A New Approach for Time Series Prediction: Fuzzy Regression Network Functions

Abstract

The fuzzy regression functions (FRFs) constructs a comprehensive model by combining a series of linear functions based on the inputs. However, the relationship between input and output is not always purely linear. The approach presents novel FRFs with nonlinear structures based on neural networks, combining the strengths of both computational models and fuzzy logic. The proposed model generates membership values by fuzzifying real-valued time series observations, utilizing the fuzzy C-means clustering algorithm. Inputs are then created from the real-valued lagged observations and transformed membership values. A set of feed-forward neural networks, corresponding to the number of fuzzy sets, produces outputs as nonlinear functions of the inputs. These outputs are combined based on the membership values, representing the degree to which each time point belongs to the respective fuzzy sets, to generate the final predictions. The proposed prediction model is referred to as Fuzzy Regression Network Functions (FRNFs). The prediction performance of FRNFs is investigated across several criteria by implementing it in various real-world time series datasets.

Keywords

• Fuzzy regression network functions
• Artificial neural networks
• Type-1 fuzzy functions
• Time series prediction

Supporting Institution

Marmara University Scientific Research Projects Coordinatorship

Project Number

FYL-2022- 10538

Ethical Statement

This study has been supported, by Marmara University Scientific Research Projects Coordinatorship, as part of the Master Science Thesis Projects (FYL-2022- 10538).

Thanks

This study has been supported, by Marmara University Scientific Research Projects Coordinatorship, as part of the Master Science Thesis Projects (FYL-2022- 10538).

Zaman Serisi Tahmini İçin Yeni Bir Yaklaşım: Bulanık Regresyon Ağı Fonksiyonları

Abstract

The fuzzy regression functions (FRFs) constructs a comprehensive model by combining a series of linear functions based on the inputs. However, the relationship between input and output is not always purely linear. The approach presents novel FRFs with nonlinear structures based on neural networks, combining the strengths of both computational models and fuzzy logic. The proposed model generates membership values by fuzzifying real-valued time series observations, utilizing the fuzzy C-means clustering algorithm. Inputs are then created from the real-valued lagged observations and transformed membership values. A set of feed-forward neural networks, corresponding to the number of fuzzy sets, produces outputs as nonlinear functions of the inputs. These outputs are combined based on the membership values, representing the degree to which each time point belongs to the respective fuzzy sets, to generate the final predictions. The proposed prediction model is referred to as Fuzzy Regression Network Functions (FRNFs). The prediction performance of FRNFs is investigated across several criteria by implementing it in various real-world time series datasets.

Keywords

• Bulanık regresyon ağı fonksiyonları
• Yapay sinir ağları
• Tip-1 bulanık fonksiyonlar
• Zaman serisi tahmini

Supporting Institution

Marmara University Scientific Research Projects Coordinatorship

Project Number

FYL-2022- 10538

Ethical Statement

This study has been supported, by Marmara University Scientific Research Projects Coordinatorship, as part of the Master Science Thesis Projects (FYL-2022- 10538).

Thanks

This study has been supported, by Marmara University Scientific Research Projects Coordinatorship, as part of the Master Science Thesis Projects (FYL-2022- 10538).

References

1. Q. Song, B.S. (1993). Chissom, Fuzzy time series and its models, Fuzzy Sets Syst 54. https://doi.org/10.1016/0165-0114(93)90372-O.
2. L.A. Zadeh, (1965). Fuzzy sets, Information and Control 8. https://doi.org/10.1016/S0019-9958(65)90241-X.
3. Q. Song, B.S. Chissom, (1993). Forecasting enrollments with fuzzy time series - Part I, Fuzzy Sets Syst 54. https://doi.org/10.1016/0165-0114(93)90355-L.
4. Q. Song, B.S. Chissom, (1994). Forecasting enrollments with fuzzy time series - part II, Fuzzy Sets Syst 62. https://doi.org/10.1016/0165-0114(94)90067-1.
5. S.M. Chen, (1996). Forecasting enrollments based on fuzzy time series, Fuzzy Sets Syst 81. https://doi.org/10.1016/0165-0114(95)00220-0.
6. S.M. Chen, (2002). Forecasting enrollments based on high-order fuzzy time series, Cybern Syst 33. https://doi.org/10.1080/019697202753306479.
7. K. Huarng, (2001). Effective lengths of intervals to improve forecasting in fuzzy time series, Fuzzy Sets Syst 123. https://doi.org/10.1016/S0165-0114(00)00057-9.
8. E. Egrioglu, C.H. Aladag, U. Yolcu, V.R. Uslu, M.A. Basaran, (2010). Finding an optimal interval length in high order fuzzy time series, Expert Syst Appl 37. https://doi.org/10.1016/j.eswa.2009.12.006.

