A novel Kumaraswamy interval type-2 TSK fuzzy logic system for subway passenger demand prediction

Document Type : Research Paper


Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran


Fuzzy logic systems (FLSs) are proper tools for learning and predicting of real-world problems. Type-2 fuzzy sets are developments of the conventional type-1 fuzzy sets which are applied for prediction problems with uncertainty. Interval type-2 fuzzy logic system (IT2 FLSs) is the most wildly used type-2 FLS due to its efficiency and simplicity. Passenger demand prediction has a crucial role in the public transportation sector. Because of the nonlinearity and instability of the passenger arrivals prediction, IT2 FLS can be an appropriate method for solving this problem. In this paper, we develop a fuzzy logic system named KIT2 TSK for passenger arrivals prediction in subway stations. In our proposed model, we utilize the Kumaraswamy distribution in the construction of an IT2 TSK FLS. Furthermore, we develop a new input selection measure that applies the Schweizer–Sklar t-conorm operator in the variable selection process. The flexibility of the Kumaraswamy distribution leads to the ability to approximate several distributions using the same equation by different values for its shape parameters. Utilizing this property, we adopt our proposed model for passenger arrivals prediction of one line of the Tehran subway system as a case study. Moreover, to see the results on unusual days, passenger demand on public holidays, weekends, and special events are also taken into account. The results demonstrate that our proposed methodology has better performance in the hourly prediction of passenger arrivals compared to the benchmarks. The results for the chaotic Mackey-Glass problem also show the superiority of our proposed model.


[1] T. Adetiloye, A. Awasthi, Predicting short-term congested traffic flow on urban motorway networks, Handbook of Neural Computation, Chapter 8, (2017), 145-165.
[2] C. Alsina, E. Trillas, L. Valverde, On some logical connectives for fuzzy set theory, Journal of Mathematical Analysis and Applications, 93 (1983), 15-26.
[3] S. Anvari, S. Tuna, M. Canci, M. Turkay, Automated Box-Jenkins forecasting tool with an application for passenger demand in urban rail systems, Journal of Advanced Transportation, 50(1) (2015), 25-49.
[4] M. Ashrafi, D. K. Prasad, C. Quek, IT2-GSETSK: An evolving interval Type-II TSK fuzzy neural system for online modeling of noisy data, Neurocomputing, 407 (2020), 1-11.
[5] A. Aziz Khater, A. M. El-Nagar, M. El-Bardini, N. M. El-Rabaie, Online learning of an interval type-2 TSK fuzzy logic controller for nonlinear systems, Journal of the Franklin Institute, 356(16) (2019), 9254-9285.
[6] F. M. Bayer, D. M. Bayer, G. Pumi, Kumaraswamy autoregressive moving average models for double bounded environmental data, Journal of Hydrology, 555 (2017), 385-396.
[7] M. B. Begian, W. W. Melek, J. M. Mendel, Stability analysis of type-2 fuzzy systems, IEEE International Conference on Fuzzy Systems, IEEE, (2008), 947-953.
[8] P. C. Chang, C. Y. Fan, A hybrid system integrating a wavelet and TSK fuzzy rules for stock price forecasting, IEEE Transactions on Systems, Man, and Cybern Part C Appl Rev., 38(6) (2008), 802-815.
[9] M. Y. Chen, D. Linkens, Rule-base self-generation and simplification for data-driven fuzzy models, Fuzzy Sets and Systems, 142(2) (2004), 243-265.
[10] Y. P. Chi, Y. Han, L. Rui, Y. P. Wei, Study of bus incident prediction based on dynamic fuzzy-neural network, IEEE 2010 International Conference on E-Product E-Service and E-Entertainment Henan China, 7-9 Nov, (2010), 1-6.
[11] O. Cosgun, Y. Ekinci, S. Yank, Fuzzy rule-based demand forecasting for dynamic pricing of a maritime company, Knowledge-Based Systems, 70 (2014), 88-96.
[12] S. P. Day, M. R. Davenport, Continuous-time temporal backpropagation with adaptable time delays, IEEE Transactions on Neural Networks, 4(2) (1993), 348-354.
[13] Z. Deng, C. Kup-Sze, C. Longbing, W. Shitong, T2FELA: Type-2 fuzzy extreme learning algorithm for fast training of interval type-2 TSK fuzzy logic system, IEEE Transactions on Neural Networks and Learning Systems, 25(4) (2014), 664-676.
