Fuzzy time series model using weighted least square estimation

Document Type : Research Paper


1 Department of Statistics, Payame Noor University Tehran 19395-3697, Iran

2 Department of Statistics, University of Birjand, Birjand, Iran


The conventional fuzzy least-squares time series models show undesirable performance when the fuzzy data set involves the outliers. By introducing a strategy to detect the outliers, this paper introduced a method for reducing the influence of outliers on the future predictions. For this purpose, according to the weighted square distance error, an estimation procedure was suggested for determining the exact coefficients in the presence of outliers. The parameters of the fuzzy time series model were then estimated using an iterative algorithm. In order to identify the potential outliers of the fuzzy data, a  quality control chart was employed based on the center of gravity criterion of fuzzy data. The defuzzification method was also employed to examine the performance of the proposed method via some  scatter plots. Several common goodness-of-fit criteria used in traditional time series models were also extended to compare the performance of the proposed fuzzy time series method to an existing method. The effectiveness of the proposed method was illustrated through two numerical examples including a simulation study. The results clearly indicated that the proposed model performs well in terms of the both scatter plot criteria and goodness-of-fit evaluations in cases where the potential outliers exist among the fuzzy data.


[1] S. S. G. Abhishekh, S. R. Singh, A score function-based method of forecasting using intuitionistic fuzzy time series, New Mathematics and Natural Computation, 14 (2018), 91-111.
[2] M. G. Akbari, G. Hesamian, Linear model with exact inputs and interval-valued fuzzy outputs, IEEE Transactions on Fuzzy Systems, 26 (2018), 518-530.
[3] S. Askari, N. Montazerin, M. H. F. Zarandi, A clustering based forecasting algorithm for multivariate fuzzy time series using linear combinations of independent variable, Applied Soft Computing, 35 (2015), 151-160.
[4] K. A’yun, A. M. N. Abadi, F. Y. Saptaningtyas, Application of weighted fuzzy time series model to forecast trans jogja’s passengers, International Journal of Applied Physics and Mathematics, 5 (2015), 76-85.
[5] E. Bas, E. Egrioglu, C. H. Aladag, U. Yolcu, Fuzzy time-series network used to forecast linear and nonlinear time series, Applied Intelligence, 43 (2015), 343-355.
[6] D. Bosq, Nonparametric statistics for stochastic process, Springer, New York, 1996.
[7] J. J. Buckley, Fuzzy statistics, studies in fuzziness and soft computing, Springer-Verlag, Berlin, 2006.
[8] O. Cagcag, U. Yolcu, E. Egrioglu, C. H. Aladag, A high order fuzzy time series forecasting method based on operation of intersection, Applied Mathematical Modelling, 40 (2016), 8750-8765.
[9] Q. Cai, D. Zhang, W. Zheng, S. C. H. Leung, A new fuzzy time series forecasting model combined with ant colony optimization and auto-regression, Knowledge-Based Systems, 74 (2015), 61-68.
[10] L. J. Cao, E. H. T. Francis, Support vector machine with adaptive parameters in financial time series forecasting, IEEE Transactions on Neural Networks, 14 (2003), 1506-1518.
[11] F. T. S. Chan, A. Samvedi, S. H. Chung, Fuzzy time series forecasting for supply chain disruptions, Industrial Management and Data Systems, 115 (2015), 419-435.
[12] S. M. Chen, S. W. Chen, Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of fuzzy logical relationships, Transactions on Cybernetics, 45 (2015), 405-417.
[13] C. H. Cheng, C. H. Chen, Fuzzy time series model based on weighted association rule for financial market forecasting, Expert System, 35 (2018), 23-30.
[14] S. H. Cheng, S. M. Chen, W. S. Jian, Fuzzy time series forecasting based on fuzzy logical relationships and similarity measures, Information Sciences, 327 (2016), 272-287.
[15] R. D. Cook, Detection of influential observations in linear regression, Technometrics, 19 (1977), 15-18.
[16] D. Dubois, P. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9 (1978), 613-626.
