{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:53:22Z","timestamp":1760237602530,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,4]],"date-time":"2020-06-04T00:00:00Z","timestamp":1591228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The fluctuation of the stock market has a symmetrical characteristic. To improve the performance of self-forecasting, it is crucial to summarize and accurately express internal fluctuation rules from the historical time series dataset. However, due to the influence of external interference factors, these internal rules are difficult to express by traditional mathematical models. In this paper, a novel forecasting model is proposed based on probabilistic linguistic logical relationships generated from historical time series dataset. The proposed model introduces linguistic variables with positive and negative symmetrical judgements to represent the direction of stock market fluctuation. Meanwhile, daily fluctuation trends of a stock market are represented by a probabilistic linguistic term set, which consist of daily status and its recent historical statuses. First, historical time series of a stock market is transformed into a fluctuation time series (FTS) by the first-order difference transformation. Then, a fuzzy linguistic variable is employed to represent each value in the fluctuation time series, according to predefined intervals. Next, left hand sides of fuzzy logical relationships between currents and their corresponding histories can be expressed by probabilistic linguistic term sets and similar ones can be grouped to generate probabilistic linguistic logical relationships. Lastly, based on the probabilistic linguistic term set expression of the current status and the corresponding historical statuses, distance measurement is employed to find the most proper probabilistic linguistic logical relationship for future forecasting. For the convenience of comparing the prediction performance of the model from the perspective of accuracy, this paper takes the closing price dataset of Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) as an example. Compared with the prediction results of previous studies, the proposed model has the advantages of stable prediction performance, simple model design, and an easy to understand platform. In order to test the performance of the model for other datasets, we use the prediction of the Shanghai Stock Exchange Composite Index (SHSECI) to prove its universality.<\/jats:p>","DOI":"10.3390\/sym12060954","type":"journal-article","created":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T03:32:21Z","timestamp":1591327941000},"page":"954","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Forecasting Model for Stock Market Based on Probabilistic Linguistic Logical Relationship and Distance Measurement"],"prefix":"10.3390","volume":"12","author":[{"given":"Aiwu","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250014, China"}]},{"given":"Junhong","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250014, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7335-5871","authenticated-orcid":false,"given":"Hongjun","family":"Guan","sequence":"additional","affiliation":[{"name":"School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250014, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1981","DOI":"10.1016\/j.eswa.2012.10.001","article-title":"Forecasting seasonal time series with computational intelligence: On recent methods and the potential of their combinations","volume":"40","author":"Martin","year":"2013","journal-title":"Expert Syst. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3606","DOI":"10.1016\/j.apenergy.2010.05.012","article-title":"Day-ahead electricity price forecasting using wavelet transform combined with ARIMA and GARCH models","volume":"87","author":"Tan","year":"2010","journal-title":"Appl. Energy"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/1912773","article-title":"Autoregressive Conditional heteroscedasticity with estimates of the variance of United Kingdom inflation","volume":"50","author":"Engle","year":"1982","journal-title":"Econometrica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1198\/073500104000000523","article-title":"A new class of multivariate skew densities, with application to generalized autoregressive conditional heteroscedasticity models","volume":"23","author":"Bauwens","year":"2005","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_5","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_6","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_7","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_8","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1016\/j.techfore.2005.07.004","article-title":"Entropy-based and trapezoid fuzzification-based fuzzy time series approaches for forecasting IT project cost","volume":"73","author":"Cheng","year":"2006","journal-title":"Technol. Forecast. Soc. Chang."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1016\/j.asoc.2014.11.043","article-title":"A new linguistic out-sample approach of fuzzy time series for daily forecasting of Malaysian electricity load demand","volume":"28","author":"Efendi","year":"2015","journal-title":"Appl. Soft Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.eswa.2017.01.049","article-title":"Improving stock index forecasts by using a new weighted fuzzy-trend time series method","volume":"76","author":"Rubio","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_11","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."},{"doi-asserted-by":"crossref","unstructured":"Chen, S.M., Manalu, G.M.T., Shih, S.C., Sheu, T.W., and Liu, H.C. (2011, January 9\u201312). A new method for fuzzy forecasting based on two-factors high-order fuzzy-trend logical relationship groups and particle swarm optimization techniques. Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, IEEE, Anchorage, AK, USA.","key":"ref_12","DOI":"10.1109\/ICSMC.2011.6084021"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.knosys.2014.11.003","article-title":"A new fuzzy time series forecasting model combined with ant colony optimization and auto-regression","volume":"74","author":"Cai","year":"2015","journal-title":"Knowl. Based Syst."},{"key":"ref_14","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."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1109\/TSMCA.2012.2190399","article-title":"TAIEX forecasting using fuzzy time series and automatically generated weights of multiple factors","volume":"42","author":"Chen","year":"2012","journal-title":"IEEE Trans. Syst. Man Cybern. Part A Syst. Hum."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.ins.2016.05.038","article-title":"A novel forecasting method based on multi-order fuzzy time series and technical analysis","volume":"367\u2013368","author":"Ye","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.1016\/j.eswa.2006.12.021","article-title":"Fuzzy time-series based on adaptive expectation model for TAIEX forecasting","volume":"34","author":"Cheng","year":"2008","journal-title":"Expert Syst. Appl."