j=1,2,\ldots,m.$ (2)

Layer-3:

Every node in the third layer is fixed node, which computes ratio of firing strength of $j$ -th rule relative to sum of firing strengths of rules.

$\displaystyle\bar{{w}}_{j}=\frac{w_{j}}{\sum\limits_{j=1}^{m}{w_{j}}}$ (3)

Layer-4:

Every node in the fourth layer is adaptive node, the output of each node

$\displaystyle\bar{{w}}_{j}Z_{t}^{(j)}=\bar{{w}}_{j}(\theta_{j1}Z_{t-1}+\theta_% {j2}Z_{t-2}+\ldots+\theta_{jp}Z_{t-p}+\theta_{j0})$ (4)

Layer-5:

Every node in the fifth layer is a fixed node which adds all of incoming signal. The output of fifth layer is the output of the whole network.

$Z_{t}=\sum\limits_{j=1}^{m}{\bar{{w}}_{j}}(\theta_{j1}Z_{t-1}+\theta_{j2}Z_{t-% 2}+\ldots+\theta_{jp}Z_{t-p}+\theta_{j0})$

The general model of ANFIS is given as follow.

$Z_{t}=\sum\limits_{j=1}^{m}{\sum\limits_{k=1}^{p}{\theta_{jk}}}(\bar{{w}}_{j}Z% _{t-k})+\sum\limits_{j=1}^{m}{\theta_{j0}}\bar{{w}}_{j}$ (5)

2.2 Estimating the consequent parameters of ANFIS

If the general ANFIS model is given as expression (5) then the estimation of consequent parameters $\theta_{j,k}(j=1,2,\ldots,m)$ , $(k=1,2,\ldots,p)$ can be determined using recursive least squares (RLS) method. To estimate the parameters, a linear equation system is constructed as below.

$\sum\limits_{j=1}^{m}{\sum\limits_{k=1}^{p}{\hat{{\theta}}_{jk}}}(\bar{{w}}_{j% }Z_{t-k})+\sum\limits_{j=1}^{m}{\hat{{\theta}}_{j0}}\bar{{w}}_{j}=Z_{t}$ (6)

If the consequent parameters had been estimated, then the premise parameters can be updated using back-propagation method. Optimal quality is obtained by minimizing the ANFIS error function.

3. Proposed procedure for selecting model in ANFIS using Lagrange Multiplier (LM) test

Lagrange Multiplier (LM) test is used for testing hypothesis of adding variables in ANFIS modeling. Some variables should be included to the model because the new inputs, number of clusters or number of rules to be added in the model.

3.1 Procedure for adding input variables

Determining input can be identified by lag plot of data or the partial autocorrelation function (PACF) plot. Lag plot can be used for testing linearity. The lag plot or PACF plot can be used for identification of autoregressive (AR) input. Based on lag plot and PACF plot, the significant lags should be tested as input variables of ANFIS. Determining input is performed by constructing models which involve a number of input variables with minimum number of clusters and minimum number of rules. Firstly, construct the models with 1 input variable, 2 clusters and 2 rules and then select the model which has the largest $R^{2}$ . Adding input is done by using LM-test procedure to get a model with optimal inputs. If given $p$ input variables $Z_{t-1}$ , $Z_{t-2},\ldots,Z_{t-p}$ with number of clusters $m$ , then the restricted model for this case can be written as:

$Z_{t}=\sum\limits_{j=1}^{m}{\sum\limits_{k=1}^{p}{\theta_{jk}(\bar{{w}}_{j}Z_{% t-k})}}+\sum\limits_{j=1}^{m}{\theta_{j0}\bar{{w}}_{j}+}\varepsilon_{t}$ (7)

where $\varepsilon_{t}\sim N(0,\sigma_{t}^{2})$ , and unrestricted model for adding by one input $Z_{t-(p+1)}$ is:

$Z_{t}=\sum\limits_{j=1}^{m}{\sum\limits_{k=1}^{p+1}{\theta_{jk}(\bar{{w}}_{j}Z% _{t-k})}}+\sum\limits_{j=1}^{m}{\theta_{j0}\bar{{w}}_{j}+}\upsilon_{t}$ (8)

where $\upsilon_{t}\sim N(0,\sigma_{t}^{2})$ .