[14] S. Dey, J. Mazucheli, S. Nadarajah, Kumaraswamy distribution: Different methods of estimation, Journal of Computational and Applied Mathematics, 37(2) (2017), DOI: 10.1007/s40314-017-0441-1.
[15] F. Dou, L. Jia, L. Wang, J. Xu, Y. Huang, Fuzzy temporal logic based railway passenger flow forecast model, Computational Intelligence and Neuroscience, 2014 (2014), 9 pages, 950371, DOI:10.1155/2014/950371.
[16] M. S. El-Deen, G. Al-Dayian, A. El-Helbawy, Statistical inference for Kumaraswamy distribution based on generalized order statistics with applications, British Journal of Mathematics and Computer Science, 4(12) (2014), 1710-1743.
[17] I. Eyoh, R. John, G. De Maere, Time series forecasting with interval type-2 intuitionistic fuzzy logic systems, IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Naples, (2017), 1-6.
[18] F. A. Gers, D. Eck, J. Schmidhuber, Applying LSTM to time series predictable through time-window approaches, Neural Nets WIRN Vietri-01, R. Tagliaferri and M. Marinaro, Eds. London, U.K.: Springer, (2002), 193-200.
[19] Y. Ghozzi, N. Baklouti, H. Hagras, M. Ben Ayed, A. Alimi, Interval type-2 beta fuzzy near set based approach to content based image retrieval, IEEE Transactions on Fuzzy Systems, (2021), DOI: 10.1109/TFUZZ.2021.3049900.
[20] I. Gosh, G. Hamedani, The Gamma-Kumaraswamy distribution: An A iternative to gamma distribution, Communication in Statistics-Theory and Methods, 47(9) (2015), DOI:10.1080/03610926.2015.1122055.
[21] Y. He, Y. Zhao, K. L. Tsui, An adapted geographically weighted LASSO (Ada-GWL) model for predicting subway ridership, Transportation, 48(3) (2021), 1185-1216.
[22] G. Hesamian, F. Torkian, M. Yarmohammadi, A fuzzy non-parametric time series model based on fuzzy data, Iranian Journal of Fuzzy Systems, 19(1) (2022), 61-72.
[24] S. Huang, M. Chen, Constructing optimized interval type-2 TSK neuro-fuzzy systems with noise reduction property by quantum inspired BFA, Neurocomputing, 173(3) (2016), 1839-1850.
[25] H. T. Huynh, V. S. Lai, I. Soumare, Stochastic simulation and applications in finance with MATLAB programs, Wiley Finance, (2011), 60-61.
[26] J. S. R. Jang, ANFIS adaptive-network-based fuzzy inference system, IEEE Transactions on Systems Man and Cybernetics, 23(3) (1993), 665-685.
[27] Z. Javanshiri, A. Habibi Rad, N. R. Arghami, Exp-Kumaraswamy distributions: Some properties and applications, Journal of Sciences, Islamic Republic of Iran, 26(1) (2015), 57-69.
[28] W. Jiang, Z. Ma, H. N. Koutsopoulos, Deep learning for short-term origin-destination passenger flow prediction under partial observability in urban railway systems, Neural Computing and Applications, (2022), DOI: 10.1007/s00521- 021-06669-1.
[29] M. C. Jones, Kumaraswamys distribution: A beta-type distribution with some tractability advantages, Statistical Methodology, 6(1) (2009), 70-81.
[30] D. R. Keshwani, D. D. Jones, G. E. Meyer, M. B. Rhonda, Rule-based Mamdani-type fuzzy modeling of skin permeability, Applied Soft Computing, 8(1) (2008), 285-294.
[31] G. J. Klir, B. Yuan, Fuzzy sets and fuzzy logic: Theory and applications, Prentice-Hall, Inc, (1995), 78-82.
[32] P. Kumaraswamy, A generalized probability density function for double-bounded random processes, Journal of Hydrology, 46(1-2) (1980), 79-88.
[33] W. H. Lai, C. Tsai, Fuzzy rule-based analysis of firms technology transfer in Taiwans machinery industry, Expert Systems with Application, 36(10) (2009), 12012-12022.
[34] J. Leski, TSK-fuzzy modeling based on ϵ-insensitive learning, IEEE Transactions on Fuzzy Systems, 13(2) (2005), 181-193.