[17] R. Efendi, M. M. Deris, Z. Ismail, Implementation of fuzzy time series in forecasting of the non-stationary data, International Journal of Computational Intelligence and Applications, 15 (2016), 1-15.
[18] R. Efendi, Z. Ismail, M. M. Deris, A new linguistic out-sample approach of fuzzy time series for daily forecasting of Malaysian electricity load demand, Applied Soft Computing, 28 (2015), 422-430.
[19] S. Efromovich, Nonparametric curve estimation: Methods, theory and applications, New York, Springer, 1999.
[20] E. Egrioglu, C. H. Aladag, U. Yolcu, A. Z. Dalar, A hybrid high order fuzzy time series forecasting approach based on PSO and ANNs methods, American Journal of Intelligent Systems, 6 (2016), 22-29.
[21] E. Egrioglu, E. Bas, C. H. Aladag, U. Yolcu, Probabilistic fuzzy time series method based on artificial neural network, American Journal of Intelligent Systems, 6 (2016), 42-47.
[22] R. L. Eubank, J. D. Hart, P. Speckman, Trigonometric series regression estimators with an application to partially linear models, Journal of Multivariate Analysis, 32 (1990), 70-83.
[23] M. Fanyong, Z. Jinxian, Z. Qiang, The shapley function for fuzzy games with fuzzy characteristic functions, International Journal of Fuzzy Systems, 25 (2013), 23-35.
[24] C. Gang, W. Peng-fei, L. I. Jin-ling, Optimization algorithm for fuzzy time series model based on autocorrelation function, Control and Decision, 30 (2015), 1977-1802.
[25] S. S. Gautam, S. Singh, A refined method of forecasting based on high-order intuitionistic fuzzy time series data, Progress in Artificial Intelligence, 7 (2018), 339-350.
[26] H. Ghosh, S. Chowdhury, Prajneshu, An improved fuzzy time-series method of forecasting based on L-R fuzzy sets and its application, Journal of Applied Statistics, 43 (2016), 1128-1139, Doi:10.1080/02664763.2015.1092111.
[27] A. Ghotmar, P. G. Khot, P. Sharma, Forecasting model based on fuzzy time series with first order diffencing, International Journal of Mathematical Archive, 7 (2016), 75-79.
[28] G. Golub, M. Heath, G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.
[29] H. Guan, Z. Dai, A. Zhao, J. He, A novel stock forecasting model based on High-order-fuzzy-fluctuation trends and back propagation neural network, PLoS ONE, 13(2) (2018), Doi:10.1371/journal.pone.0192366.
[30] C. Gupta, G. Jain, D. K. Tayal, O. Castillo, ClusFuDE: Forecasting low dimensional numerical data using an improved method based on automatic clustering, fuzzy relationships and differential evolution, Engineering Applications of Artificial Intelligence, 71 (2018), 175-189.
[31] G. Hesamian, M. G. Akbari, A semi-parametric model for time series based on fuzzy data, IEEE Transactions on Fuzzy Systems, 26 (2018), 2953-2966.
[32] G. Hesamian, M. G. Akbari, Fuzzy absolute error distance measure based on a generalised difference operation, International Journal of Systems Science, Taylor and Francis Journals, 49(11) (2018), 2454-2462.
[33] M. Hudec, D. Praenka, Collecting and managing fuzzy data in statistical relational databases, Statistical Journal of the IAOS, 32 (2016), 245-255.
[34] W. S. Jian, Fuzzy forecasting based on fuzzy logical relationships, fuzzy trend logical relationship groups, k-means clustering algorithm, similarity measures and particle swarm optimization techniques, Information Sciences, 327 (2016), 272-287.
[35] C. T. Kelley, Iterative methods for optimization, SIAM, (1999).
[36] K. H. Lee, First course on fuzzy theory and applications, Springer-Verlag, Berlin, 2005.
[37] R. Li, Water quality forecasting of Haihe River based on improved fuzzy time series model, Desalination and Water Treatment, 106 (2018), 285-291.
[38] R. A. Maronna, D. R. Martin, V. J. Yohai, Robust statistics: Theory and methods, New York, John Wiley and Sons, 2006.
[39] T. C. Mills, Applied time series analysis: A practical guide to modelling and forecasting, London: Academic Press, 2019.
[40] D. C. Montgomery, Introduction to statistical quality control, John Wiley and Sons, New York, 2009.
[41] V. Novák, Detection of structural breaks in time series using fuzzy techniques, The International Journal of Fuzzy Logic and Intelligent Systems, 18 (2018), 1-12.