The null hypothesis for testing of adding variables can be formulated as follow:

$H_{0}:\theta_{1(p+1)}=\theta_{2(p+1)}=\ldots=\theta_{m(p+1)}=0$

Procedure of hypothesis test using LM-test can be done as the following steps:

Step-1: Step-1:

Estimate the parameters of restricted model

$\hat{{\theta}}_{11},\hat{{\theta}}_{12},\ldots,\hat{{\theta}}_{1p},\hat{{% \theta}}_{10},\hat{{\theta}}_{21},\hat{{\theta}}_{22},\ldots,\hat{{\theta}}_{2% p},\hat{{\theta}}_{20},\ldots,\hat{{\theta}}_{m1},\hat{{\theta}}_{m2},\ldots,% \hat{{\theta}}_{mp},\hat{{\theta}}_{m0}.$

Step-2:

Determine the estimate of residual:

$\hat{\varepsilon}_{t}=Z_{t}-\sum\limits_{j=1}^{m}{\sum\limits_{k=1}^{p}{\hat{% \theta}_{jk}(\bar{{w}}_{j}Z_{t-k})}}-\sum\limits_{j=1}^{m}{\hat{\theta}_{j0}% \bar{{w}}_{j}}$

Step-3:

Regress the residual $\hat{{\varepsilon}}_{r}$ with: a constant, $(\bar{{w}}_{1}Z_{t-1}),(\bar{{w}}_{1}Z_{t-2}),\ldots,(\bar{{w}}_{1}Z_{t-p}),(% \bar{{w}}_{1}Z_{t-p-1}),\bar{{w}}_{1},$ $(\bar{{w}}_{2}Z_{t-1}),(\bar{{w}}_{2}Z_{t-2}),\ldots,(\bar{{w}}_{2}Z_{t-p}),(% \bar{{w}}_{2}Z_{t-p-1}),\bar{{w}}_{2},\ldots,(\bar{{w}}_{m}Z_{t-1}),(\bar{{w}}% _{m}Z_{t-2}),\ldots,(\bar{{w}}_{m}Z_{t-p}),$ $(\bar{{w}}_{m}Z_{t-p-1}),\bar{{w}}_{m}$ and calculate $\textit{nR}_{\hat{{\varepsilon}}}^{2}$ . It was known that $\textit{LM}=\textit{nR}_{\hat{{\varepsilon}}}^{2}\sim\chi^{2}$ where the degree of freedom is equal to number of additional variables in unrestricted model (Engle, 1982). In this case $\textit{nR}_{\hat{{\varepsilon}}}^{2}\sim\chi_{m}^{2}$ . Null hypothesis Ho is rejected when $\textit{nR}_{\hat{{\varepsilon}}}^{2}>\chi_{m}^{2}(\alpha)$ .

3.2 Procedure for adding number of clusters of inputs

When the ANFIS model with optimal inputs was obtained, adding number of clusters can be executed using LM-test procedure to get a model with optimal number of clusters. If given p input variables $Z_{t-1}$ , $Z_{t-2},\ldots,Z_{t-p}$ with number of clusters m, and the restricted model (7), then the unrestricted model for adding by one cluster

$Z_{t}=\sum\limits_{j=1}^{m+1}{\sum\limits_{k=1}^{p}{\theta_{jk}(\bar{{w}}_{j}Z% _{t-k})}}+\sum\limits_{j=1}^{m}{\theta_{j0}\bar{{w}}_{j}+}\upsilon_{t}$ (9)

where $\upsilon_{t}\sim N(0,\sigma_{\upsilon}^{2})$ .

The null hypothesis for testing adding variables $(\bar{{w}}_{m+1}Z_{t-1}),(\bar{{w}}_{m+1}Z_{t-2}),\ldots,(\bar{{w}}_{m+1}Z_{t-% p})$ can be formulated as follow:

$H_{0}:\theta_{(m+1)1}=\theta_{(m+1)2}=\ldots=\theta_{(m+1)p}=0.$

Procedure of hypothesis test using LM-test can be done as the following steps:

Step-1: Step-1:

Estimate the parameters of restricted model:

Step-2:

Determine the estimate of residual:

$\hat{{\varepsilon}}_{t}=Z_{t}-\sum\limits_{j=1}^{m}{\sum\limits_{k=1}^{p}{\hat% {{\theta}}_{jk}(\bar{{w}}_{j}Z_{t-k})}}+\sum\limits_{j=1}^{m}{\hat{{\theta}}_{% j0}\bar{{w}}_{j}}$

Step-3:

Regress the residual $\hat{{\varepsilon}}_{r}$ with a constant, $(\bar{{w}}_{j}Z_{t-k}),\bar{{w}}_{j}$ ; $j=1,2,\ldots,m$ ; $k=1,2,\ldots,p$ , and calculate $\textit{LM}=\textit{nR}_{\hat{{\varepsilon}}}^{2}$ . In this case, it is known that $\textit{LM}=\textit{nR}_{\hat{{\varepsilon}}}^{2}\sim\chi_{p}^{2}$ . Null hypothesis Ho is rejected when $\textit{nR}_{\hat{{\varepsilon}}}^{2}>\chi_{p}^{2}(\alpha)$ .

Figure 2.

Time series, ACF and PACF plots of cayenne production data.

3.3 Procedure for generating rule bases

The rule-bases of ANFIS are generated based on the optimal input variables and optimal number of clusters. The all possible combination of rules which can be generated is $m^{p}$ , where $m$ is number of membership functions and $p$ is number of inputs. For instance, given $p$ inputs $Z_{t-1},Z_{t-2},\ldots,Z_{t-p}$ with $m$ number of clusters, and one output $Z_{t}$ by assuming first order Sugeno fuzzy system with $m$ rules:

$\displaystyle R_{1}\text{: If }Z_{t-1}\text{ is }A_{11}\text{ and }Z_{t-2}% \text{ is }A_{21}\ldots\text{ and }Z_{t-p}\text{ is }A_{p1}\text{ then }$ $\displaystyle\quad∼{}Z_{t}^{(1)}=\theta_{11}Z_{t-1}+\theta_{12}Z_{t-2}+\ldots+% \theta_{1p}Z_{t-p}+\theta_{10};$ $\displaystyle R_{2}\text{: If }Z_{t-1}\text{ is }A_{12}\text{ and }Z_{t-2}% \text{ is }A_{22}\ldots\text{ and }Z_{t-p}\text{ is }A_{p2}\text{ then }$ $\displaystyle\quad∼{}Z_{t}^{(2)}=\theta_{21}Z_{t-1}+\theta_{22}Z_{t-2}+\ldots+% \theta{}_{2p}Z_{t-p}+\theta_{20};$ $\displaystyle\quad∼{}\vdots$ $\displaystyle R_{m}\text{: If }Z_{t-1}\text{ is }A_{1m}\text{ and }Z_{t-2}% \text{ is }A_{2m}\ldots\text{ and }Z_{t-p}\text{ is }A_{pm}\text{ then }$ $\displaystyle\quad∼{}Z_{t}^{(m)}=\theta_{m1}Z_{t-1}+\theta_{m2}Z_{t-2}+\ldots+% \theta_{mp}Z_{t-p}+\theta_{m0};$

then the output of ANFIS is given by equation:

$Z_{t}=\sum\limits_{j=1}^{m}{\bar{{w}}_{j}}(\theta_{j1}Z_{t-1}+\theta_{j2}Z_{t-% 2}+\ldots+\,\theta_{jp}Z_{t-p}+\theta_{j0}).$ (10)

4. Application of model for forecasting cayenne production data

As an implementation of ANFIS modeling for forecasting time series data, we construct ANFIS model for forecasting cayenne production as a case study. In this research, the monthly cayenne production data from January 2003 up to December 2014 (www.bps.go.id) are used for constructing the ANFIS model. Based on time series plot and ACF plot (see Fig. 2), Indonesia cayenne production data are stationary in mean, but non-stationary in variance.

Figure 3.

Cayenne production data and its prediction.