9. E. Egrioglu, C.H. Aladag, M.A. Basaran, U. Yolcu, V.R. Uslu, (2011). A new approach based on the optimization of the length of intervals in fuzzy time series, Journal of Intelligent and Fuzzy Systems 22. https://doi.org/10.3233/IFS-2010-0470.
10. K. Huarng, T.H.K. Yu, (2006). Ratio-based lengths of intervals to improve fuzzy time series forecasting, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 36. https://doi.org/10.1109/TSMCB.2005.857093.
11. U. Yolcu, E. Egrioglu, V.R. Uslu, M.A. Basaran, C.H. Aladag, (2009). A new approach for determining the length of intervals for fuzzy time series, Applied Soft Computing Journal 9. https://doi.org/10.1016/j.asoc.2008.09.002.
12. S. Panigrahi, H.S. Behera, (2020). A study on leading machine learning techniques for high order fuzzy time series forecasting, Eng Appl Artif Intell 87 https://doi.org/10.1016/j.engappai.2019.103245.
13. L.W. Lee, L.H. Wang, S.M. Chen, (2007). Temperature prediction and TAIFEX forecasting based on fuzzy logical relationships and genetic algorithms, Expert Syst Appl 33. https://doi.org/10.1016/j.eswa.2006.05.015
14. L.W. Lee, L.H. Wang, S.M. Chen, (2008). Temperature prediction and TAIFEX forecasting based on high-order fuzzy logical relationships and genetic simulated annealing techniques, Expert Syst Appl 34. https://doi.org/10.1016/j.eswa.2006.09.007.
15. I.H. Kuo, S.J. Horng, T.W. Kao, T.L. Lin, C.L. Lee, Y. Pan, (2009). An improved method for forecasting enrollments based on fuzzy time series and particle swarm optimization, Expert Syst Appl 36. https://doi.org/10.1016/j.eswa.2008.07.043.
16. I.H. Kuo, S.J. Horng, Y.H. Chen, R.S. Run, T.W. Kao, R.J. Chen, J.L. Lai, T.L. Lin, (2010). Forecasting TAIFEX based on fuzzy time series and particle swarm optimization, Expert Syst Appl 37. https://doi.org/10.1016/j.eswa.2009.06.102.
17. S. Davari, M.H.F. Zarandi, I.B. Turksen, (2009). An improved fuzzy time series forecasting model based on particle swarm intervalization, in: Annual Conference of the North American Fuzzy Information Processing Society - NAFIPS, https://doi.org/10.1109/NAFIPS.2009.5156420
18. L.Y. Hsu, S.J. Horng, T.W. Kao, Y.H. Chen, R.S. Run, R.J. Chen, J.L. Lai, I.H. Kuo, (2010). Temperature prediction and TAIFEX forecasting based on fuzzy relationships and MTPSO techniques, Expert Syst Appl 37. https://doi.org/10.1016/j.eswa.2009.09.015.
19. C.H. Aladag, (2013). Using multiplicative neuron model to establish fuzzy logic relationships, in: Expert Syst Appl, https://doi.org/10.1016/j.eswa.2012.05.039.
20. O. Cagcag Yolcu, H.K. Lam, (2017). A combined robust fuzzy time series method for prediction of time series, Neurocomputing 247. https://doi.org/10.1016/j.neucom.2017.03.037.
21. U. Yolcu, O. Cagcag, C.H. Aladag, E. Egrioglu, (2014). An enhanced fuzzy time series forecasting method based on artificial bee colony, Journal of Intelligent and Fuzzy Systems 26. https://doi.org/10.3233/IFS-130933.
22. Q. Cai, D. Zhang, W. Zheng, S.C.H. Leung, (2015). A new fuzzy time series forecasting model combined with ant colony optimization and auto-regression, Knowl Based Syst 74. https://doi.org/10.1016/j.knosys.2014.11.003.
23. M. Jiang, L. Jia, Z. Chen, W. Chen, (2022). The two-stage machine learning ensemble models for stock price prediction by combining mode decomposition, extreme learning machine and improved harmony search algorithm, Ann Oper Res 309. https://doi.org/10.1007/s10479-020-03690-w.