[35] R. Li, C. Jiang, F. Zhu, X. Chen, Traffic flow data forecasting based on interval type-2 fuzzy sets theory, IEEE/CAA Journal of Automatica Sinica, 3(2) (2016), 141-148.
[36] L. Li, W. Lin, H. Liu, Type-2 fuzzy logic approach for short-term traffic forecasting, IEEE Proceedings-Intelligent Transport Systems, 153(1) (2006), 33-40.
[37] Y. Li, X. Wang, S. Sun, X. Ma, G. Lu, Forecasting short-term subway passenger flow under special events scenarios using multiscale radial basis function networks, Transportation Research Part C, 77 (2017), 306-328.
[38] H. Li, Y. Wang, X. Xu, L. Qin, H. Zhang, Short-term passenger flow prediction under passenger flow control using a dynamic radial basis function network, Applied Soft Computing Journal, 83 (2019), 105620, DOI: 10.1016/j.asoc.2019.105620.
[39] J. Li, L. Yang, X. Fu, F. Chao, Y. Qu, Interval Type-2 TSK+ fuzzy inference system, IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Rio de Janeiro, (2018), 1-8.
[40] J. Li, L. Yang, Y. Qu, G. Sexton, An extended Takagi-Sugeno-Kang inference system (TSK+) with fuzzy interpolation and its rule base generation, Soft Computing, 22 (2018), 3155-3170.
[41] X. Liang, G. Wang, M. Min, Y. Qi, Z. Han, A deep Spatio-Temporal fuzzy neural network for passenger demand prediction, Proceedings of the 2019 SIAM International Conference on Data Mining, (2019), 9 pages, DOI: 10.1137/1.9781611975673.12.
[42] L. Liu, R. C. Chen, A novel passenger flow prediction model using deep learning methods, Transportation Research Part C: Emerging Technologies, 84 (2017), 74-91, DOI:10.1016/j.trc.2017.08.001.
[43] E. H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proceedings of the Institution of Electrical Engineers, 121(12) (2009), 1585-1588.
[44] MATLAB and Statistics Toolbox Release 2014a, The MathWorks, Inc., Natick, Massachusetts, United States.
[45] J. B. McDonald, Some generalized functions for the size distribution of income, Econometrica, 52(3) (1984), 647- 664.
[46] M. Milenkovic, L. Svadlenka, V. Melichar, N. Bojovic, Z. Avramovic, ŠARIMA modeling approach for railway passenger flow forecasting, Transport, 33(5) (2018), 1113-1120.
[47] P. A. Mitnik, New properties of the Kumaraswamy distribution, Communications in Statistics-Theory and Methods, 42(5) (2013), 741-755.
[48] K. Mittal, A. Jain, K. S. Vaisla, O. Castillo, J. Kacprzyk, A comprehensive review on type 2 fuzzy logic applications: Past, present and future, Engineering Applications of Artificial Intelligence, 95 (2020), 103916.
[49] J. E. Moreno, M. A. Sanchez, O. Mendoza, A. Rodríguez-Díaz, O. Castillo, P. Melin, J. R. Castro, Design of an interval type-2 fuzzy model with justifiable uncertainty, Information Sciences, 513 (2020), 206-221.
[50] S. Mousavi, A. Esfahanipour, M. H. Zarandi, MGP-INTACTSKY: Multitree genetic programming-based learning of INTerpretable and ACcurate TSK sYstems for dynamic portfolio trading, Applied Soft Computing, 34 (2015), 449-462.
[51] S. Nadarajah, Discussion on the distribution of Kumaraswamy, Journal of Hydrology, 348 (2008), 568-569.
[52] N. Naik, R. Diao, Q. Shen, Genetic algorithm-aided dynamic fuzzy rule interpolation, IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Beijing, China, 6-11 July 2014, 2198-2205.
[53] A. K. Nandi, F. Klawonn, Detecting ambiguities in regression problems using TSK models, Soft Computing, 11(5) (2007), 467-478.
[54] P. Noursalehi, H. Koutsopoulos, J. N. Zhao, Real time transit demand prediction capturing station interactions and impact of special events, Transportation Research Part C., 97 (2018), 277-300.
[55] A. Piegat, M. Landowski, Multidimensional interval type 2 epistemic fuzzy arithmetic, Iranian Journal of Fuzzy Systems, 18(5) (2021), 19-36.