Based on the partial autocorrelation function (PACF) plot of cayenne production data (see Fig. 2), lag-1 and lag-3 are significant different to zero. Based on LM test, the variables: lag-1 and lag-3 can be selected as inputs ANFIS. The optimal cluster number of the inputs variables is two clusters (membership functions). The result of optimal model selection can be written as follow:

$Z_{t}=\bar{{w}}_{1,t}Z_{t-1}-0,1\bar{{w}}_{1,t}Z_{t-3}+17.048\bar{{w}}_{1,t}+% \bar{{w}}_{2,t}Z_{t-1}-0,1\bar{{w}}_{2,t}Z_{t-3}+16.216\bar{{w}}_{2,t}$ (11)

where

$\displaystyle\bar{{w}}_{1,t}=\frac{w_{1,t}}{w_{1,t}+w_{2,t}},∼{}∼{}\bar{{w}}_{% 2,t}=\frac{w_{2,t}}{w_{1,t}+w_{2,t}},$ $\displaystyle w_{1,t}=\exp\left[{-\frac{1}{2}\left\{{\left({\frac{Z_{t-1}-94.7% 21}{27.970}}\right)^{2}+\left({\frac{Z_{t-3}-83.266}{26.458}}\right)^{2}}% \right\}}\right],$ $\displaystyle w_{2,t}=\exp\left[{-\frac{1}{2}\left\{{\left({\frac{Z_{t-1}-38.5% 11}{30.020}}\right)^{2}+\left({\frac{Z_{t-3}-40.954}{34.281}}\right)^{2}}% \right\}}\right].$

The RMSE value of forecasting for in sample data using ANFIS is 20558. Whereas the RMSE value using ARIMA ([1,3],0,0) is 20739.9. Therefore, the performance of ANFIS model is better than ARIMA model for forecasting cayenne production data in Central Java Province. The predicted value of cayenne production data based on model (11) can be seen as Fig. 3.

5. Conclusion

The proposed procedure of ANFIS modeling using statistical inference based on LM-test can work well for forecasting time series data. Based on the empirical study can be concluded that the developed model has a good performance for forecasting cayenne production data. An optimal ANFIS model for forecasting cayenne production is a model with 2 inputs (lag-1 and lag-3), 2 membership functions and 2 rules. Based on RMSE values, the performance of ANFIS model is better than ARIMA model for forecasting cayenne production data in Central Java Province.

Footnotes

Acknowledgments

We would like to give thank to Rector of Universitas Diponegoro, Chief of Research and Public Services Universitas Diponegoro for their support to this research.

References

Abdollahzade

Miranian

Hossein

, & Iranmanesh

(2015). A new hybrid enhanced local linear neuro-fuzzy model based on the optimized singular spectrum analysis and its application for nonlinear and chaotic time series forecasting. Information Sciences, 295, 107125.

Abiyev

V. H.

, & Ardill

(2005). Electricity consumption prediction model using neuro-fuzzy system. World Academy of Science, Engineering and Technology, 8.

Alakhras

M. N. Y.

(2005). Neural network-based fuzzy inference system for exchange rate prediction. Journal of Computer Science (Special Issue), Science Publications, 112-120.

Alizadeh

Rada

Balagh

A. K. G.

, & Esfahani

M. M. S.

(2009). Forecasting exchange rates: A neuro-fuzzy approach. IFSA-EUSFLAT, 1745-1750.

Anders

, & Korn

(1999). Model selection in neural networks. Neural Networks, 12, 309-323.

Behmanesh

Mohammadi

, & Naeini

V. S.

(2014). Chaotic time series prediction using improved ANFIS with imperialist competitive learning algorithm. International Journal of Soft Computing and Engineering (IJSCE), 4(4), 25-33.

Bollerslev

(1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31, 307-327.

Box

G. E. P.

Jenkins

G. M.

, & Reinsel

G. C.

(1994). Time Series Analysis, Forecasting and Control, 3rd edition. Englewood Cliffs: Prentice Hall.

Boyacioglu

M. A.

, & Avci

(2010). An adaptive network-based fuzzy inference system (ANFIS) for the prediction of stock market return: The case of the Istanbul stock exchange. Expert Systems with Applications, 37, 7908-7912.

10.

Brockwell

P. J.

, & Davis

R. A.

(1991). Time series: Theory and methods, 2nd edition. New York: Springer Verlag.

11.

Chen

Lin

, & Huang

(2009). The prediction of Taiwan 10-year Government bond yield. WSEAS TRANSACTIONS on SYSTEMS, 8(9), 1051-1060.

12.

Cheng

S.-H.

Chen

S.-M.

, & Jian

W-S..