24. C.H. Cheng, G.W. Cheng, J.W. Wang, (2008). Multi-attribute fuzzy time series method based on fuzzy clustering, Expert Syst Appl 34. https://doi.org/10.1016/j.eswa.2006.12.013.
25. S.T. Li, Y.C. Cheng, S.Y. Lin, (2008). A FCM-based deterministic forecasting model for fuzzy time series, Computers and Mathematics with Applications 56. https://doi.org/10.1016/j.camwa.2008.07.033.
26. F. Alpaslan, O. Cagcag Yolcu, (2012). A Seasonal Fuzzy Time Series Forecasting Method Based On Gustafson-Kessel Fuzzy Clustering, Journal of Social and Economic Statistics 1 1–13.
27. E. Egrioglu, C.H. Aladag, U. Yolcu, (2013). Fuzzy time series forecasting with a novel hybrid approach combining fuzzy c-means and neural networks, in: Expert Syst Appl, https://doi.org/10.1016/j.eswa.2012.05.040.
28. L.Y. Wei, C.H. Cheng, H.H. Wu, (2014). A hybrid ANFIS based on n-period moving average model to forecast TAIEX stock, Applied Soft Computing Journal 19. https://doi.org/10.1016/j.asoc.2014.01.022.
29. S.H. Cheng, S.M. Chen, W.S. Jian, (2016). Fuzzy time series forecasting based on fuzzy logical relationships and similarity measures, Inf Sci (N Y) 327. https://doi.org/10.1016/j.ins.2015.08.024.
30. B.Q. Sun, H. Guo, H. Reza Karimi, Y. Ge, S. Xiong, (2015). Prediction of stock index futures prices based on fuzzy sets and multivariate fuzzy time series, Neurocomputing 151. https://doi.org/10.1016/j.neucom.2014.09.018.
31. W. Wang, X. Liu, (2015). Fuzzy forecasting based on automatic clustering and axiomatic fuzzy set classification, Inf Sci (N Y) 294. https://doi.org/10.1016/j.ins.2014.09.027.
32. O. Cagcag Yolcu, (2018). F. Alpaslan, Prediction of TAIEX based on hybrid fuzzy time series model with single optimization process, Applied Soft Computing Journal 66. https://doi.org/10.1016/j.asoc.2018.02.007.
33. J. Sullivan, W.H. Woodall, (1994). A comparison of fuzzy forecasting and Markov modeling, Fuzzy Sets Syst 64. https://doi.org/10.1016/0165-0114(94)90152-X.
34. C. Kocak, (2013). First-order ARMA type fuzzy time series method based on fuzzy logic relation tables, Math Probl Eng 2013.https://doi.org/10.1155/2013/769125.
35. C. Kocak, (2017). ARMA(p,q) type high order fuzzy time series forecast method based on fuzzy logic relations, Applied Soft Computing Journal 58. https://doi.org/10.1016/j.asoc.2017.04.021.
36. K. Huarng, T.H.K. Yu, (2006). The application of neural networks to forecast fuzzy time series, Physica A: Statistical Mechanics and Its Applications 363. https://doi.org/10.1016/j.physa.2005.08.014.
37. C.H. Aladag, U. Yolcu, E. Egrioglu, (2010). A high order fuzzy time series forecasting model based on adaptive expectation and artificial neural networks, Math Comput Simul 81. https://doi.org/10.1016/j.matcom.2010.09.011.
38. C. Kocak, A.Z. Dalar, O. Cagcag Yolcu, E. Bas, E. Egrioglu, (2020). A new fuzzy time series method based on an ARMA-type recurrent Pi-Sigma artificial neural network, Soft Comput 24. https://doi.org/10.1007/s00500-019-04506-1.
39. S.M. Chen, S.W. Chen, (2015). Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of fuzzy logical relationships, IEEE Trans Cybern 45. https://doi.org/10.1109/TCYB.2014.2326888.
40. S.N. Arslan, O. Cagcag Yolcu, (2022). A hybrid sigma-pi neural network for combined intuitionistic fuzzy time series prediction model, Neural Comput Appl 34. https://doi.org/10.1007/s00521-022-07138-z.
41. O. Cagcag Yolcu, (2013). A hybrid fuzzy time series approach based on fuzzy clustering and artificial neural network with single multiplicative neuron model, Math Probl Eng 2013. https://doi.org/10.1155/2013/560472.