[56] B. Rezaee, M. H. Zarandi, Data-driven fuzzy modeling for Takagi- Sugeno-Kang fuzzy system, Information Sciences, 180(2) (2010), 241-255.
[57] Z. Saghian, A. Esfahanipour, B. Karimi, Passenger flow prediction of subway systems utilizing TSK fuzzy modeling based on Gustafson-Kessel Possibilistic c-Means Clustering approach, 17th Iranian International Industrial Engineering Conference held in Mashhad, (2021), 7 pages.
[58] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific Journal of Mathematics, 10(1) (1960), 313-334.
[59] C. Syms, Principal components analysis, In: Jorgensen, Sven Erik, and Fath, Brian D., (eds.) Encyclopedia of Ecology, Elsevier, Oxford, (2008), 2940-2949.
[60] K. Tahera, R. N. Ibrahim, P. B. Lochert, A fuzzy logic approach for dealing with qualitative quality characteristics of a process, Expert Systems with Applications, 34(4) (2008), 2630-2638.
[61] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man, and Cybernetics, 15(1) (1985), 116-132.
[62] F. Toque, E. Cŏme, M. K. E. Mahrsi, L. Oukhellou, Forecasting dynamic public transport origin-destination matrices with long-short term memory recurrent neural networks, In IEEE Conference on Intelligent Transportation Systems, Proceedings, 2016 IEEE 19th International Conference on Intelligent Transportation Systems (ITSC), Rio de Janeiro, Brazil, (2016), 1071-1076.
[63] N. L. Tsakiridis, J. B. Theocharis, G. C. Zalidis, DECO 3 RUM: A differential evolution learning approach for generating compact Mamdani fuzzy rule-based models, Expert Systems with Applications, 83 (2017), 257-272.
[64] A. Ustundag, M. S. Kilinc, E. Cevikcan, Fuzzy rule-based system for the economic analysis of RFID investments, Expert Systems with Applications, 37(7) (2010), 5300-5306.
[65] B. B. Ustundag, A. Kulaglic, High-performance time series prediction with predictive error compensated wavelet neural networks, IEEE Access, 8 (2020), 210532-210541.
[66] L. Wang, S. Dey, Y. M. Tripathi, S. J. Wu, Reliability inference for a multicomponent stress-strength model based on Kumaraswamy distribution, Journal of Computational and Applied Mathematics, 376(1) (2020), 112823.
[67] P. Wang, P. Liu, Some Maclaurin symmetric mean aggregation operators based on Schweizer-Sklar operations for intuitionistic fuzzy numbers and their application to decision making, Journal of Intelligent and Fuzzy Systems, 36 (2019), 3801-3824.
[68] Y. Wei, M. C. Chen, Forecasting the short-term metro passenger flow with empirical mode decomposition and neural networks, Transportation Research Part C: Emerging Technologies, 21(1) (2012), 148-162.
[69] D. Wu, J. M. Mendel, Recommendations on designing practical interval type-2 fuzzy systems, Engineering Applications of Artificial Intelligence, 85 (2019), 182-193.
[70] Y. Xiao, J. J. Liu, Y. Hu, Y. F. Wang, K. K. Lai, S. Wang, A neuro-fuzzy combination model based on singular spectrum analysis for air transport demand forecasting, Journal of Air Transport Management, 39 (2014), 1-11.
[71] R. R. Yager, Generalized triangular norm and conorm aggregation operators on ordinal spaces, International Journal of General Systems, 32(5) (2003), 475-490.
[72] H. T. Yu, C. J. Jiang, R. D. Xiao, H. O. Liu, W. Lv, Passenger flow prediction for new line using region dividing and fuzzy boundary processing, IEEE Transactions on Fuzzy Systems, 27(5) (2019), 994-1007.
[73] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353.
[74] D. P. Zhang, X. K. Wang, Transit ridership estimation with network Kriging: A case study of second avenue subway, NYC, Journal of Transport Geography, 41 (2014), 107-115.
[75] C. Zhong, M. Batty, E. Manley, J. Wang, Z. Wang, F. Chen, G. Schmitt, Variability in regularity: Mining temporal mobility patterns in London, Singapore and Beijing using smart-card data, PLoS One, 11(2) (2016), DOI: 10.1371/journal.pone.0149222.