(2016). Fuzzy time series forecasting based on fuzzy logical relationships and similarity measures. Information Sciences, 327, 272-287.

13.

Cheng

C. H.

, & Wei

L. Y.

(2010). One step-ahead ANFIS time series model for forecasting electricity loads. Optimization and Engineering, 11, 303-317.

14.

Daneshmand

Tavousi

Khosravi

, & Tavakoli

(2015). Modeling mini-mum temperature using adaptive neuro-fuzzy inference system based on spectral analysis of climate indices: A case study in Iran. Journal of the Saudi Society of Agricultural Sciences, 14, 3340.

15.

Engle

R. F.

(1982). Autoregressive conditional heteroscedasticity with estimates of variance United Kingdom inflation. Econometrica, 50(4), 987-1007.

16.

Fahimifard

S. M.

Homayounifar

Sabouhi

, & Moghaddamnia

A. R.

(2009). Comparison of ANFIS, ANN, GARCH and ARIMA techniques to exchange rate forecasting. Journal of Applied Sceinces, 9, 3641-3651.

17.

Fausset

(1994). Fundamental of neural networks: Architectures, algorithms and applications. Prentice Hall, Englewood Cliffs, New Jersey.

18.

(1999). Neural Networks: A Comprehensive Foundation, Second Edition. Prentice Hall International, Inc.

19.

Hung

J-C.

(2009). A fuzzy asymmetric GARCH model applied to stock markets. Information Sciences, 179, 3930-3943.

20.

Jang

J-S.. R.

(1993). ANFIS: Adaptive-network-based fuzzy inference system. IEEE Transactions on Systems, Man, and Cybernetics, 23(3), 665-685.

21.

Jang

J-S.. R.

Sun

C.-T.

, & Mizutani

(1997). Neuro-fuzzy and soft computing: A computational approach to learning and machine intelligent. Prentice Hall International.

22.

Kablan

(2009). Adaptive neuro-fuzzy inference system for financial trading using intraday seasonality observation model. World Academy of Science, Engineering and Technology, 58.

23.

Luna

, & Ballini

(2012). Adaptive fuzzy system to forecast financial time series volatility. Journal of Fuzzy and Intelligent Systems, 23, 27-38.

24.

Mombeini

, & Yazdani-Chamzini

(2014). Developing a new approach for forecasting the trends of oil price. The Business Management Review, 4(3), 120-132.

25.

Maciel

L. S.

(2012). A hybrid fuzzy GJR-GARCH modeling approach for stock market volatility forecasting. Rev Bras Financas, Rio de Janeiro, 10(3), 337-367.

26.

Maciel

L. S.

Gomide

, & Ballini

(2012). An evolving fuzzy-GARCH approach for financial volatility and forecasting. 1-16, unpublished.

27.

Makridakis

Wheelwright

S. C.

, & Hyndman

R. J.

(1998). Forecasting: Methods and Applications. John Wiley and Sons Inc, New York.

28.

Samanta

(2011). Prediction of chaotic time series using computational intelligence. Expert Systems with Applications, 38, 11406-11411.

29.

Tsay

R. S.

(2005). Analysis of Financial Time Series, Second Edition. Wiley Interscience, A John Wiley and Sons Inc Publication, USA.

30.

Talebizadeh

, & Moridnejad

(2011). Uncertainty analysis for the forecast of lake level fluctuations using ensembles of ANN and ANFIS models. Expert Systems with Applications, 38, 4126-4135.

31.

Wang

F. K.

Chang

K. K.

, & Tzeng

C. W.

(2011). Using adaptive network-based fuzzy inference system to forecast automobile sales. Expert Systems with Applications, 38, 10587-10593.

32.

Wei

W. W. S.

(2006). Time Series Analysis: Univariate and Multivariate Methods, Second Edition. Pearson Education Inc Boston.

33.

Wei

L. Y.

Chen

T. L.

, & Ho

T. H.

(2011). A hybrid model based on adaptive-network-based fuzzy inference system to forecast Taiwan stock market. Systems with Applications, 38, 13625-1363.

34.

Wei

L. Y.

Cheng

C. H.

, & Wu

H. H.

(2011). Fusion ANFIS model based on AR for forecasting EPS of leading industries. International Journal of Innovative Computing Information and Control, 7, 5445-5458.