42. T.H.K. Yu, K.H. Huarng, (2008). A bivariate fuzzy time series model to forecast the TAIEX, Expert Syst Appl 34. https://doi.org/10.1016/j.eswa.2007.05.016.
43. F. Alpaslan, O. Cagcag, C.H. Aladag, U. Yolcu, E. Egrioglu, (2012). A novel seasonal fuzzy time series method, Hacettepe Journal of Mathematics and Statistics 41.
44. U. Yolcu, C.H. Aladag, E. Egrioglu, V.R. Uslu, (2013). Time-series forecasting with a novel fuzzy time-series approach: An example for Istanbul stock market, J Stat Comput Simul 83. https://doi.org/10.1080/00949655.2011.630000.
45. S. ARSLAN, (2023). Gated recurrent unit network-based fuzzy time series forecasting model, Afyon Kocatepe University Journal of Sciences and Engineering 23. https://doi.org/10.35414/akufemubid.1175297.
46. S.M. Chen, N.Y. Chung, (2006). Forecasting enrollments using high-order fuzzy time series and genetic algorithms, International Journal of Intelligent Systems 21. https://doi.org/10.1002/int.20145.
47. C.H. Aladag, U. Yolcu, E. Egrioglu, A.Z. Dalar, (2012). A new time invariant fuzzy time series forecasting method based on particle swarm optimization, Applied Soft Computing Journal 12. https://doi.org/10.1016/j.asoc.2012.05.002.
48. B. Sarıca, E. Eğrioğlu, B. Aşıkgil, (2018). A new hybrid method for time series forecasting: AR–ANFIS, Neural Comput Appl 29. https://doi.org/10.1007/s00521-016-2475-5.
49. J.P.S. Catalão, H.M.I. Pousinho, V.M.F. Mendes, (2011). Hybrid wavelet-PSO-ANFIS approach for short-term electricity prices forecasting, IEEE Transactions on Power Systems 26. https://doi.org/10.1109/TPWRS.2010.2049385.
50. B.R. Chang, (2008). Resolving the forecasting problems of overshoot and volatility clustering using ANFIS coupling nonlinear heteroscedasticity with quantum tuning, Fuzzy Sets Syst 159. https://doi.org/10.1016/j.fss.2008.04.003.
51. C.H. Cheng, L.Y. Wei, Y.S. Chen, (2009). Fusion ANFIS models based on multi-stock volatility causality for TAIEX forecasting, Neurocomputing 72. https://doi.org/10.1016/j.neucom.2008.09.027.
52. C.H. Cheng, L.Y. Wei, J.W. Liu, T.L. Chen, (2013). OWA-based ANFIS model for TAIEX forecasting, Econ Model 30. https://doi.org/10.1016/j.econmod.2012.09.047.
53. Y.C. Ho, C.T. Tsai, (2011). Comparing ANFIS and SEM in linear and nonlinear forecasting of new product development performance, Expert Syst Appl 38. https://doi.org/10.1016/j.eswa.2010.11.095.
54. H.M.I. Pousinho, V.M.F. Mendes, J.P.S. Catalão, (2012). Short-term electricity prices forecasting in a competitive market by a hybrid PSO-ANFIS approach, International Journal of Electrical Power and Energy Systems 39. https://doi.org/10.1016/j.ijepes.2012.01.001.
55. I.B. Türkşen, (2008). Fuzzy functions with LSE, Applied Soft Computing Journal 8. https://doi.org/10.1016/j.asoc.2007.12.004.
56. C.H. Aladag, I.B. Turksen, A.Z. Dalar, E. Egrioglu, U. Yolcu, (2014). Application of Type-1 Fuzzy Functions Approach for Time Series Forecasting, TJFS: Turkish Journal of Fuzzy Systems An Official Journal of Turkish Fuzzy Systems Association 5.
57. C.H. Aladag, U. Yolcu, E. Egrioglu, I.B. Turksen, (2016).Type-1 fuzzy time series function method based on binary particle swarm optimisation, in: International Journal of Data Analysis Techniques and Strategies, https://doi.org/10.1504/IJDATS.2016.075970.
58. N. Tak, A.A. Evren, M. Tez, E. Egrioglu, (2018). Recurrent type-1 fuzzy functions approach for time series forecasting, Applied Intelligence 48. https://doi.org/10.1007/s10489-017-0962-8.