},{"doi-asserted-by":"crossref","unstructured":"Jia, J.Y., Zhao, A.W., and Guan, S. (2017). Forecasting based on high-order fuzzy-fluctuation trends and particle swarm optimization machine learning. Symmetry, 9.","key":"ref_19","DOI":"10.20944\/preprints201707.0006.v1"},{"doi-asserted-by":"crossref","unstructured":"Guan, H., Dai, Z., Zhao, A., and He, J. (2018). A novel stock forecasting model based on high-order-fuzzy-fluctuation trends and back propagation neural network. PLoS ONE, 13.","key":"ref_20","DOI":"10.1371\/journal.pone.0192366"},{"key":"ref_21","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_22","doi-asserted-by":"crossref","first-page":"2121","DOI":"10.1016\/j.eswa.2014.09.036","article-title":"A high-order multi-variable fuzzy time series forecasting algorithm based on fuzzy clustering","volume":"42","author":"Askari","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.jocs.2015.11.011","article-title":"Intraday stock price forecasting based on variational mode decomposition","volume":"12","author":"Lahmiri","year":"2016","journal-title":"J. Comput. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.eswa.2016.02.025","article-title":"A variational mode decompoisition approach for analysis and forecasting of economic and financial time series","volume":"55","author":"Lahmiri","year":"2016","journal-title":"Expert Syst. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.physa.2004.11.006","article-title":"Weighted fuzzy time series models for TAIEX forecasting","volume":"349","author":"Yu","year":"2005","journal-title":"Phys. A Stat. Mech. Appl."},{"doi-asserted-by":"crossref","unstructured":"Guan, H., Dai, Z., Zhao, A., Dai, Z., and Guan, S. (2018). A forecasting model based on multi-valued neutrosophic sets and two-factor, third-order fuzzy fluctuation logical relationships. Symmetry, 10.","key":"ref_26","DOI":"10.3390\/sym10070245"},{"key":"ref_27","first-page":"385","article-title":"A unifying field in logics: Neutrosophic logic","volume":"8","author":"Florentin","year":"2002","journal-title":"Mult. Valued Log."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.ins.2016.06.021","article-title":"Probabilistic linguistic term sets in multi-attribute group decision making","volume":"369","author":"Pang","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"105572","DOI":"10.1016\/j.asoc.2019.105572","article-title":"Inclusion measures of probabilistic linguistic term sets and their application in classifying cities in the Economic Zone of Chengdu Plain","volume":"82","author":"Tang","year":"2019","journal-title":"Appl. Soft Comput. J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1080\/01605682.2018.1510806","article-title":"Group decision-making approach for evaluating the sustainability of constructed wetlands with probabilistic linguistic preference relations","volume":"12","author":"Luo","year":"2019","journal-title":"J. Oper. Res. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1007\/s40815-019-00705-y","article-title":"Bidirectional projection method for probabilistic linguistic multi-criteria group decision-making based on power average operator","volume":"21","author":"Liu","year":"2019","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s10700-019-09309-5","article-title":"A survey of decision-making methods with probabilistic linguistic information: Bibliometrics, preliminaries, methodologies, applications and future directions","volume":"19","author":"Liao","year":"2020","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1002\/cplx.21625","article-title":"An extended TODIM method for multiple attribute group decision-making based on 2-dimension uncertain linguistic variable","volume":"21","author":"Liu","year":"2016","journal-title":"Complexity"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1007\/s12559-019-09648-w","article-title":"A novel decision-making method based on probabilistic linguistic information","volume":"11","author":"Liu","year":"2019","journal-title":"Cogn. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0165-0114(95)00107-7","article-title":"A model of consensus in group decision making under linguistic assessment","volume":"78","author":"Herrera","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1016\/j.asoc.2017.08.009","article-title":"A multi-criteria decision-making model for hotel selection with linguistic distribution assessments","volume":"67","author":"Yu","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1016\/j.asoc.2010.04.010","article-title":"A hybrid ANFIS model based on AR and volatility for TAIEX forecasting","volume":"11","author":"Chang","year":"2011","journal-title":"Appl. Soft Comput. J."},{"key":"ref_38","first-page":"65","article-title":"Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups, similarity measures and PSO techniques","volume":"391\u2013392","author":"Chen","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1016\/j.econmod.2012.09.047","article-title":"OWA-based ANFIS model for TAIEX forecasting","volume":"30","author":"Cheng","year":"2013","journal-title":"Econ. Model."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2510","DOI":"10.1016\/j.asoc.2010.09.007","article-title":"Forecasting stock markets using wavelet transforms and recurrent neural networks: An integrated system based on artificial bee colony algorithm","volume":"11","author":"Hsieh","year":"2011","journal-title":"Appl. Soft Comput. J."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1109\/TSMCB.2012.2223815","article-title":"Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and particle swarm optimization techniques","volume":"43","author":"Chen","year":"2013","journal-title":"IEEE Trans. Cybern."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1109\/TCYB.2014.2326888","article-title":"Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of fuzzy logical relationships","volume":"45","author":"Chen","year":"2015","journal-title":"IEEE Trans. Cybern."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.neucom.2018.04.014","article-title":"Fuzzy time-series model based on rough set rule induction for forecasting stock price","volume":"302","author":"Cheng","year":"2018","journal-title":"Neurocomputing"},{"doi-asserted-by":"crossref","unstructured":"Guan, S., and Zhao, A. (2017). A two-factor autoregressive composite moving average model based on fuzzy fluctuation logic relationships. Symmetry, 9.","key":"ref_44","DOI":"10.3390\/sym9100207"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.3846\/tede.2019.10766","article-title":"Hotel selection utilizing online reviews: A novel decision support model based on sentiment analysis and DL-VIKOR method","volume":"25","author":"Liang","year":"2019","journal-title":"Technol. Econ. Dev. Econ."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/954\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:35:36Z","timestamp":1760175336000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/954"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,4]]},"references-count":45,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12060954"],"URL":"https:\/\/doi.org\/10.3390\/sym12060954","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,6,4]]}}}