{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T18:02:09Z","timestamp":1783620129984,"version":"3.55.0"},"reference-count":54,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T00:00:00Z","timestamp":1747180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71272161"],"award-info":[{"award-number":["71272161"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we propose a novel fuzzy time series forecasting model that integrates fuzzy C-means (FCM) clustering, the principle of justifiable granularity (PJG), and particle swarm optimization (PSO), with a focus on leveraging symmetry in subinterval partitioning to enhance model interpretability and forecasting accuracy. First, the FCM method is employed to partition the universe of discourse, generating an initial division of subintervals. To ensure symmetric information representation, triangular fuzzy information granules are constructed for these subintervals in accordance with the principle of justifiable granularity. Then, an objective function is formulated for the entire universe of discourse, and the PSO algorithm is utilized to optimize the subinterval division, resulting in the final optimal partition. This process ensures that the subintervals achieve a balance between coverage and specificity, thereby introducing a form of symmetry in the partitioning of the universe of discourse. Leveraging the optimized symmetric partition, the framework of the fuzzy time series model is implemented for forecasting. Finally, the proposed approach is carried out on the Taiwan Weighted Stock Index (TAIEX) datasets and the Shanghai Composite Index (SHCI) datasets. The forecasting results demonstrate that the proposed approach achieves higher prediction accuracy and semantic accuracy compared with other methods.<\/jats:p>","DOI":"10.3390\/sym17050753","type":"journal-article","created":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T10:27:41Z","timestamp":1747218461000},"page":"753","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["An Enhanced Fuzzy Time Series Forecasting Model Integrating Fuzzy C-Means Clustering, the Principle of Justifiable Granularity, and Particle Swarm Optimization"],"prefix":"10.3390","volume":"17","author":[{"given":"Hailan","family":"Chen","sequence":"first","affiliation":[{"name":"School of Business, Sichuan Normal University, Chengdu 610101, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuedong","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qi","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Finance, Hebei University of Economics and Business, Shijiazhuang 050061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"110356","DOI":"10.1016\/j.asoc.2023.110356","article-title":"Stock index forecasting based on multivariate empirical mode decomposition and temporal convolutional networks","volume":"142","author":"Yao","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Palash, W., Akanda, A.S., and Islam, S. (2024). A data-driven global flood forecasting system for medium to large rivers. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-59145-w"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"119652","DOI":"10.1016\/j.eswa.2023.119652","article-title":"A novel fractional grey system model with non-singular exponential kernel for forecasting enrollments","volume":"219","author":"Xie","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"8399","DOI":"10.1007\/s10489-024-05504-z","article-title":"Machine learning-based spatial downscaling and bias-correction framework for high-resolution temperature forecasting","volume":"54","author":"Meng","year":"2024","journal-title":"Appl. Intell."},{"key":"ref_5","unstructured":"Box, G.E.P., and Jenkins, G.M. (1990). Time Series Analysis: Forecasting and Control, Holden-Day."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107730","DOI":"10.1016\/j.asoc.2021.107730","article-title":"Mohapatra, Data driven day-ahead electrical load forecasting through repeated wavelet transform assisted SVM model","volume":"111","author":"Aasim","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_7","first-page":"529","article-title":"Analysis and applications of time series forecasting model via support vector machines","volume":"27","author":"Wei","year":"2005","journal-title":"Syst. Eng. Electron."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1186\/s40537-022-00599-y","article-title":"Time-series analysis with smoothed Convolutional Neural Network","volume":"9","author":"Wibawa","year":"2022","journal-title":"J. Big Data"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3357","DOI":"10.1007\/s11063-022-10767-z","article-title":"A new CNN-based model for financial time series: TAIEX and FTSE stocks forecasting","volume":"54","author":"Kirisci","year":"2022","journal-title":"Neural Process. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Siami-Namini, S., Tavakoli, N., and Namin, A.S. (2019, January 9\u201312). The performance of LSTM and BiLSTM in forecasting time series. Proceedings of the 2019 IEEE International Conference on Big Data, Los Angeles, CA, USA.","DOI":"10.1109\/BigData47090.2019.9005997"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"118601","DOI":"10.1016\/j.apenergy.2022.118601","article-title":"Carbon price forecasting based on CEEMDAN and LSTM","volume":"311","author":"Zhou","year":"2022","journal-title":"Appl. Energy"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1109\/TSMC.1985.6313399","article-title":"Fuzzy Identification of Systems and its Applications to Modeling and Control","volume":"15","author":"Takagi","year":"1985","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0165-0114(93)90355-L","article-title":"Forecasting enrollments with fuzzy time series\u2014Part I","volume":"54","author":"Song","year":"1993","journal-title":"Fuzzy Sets Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0165-0114(94)90067-1","article-title":"Forecasting enrollments with fuzzy time series\u2014Part II","volume":"62","author":"Song","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_16","first-page":"159","article-title":"A Novel Fuzzy Time Series Forecasting Model Based on Optimal Partitioning of the Universe of Discourse","volume":"114","author":"Chen","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0165-0114(93)90372-O","article-title":"Fuzzy time series and its models","volume":"54","author":"Song","year":"1993","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/S0165-0114(00)00057-9","article-title":"Effective Lengths of Intervals to Improve Forecasting in Fuzzy Time Series","volume":"123","author":"Huarng","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1109\/TSMCB.2005.857093","article-title":"Ratio-Based Lengths of Intervals to Improve Fuzzy Time Series Forecasting","volume":"36","author":"Huarng","year":"2006","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"ref_20","first-page":"234","article-title":"A New Method to Forecast Enrollments Using Fuzzy Time Series","volume":"2","author":"Chen","year":"2004","journal-title":"Int. J. Appl. Sci. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.ins.2014.09.038","article-title":"A Hybrid Fuzzy Time Series Model Based on Granular Computing for Stock Price Forecasting","volume":"294","author":"Chen","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5673","DOI":"10.1016\/j.eswa.2013.04.026","article-title":"Effective Intervals Determined by Information Granules to Improve Forecasting in Fuzzy Time Series","volume":"40","author":"Wang","year":"2013","journal-title":"Expert Syst. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3134","DOI":"10.1016\/j.eswa.2013.10.046","article-title":"Determination of Temporal Information Granules to Improve Forecasting in Fuzzy Time Series","volume":"41","author":"Wang","year":"2014","journal-title":"Expert Syst. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109574","DOI":"10.1016\/j.asoc.2022.109574","article-title":"Interval type-2 fuzzy C-means forecasting model for fuzzy time series","volume":"129","author":"Yin","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TFUZZ.2015.2453393","article-title":"Designing fuzzy sets with the use of the parametric principle of justifiable granularity","volume":"24","author":"Pedrycz","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.ins.2019.10.042","article-title":"Design of an interval Type-2 fuzzy model with justifiable uncertainty","volume":"513","author":"Moreno","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3456","DOI":"10.1109\/TFUZZ.2020.3023758","article-title":"Design of Interval Type-2 Information Granules Based on the Principle of Justifiable Granularity","volume":"29","author":"Zhang","year":"2021","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_28","first-page":"253","article-title":"A Methodology for Building of Interval and General Type-2 Fuzzy Systems Based on the Principle of Justifiable Granularity","volume":"40","author":"Castillo","year":"2023","journal-title":"J. Mult. Valued Log. Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4228","DOI":"10.1016\/j.eswa.2008.04.001","article-title":"Forecasting in High Order Fuzzy Times Series by Using Neural Networks to Define Fuzzy Relations","volume":"36","author":"Aladag","year":"2009","journal-title":"Expert Syst. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1016\/j.matcom.2010.09.011","article-title":"A High Order Fuzzy Time Series Forecasting Model Based on Adaptive Expectation and Artificial Neural Networks","volume":"81","author":"Aladag","year":"2010","journal-title":"Math. Comput. Simul."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5052","DOI":"10.1016\/j.eswa.2009.12.006","article-title":"Finding an Optimal Interval Length in High Order Fuzzy Time Series","volume":"37","author":"Egrioglu","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1494","DOI":"10.1016\/j.eswa.2009.06.102","article-title":"Forecasting TAIFEX Based on Fuzzy Time Series and Particle Swarm Optimization","volume":"37","author":"Kuo","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"8014","DOI":"10.1016\/j.eswa.2010.12.127","article-title":"A Hybrid Forecasting Model for Enrollments Based on Aggregated Fuzzy Time Series and Particle Swarm Optimization","volume":"38","author":"Huang","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1007\/s41066-021-00300-3","article-title":"Fuzzy time series forecasting based on hesitant fuzzy sets, particle swarm optimization and support vector machine-based hybrid method","volume":"7","author":"Pant","year":"2022","journal-title":"Granul. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1080\/03081079.2024.2405688","article-title":"VWFTS-PSO: A novel method for time series forecasting using variational weighted fuzzy time series and particle swarm optimization","volume":"54","author":"Didugu","year":"2024","journal-title":"Int. J. Gen. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3907","DOI":"10.1007\/s00500-017-2601-z","article-title":"A novel fuzzy time series forecasting method based on the improved artificial fish swarm optimization algorithm","volume":"22","author":"Xian","year":"2018","journal-title":"Soft Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0098-3004(84)90020-7","article-title":"FCM: The fuzzy C-means clustering algorithm","volume":"10","author":"Bezdek","year":"1984","journal-title":"Comput. Geosci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1109\/3477.907568","article-title":"Abstraction and specialization of information granules","volume":"31","author":"Pedrycz","year":"2001","journal-title":"IEEE Trans. Syst. Man Cybern. B Cybern."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4209","DOI":"10.1016\/j.asoc.2013.06.017","article-title":"Building the fundamentals of granular computing: A principle of justifiable granularity","volume":"13","author":"Pedrycz","year":"2013","journal-title":"Appl. Soft Comput."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Geng, G., He, Y., Zhang, J., Qin, T., and Yang, B. (2023). Short-Term Power Load Forecasting Based on PSO-Optimized VMD-TCN-Attention Mechanism. Energies, 16.","DOI":"10.3390\/en16124616"},{"key":"ref_41","unstructured":"Lu, W. (2015). Time Series Analysis and Modeling Method Research Based on Granular Computing, Dalian University of Technology. (In Chinese)."},{"key":"ref_42","unstructured":"Shao, G.H. (2017). Modeling and Forecasting Based on Multivariate Granular Time Series, Dalian University of Technology. (In Chinese)."},{"key":"ref_43","unstructured":"Zhou, W. (2019). Modeling Methods for Interval-Valued Time Series Based on Granular Computing, Dalian University of Technology. (In Chinese)."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/0165-0114(94)90152-X","article-title":"A Comparison of Fuzzy Forecasting and Markov Modeling","volume":"64","author":"Sullivan","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/0165-0114(95)00220-0","article-title":"Forecasting Enrollments Based on Fuzzy Time Series","volume":"81","author":"Chen","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/S0165-0114(00)00093-2","article-title":"Heuristic Models of Fuzzy Time Series for Forecasting","volume":"123","author":"Huarng","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.physa.2004.11.006","article-title":"Weighted Fuzzy Time Series Model for TAIEX Forecasting","volume":"349","author":"Yu","year":"2005","journal-title":"Physica A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/j.physa.2005.08.014","article-title":"The Application of Neural Networks to Forecast Fuzzy Time Series","volume":"363","author":"Huarng","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_49","first-page":"10594","article-title":"Fuzzy Forecasting Based on Fuzzy-trend Logical Relationship Groups","volume":"40","author":"Chen","year":"2010","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TFUZZ.2010.2073712","article-title":"TAIEX Forecasting Based on Fuzzy Time Series and Fuzzy Variation Groups","volume":"19","author":"Chen","year":"2011","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.eswa.2016.01.053","article-title":"A hybrid forecasting model based on automatic clustering, axiomatic fuzzy set classification, and autoregressive integrated moving average (ARIMA) for stock market trends","volume":"55","author":"Wang","year":"2016","journal-title":"Expert Syst. Appl."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"836","DOI":"10.1109\/TSMCB.2006.890303","article-title":"A Multivariate Heuristic Model for Fuzzy Time-Series Forecasting","volume":"37","author":"Huarng","year":"2007","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2945","DOI":"10.1016\/j.eswa.2007.05.016","article-title":"A Bivariate Fuzzy Time Series Model to Forecast the TAIEX","volume":"34","author":"Yu","year":"2008","journal-title":"Expert Syst. Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"4772","DOI":"10.1016\/j.ins.2010.08.026","article-title":"Multi-Variable Fuzzy Forecasting Based on Fuzzy Clustering and Fuzzy Rule Interpolation Techniques","volume":"180","author":"Chen","year":"2010","journal-title":"Inf. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/5\/753\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:32:19Z","timestamp":1760031139000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/5\/753"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,14]]},"references-count":54,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["sym17050753"],"URL":"https:\/\/doi.org\/10.3390\/sym17050753","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,14]]}}}