59. N. Tak, (2021). Forecast combination with meta possibilistic fuzzy functions, Inf Sci (N Y) 560. https://doi.org/10.1016/j.ins.2021.01.024.
60. S. Goudarzi, M.B. Khodabakhshi, M.H. Moradi, (2016). Interactively recurrent fuzzy functions with multi objective learning and its application to chaotic time series prediction, Journal of Intelligent and Fuzzy Systems 30. https://doi.org/10.3233/IFS-151839.
61. M.H.F. Zarandi, M. Zarinbal, N. Ghanbari, I.B. Turksen, (2013). A new fuzzy functions model tuned by hybridizing imperialist competitive algorithm and simulated annealing. Application: Stock price prediction, Inf Sci (N Y) 222. https://doi.org/10.1016/j.ins.2012.08.002.
62. S. YALAZ, A. ATAY, (2016). Fuzzy Linear Regression for the Time Series Data which is Fuzzified with SMRGT Method, SDÜ Fen Bilimleri Enstitüsü Dergisi 20. https://doi.org/10.19113/sdufbed.49849.
63. J.C. Bezdek, (1981). Pattern Recognition with Fuzzy Objective Function Algorithms, https://doi.org/10.1007/978-1-4757-0450-1.
64. I.B. Türkşen, (2008). Fuzzy functions with LSE, Applied Soft Computing Journal 8. https://doi.org/10.1016/j.asoc.2007.12.004.
65. M. Kirisci, O. Cagcag Yolcu, (2022). A New CNN-Based Model for Financial Time Series: TAIEX and FTSE Stocks Forecasting, Neural Process Lett 54. https://doi.org/10.1007/s11063-022-10767-z.
66. E. Bas, E. Egrioglu, U. Yolcu, C. Grosan, (2019). Type 1 fuzzy function approach based on ridge regression for forecasting, Granular Computing 4. https://doi.org/10.1007/s41066-018-0115-4.
67. N. Tak, D. İnan, (2022). Type-1 fuzzy forecasting functions with elastic net regularization, Expert Syst Appl 199. https://doi.org/10.1016/j.eswa.2022.116916.
68. K.H. Huarng, T.H.K. Yu, Y.W. Hsu, (2007). A multivariate heuristic model for fuzzy time-series forecasting, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 37. https://doi.org/10.1109/TSMCB.2006.890303.
69. S.M. Chen, Y.C. Chang, (2010). Multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques, Inf Sci (N Y) 180. https://doi.org/10.1016/j.ins.2010.08.026.
70. S.M. Chen, C.D. Chen, (2011). TAIEX forecasting based on fuzzy time series and fuzzy variation groups, IEEE Transactions on Fuzzy Systems 19. https://doi.org/10.1109/TFUZZ.2010.2073712.
71. S.M. Chen, G.M.T. Manalu, J.S. Pan, H.C. Liu, (2013). Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and particle swarm optimization techniques, IEEE Trans Cybern 43. https://doi.org/10.1109/TSMCB.2012.2223815.
72. Y.S. Chen, C.H. Cheng, W.L. Tsai, (2014). Modeling fitting-function-based fuzzy time series patterns for evolving stock index forecasting, Applied Intelligence 41. https://doi.org/10.1007/s10489-014-0520-6.
73. E. Bas, E. Egrioglu, C.H. Aladag, U. Yolcu, (2015). Fuzzy-time-series network used to forecast linear and nonlinear time series, Applied Intelligence 43. https://doi.org/10.1007/s10489-015-0647-0.
74. Y.S. Chen, C.H. Cheng, C.L. Chiu, S.T. Huang, (2016). A study of ANFIS-based multi-factor time series models for forecasting stock index, Applied Intelligence 45. https://doi.org/10.1007/s10489-016-0760-8.
75. S.M. Chen, W.S. Jian, (2017). Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups, similarity measures and PSO techniques, Inf Sci (N Y) 391–392. https://doi.org/10.1016/j.ins.2016.11.004.
76. S.M. Chen, B.D.H. Phuong, (2017). Fuzzy time series forecasting based on optimal partitions of intervals and optimal weighting vectors, Knowl Based Syst 118. https://doi.org/10.1016/j.knosys.2016.11.019.