[42] T. O. Olatayo, A. I. Taiwo, Statistical modelling and prediction of rainfall time series data, Global Journal of Computer Science and Technology, 14 (2014), 1-9.
[43] T. T. H. Phan, A. Bigand, E. P. Caillault, A new fuzzy logic-based similarity measure applied to large gap imputation for uncorrelated multivariate time series, Applied Computational Intelligence and Soft Computing, (2018), 1-15.
[44] W. Qiu, C. Zhang, Z. Ping, Generalized fuzzy time series forecasting model enhanced with particle swarm optimization, International Journal of u- and e- Service, Science and Technology, 8 (2015), 129-140.
[45] W. Qiu, P. Zhang, Y. Wang, Fuzzy time series forecasting model based on automatic clustering techniques and generalized fuzzy logical relationship, Mathematical Problems in Engineering, (2015), 1-8.
[46] N. F. Rahim, M. Othman, R. Sokkalingam, E. A. Kadir, Forecasting crude palm oil prices using fuzzy rule-based time series method, IEEE Access, 6 (2018), 32216-32224.
[47] P. M. Robinson, Root-n-consistent semi-parametric regression, Econometrica, 56 (1988), 931-954.
[48] A. Roy, A novel multivariate fuzzy time series based forecasting algorithm incorporating the effect of clustering on prediction, Soft Computing, 20 (2016), 1991-2019.
[49] S. Rubinstein, A. Goor, A. Rotshtein, Time series forecasting of crude oil consumption using neuro-fuzzy inference, Journal of Industrial and Intelligent Information, 3 (2015), 84-90.
[50] K. Sabzi, T. Allahviranloo, S. Abbasbandy, A fuzzy generalized power series method under generalized Hukuhara differentiability for solving fuzzy Legendre differential equation, Soft Computing, 24 (2020), 8763-8779.
[51] A. Sachdeva, V. Sharma, A survey on stock forecasting model based on combined fuzzy genetic algorithm, International Journal of Advanced Research in Computer Science and Software Engineering, 5 (2015), 397-401.
[52] H. J. Sadaei, R. Enayatifar, F. G. Guimaraes, M. Mahmud, Z. A. Alzamil, Combining ARFIMA models and fuzzy time series for the forecast of long memory time series, Neurocomputing, 175 (2016), 782-796.
[53] S. Sakhuja, V. Jain, S. Kumar, C. Chandra, S. K. Ghildayal, Genetic algorithm based fuzzy time series tourism demand forecast model, Industrial Management and Data Systems, 116 (2016), 1-24.
[54] S. Sharma, M. Chouhan, A review: Fuzzy time series model for forecasting, International Journal of Advanced Science and Technology, 2 (2014), 32-35.
[55] B. W. Silverman, Density estimation, Chapman and Hall, 1986.
[56] P. Singh, A brief review of modeling approaches based on fuzzy time series, International Journal of Machine Learning and Cybernetics, 8 (2017), 397-420.
[57] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems, 161 (2010), 1564-1584.
[58] B. Sun, H. Guo, H. R. Karimi, Y. Ge, S. Xiong, Prediction of stock index futures prices based on fuzzy sets and multivariate fuzzy time series, Neurocomputing, 151 (2015), 1528-1536.
[59] F. M. Tseng, G. H. Tzeng, A fuzzy seasonal ARIMA model for forecasting, Fuzzy Sets and Systems, 126 (2002), 367-376.
[60] Y. Wang, Y. Lei, X. Fan, Y. Wang, Intuitionistic fuzzy time series forecasting model based on intuitionistic fuzzy reasoning, Mathematical Problems in Engineering, (2016), 1-12.
[61] L. Wasserman, All of nonparametric statistics, New York, Springer, 2007.
[62] U. Yolcu, A new approach based on optimation of ratio for seasonal fuzzy time series, Iranian Journal of Fuzzy Systems, 13 (2016), 19-36.
[63] U. Yolcu, E. Bas, The forecasting of labour force participation and the unemployment rate in poland and turkey using fuzzy time series methods, Comparative Economic Research, 19 (2016), 5-25.
[64] O. C. Yolcu, H. K. Lam, A combined robust fuzzy time series method for prediction of time series, Neurocomputing, 247 (2017), 87-101.
[65] R. Zarei, M. Gh. Akbari, J. Chachi, Modeling autoregressive fuzzy time series data based on semiparametricmethods, Soft Computing, 24(10) (2020), 7295-7304.
[66] K. Zhang, Z. Li, H. F. Wang, H. X. Wang, Fuzzy time series prediction model and application based on fuzzy inverse, International Journal of Signal Processing, Image Processing and Pattern Recognition, 8 (2015), 121-128.