77. J.S.R. Jang, (1993). ANFIS: Adaptive-Network-Based Fuzzy Inference System, IEEE Trans Syst Man Cybern 23. https://doi.org/10.1109/21.256541.
78. E. Egrioglu, C.H. Aladag, U. Yolcu, E. Bas, (2015). A New Adaptive Network Based Fuzzy Inference System for Time Series Forecasting, Aloy Journal of Soft Computing and Applications 2.
79. O. Cagcag Yolcu, E. Bas, E. Egrioglu, U. Yolcu, (2020). A new intuitionistic fuzzy functions approach based on hesitation margin for time-series prediction, Soft Comput. https://doi.org/10.1007/s00500-019-04432-2.
80. N. Tak, (2020). Type-1 recurrent intuitionistic fuzzy functions for forecasting, Expert Syst Appl 140. https://doi.org/10.1016/j.eswa.2019.112913.
81. K. Huarng, (2001). Heuristic models of fuzzy time series for forecasting, Fuzzy Sets Syst 123. https://doi.org/10.1016/S0165-0114(00)00093-2.
82. H.K. Yu, (2005). Weighted fuzzy time series models for TAIEX forecasting, Physica A: Statistical Mechanics and Its Applications 349. https://doi.org/10.1016/j.physa.2004.11.006.
83. S.M. Chen, K. Tanuwijaya, (2011). Fuzzy forecasting based on high-order fuzzy logical relationships and automatic clustering techniques, Expert Syst Appl 38. https://doi.org/10.1016/j.eswa.2011.06.019.
84. C.H. Aladag, U. Yolcu, E. Egrioglu, E. Bas, (2014). Fuzzy lagged variable selection in fuzzy time series with genetic algorithms, Applied Soft Computing Journal 22. https://doi.org/10.1016/j.asoc.2014.03.028.
85. S. Askari, N. Montazerin, M.H.F. Zarandi, (2015). A clustering based forecasting algorithm for multivariable fuzzy time series using linear combinations of independent variables, Applied Soft Computing Journal 35. https://doi.org/10.1016/j.asoc.2015.06.028.
86. O. Cagcag Yolcu, E. Bas, E. Egrioglu, U. Yolcu, (2020). A new intuitionistic fuzzy functions approach based on hesitation margin for time-series prediction, Soft Comput 24. https://doi.org/10.1007/s00500-019-04432-2.
87. H.J. Sadaei, R. Enayatifar, M.H. Lee, M. Mahmud, (2016). A hybrid model based on differential fuzzy logic relationships and imperialist competitive algorithm for stock market forecasting, Applied Soft Computing Journal 40. https://doi.org/10.1016/j.asoc.2015.11.026.
88. F. Ye, L. Zhang, D. Zhang, H. Fujita, Z. Gong, (2016). A novel forecasting method based on multi-order fuzzy time series and technical analysis, Inf Sci (N Y) 367–368. https://doi.org/10.1016/j.ins.2016.05.038.
89. Y. Wan, Y.W. Si, (2017). Adaptive neuro fuzzy inference system for chart pattern matching in financial time series, Applied Soft Computing Journal 57. https://doi.org/10.1016/j.asoc.2017.03.023.
90. C.H. Cheng, J.H. Yang, (2018). Fuzzy time-series model based on rough set rule induction for forecasting stock price, Neurocomputing 302. https://doi.org/10.1016/j.neucom.2018.04.014.
91. H. Wu, H. Long, J. Jiang, (2019). Handling forecasting problems based on fuzzy time series model and model error learning, Applied Soft Computing Journal 78. https://doi.org/10.1016/j.asoc.2019.02.021.
92. G.E.P., Box, G.M., and Jenkins, G. Reinsel, (2008). Time Series Analysis, Forecasting and Control, 4th edition. Wiley, Prentice-Hall, Englewood Cliffs.
93. R.G. Brown, (1957). Exponential Smoothing for predicting demand, in: Oper Res.
94. D.E. Rumelhart, G.E. Hinton, R.J. Williams, (1986). Learning representations by back-propagating errors, Nature 323. https://doi.org/10.1038/323533a0.