{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T11:41:54Z","timestamp":1778067714746,"version":"3.51.4"},"reference-count":63,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"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>Accurate stock price forecasting is crucial for supporting informed investment decisions, effective risk management, and the identification of profitable market opportunities. Financial time series present considerable challenges for prediction due to their complex, nonlinear dynamics and sensitivity to a wide range of economic factors. Although various statistical methods have been developed to model the multidimensional relationships inherent in such datasets, advancements in big data technologies have greatly facilitated the recording, analysis, and interpretation of large-scale financial data, thereby accelerating the adoption of deep learning (DL) algorithms in this domain. In the present study, RNN-, LSTM-, and GRU-based models were developed to forecast the closing prices of two airline stocks, with hyperparameter optimization conducted via the Bayesian optimization algorithm. The dataset consisted of daily closing prices of THYAO and PGSUS stocks obtained from Yahoo Finance. Comparative analysis demonstrated that the GRU model yielded the highest accuracy for THYAO stock price prediction, achieving a MAPE of 3.05% and an RMSE of 3.195, whereas for PGSUS, the model achieved a MAPE of 3.97% and an RMSE of 3.232. Beyond its empirical contribution, this study also emphasizes the conceptual relevance of symmetry in financial forecasting. The proposed deep learning framework captures the balanced relationships and nonlinear interactions inherent in stock market behavior, reflecting both symmetry and asymmetry in market responses to economic factors.<\/jats:p>","DOI":"10.3390\/sym17111905","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T10:56:45Z","timestamp":1762513005000},"page":"1905","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Hyperparameter-Optimized RNN, LSTM, and GRU Models for Airline Stock Price Prediction: A Comparative Study on THYAO and PGSUS"],"prefix":"10.3390","volume":"17","author":[{"given":"Funda H.","family":"Sezgin","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, Faculty of Engineering, Istanbul University-Cerrahpa\u015fa, 34320 Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2016-8674","authenticated-orcid":false,"given":"\u00d6mer","family":"Algorabi","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, Faculty of Engineering, Istanbul University-Cerrahpa\u015fa, 34320 Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gamze","family":"Sart","sequence":"additional","affiliation":[{"name":"Department of Educational Sciences, Hasan Ali Yucel Faculty of Education, Istanbul University-Cerrahpa\u015fa, 34500 Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9289-5088","authenticated-orcid":false,"given":"Mustafa","family":"G\u00fcler","sequence":"additional","affiliation":[{"name":"Department of Engineering Science, Faculty of Engineering, Istanbul University-Cerrahpa\u015fa, 34320 Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"419","DOI":"10.18657\/yonveek.1208807","article-title":"Derin \u00d6\u011frenme ve ARIMA Y\u00f6ntemlerinin Tahmin Performanslar\u0131n\u0131n K\u0131yaslanmas\u0131: Bir Borsa \u0130stanbul Hissesi \u00d6rne\u011fi","volume":"30","author":"Erden","year":"2023","journal-title":"Y\u00f6netim Ekon. Derg."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shahi, T.B., Shrestha, A., Neupane, A., and Guo, W. (2020). Stock price forecasting with deep learning: A comparative study. Mathematics, 8.","DOI":"10.3390\/math8091441"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109830","DOI":"10.1016\/j.asoc.2022.109830","article-title":"Novel optimization approach for stock price forecasting using multi-layered sequential LSTM","volume":"134","author":"Md","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"121080","DOI":"10.1016\/j.eswa.2023.121080","article-title":"A novel hybrid model for stock price forecasting integrating Encoder Forest and Informer","volume":"234","author":"Ren","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"284","DOI":"10.3846\/jbem.2022.16094","article-title":"Oil price volatility and airlines\u2019 stock returns: Evidence from the global aviation industry","volume":"23","author":"Horobet","year":"2022","journal-title":"J. Bus. Econ. Manag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"160","DOI":"10.30586\/pek.1419904","article-title":"Time Series Analysis of Long-Term Stock Performance of Airlines: The case of Turkish Airlines","volume":"8","author":"Akusta","year":"2024","journal-title":"Polit. Ekon. Kuram"},{"key":"ref_7","first-page":"4758698","article-title":"Research on stock price time series prediction based on deep learning and autoregressive integrated moving average","volume":"2022","author":"Xiao","year":"2022","journal-title":"Sci. Program."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1108\/IJCS-05-2020-0012","article-title":"A stock price prediction method based on deep learning technology","volume":"5","author":"Ji","year":"2021","journal-title":"Int. J. Crowd Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1049\/cit2.12059","article-title":"Stock market prediction using deep learning algorithms","volume":"8","author":"Mukherjee","year":"2021","journal-title":"CAAI Trans. Intell. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"163815","DOI":"10.1109\/ACCESS.2021.3134138","article-title":"Impact of hyperparameter tuning on machine learning models in stock price forecasting","volume":"9","author":"Hoque","year":"2021","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"101969","DOI":"10.1016\/j.jairtraman.2020.101969","article-title":"Influential factors on Chinese airlines\u2019 profitability and forecasting methods","volume":"91","author":"Xu","year":"2020","journal-title":"J. Air Transp. Manag."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Chang, V., Xu, Q.A., Chidozie, A., and Wang, H. (2024). Predicting Economic Trends and Stock Market Prices with Deep Learning and Advanced Machine Learning Techniques. Electronics, 13.","DOI":"10.3390\/electronics13173396"},{"key":"ref_13","first-page":"359","article-title":"Multi-step time series analysis and forecasting strategy using ARIMA and evolutionary algorithms","volume":"14","author":"Kumar","year":"2021","journal-title":"Int. J. Inf. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"116941","DOI":"10.1016\/j.eswa.2022.116941","article-title":"An adaptive feature selection schema using improved technical indicators for predicting stock price movements","volume":"200","author":"Ji","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107760","DOI":"10.1016\/j.asoc.2021.107760","article-title":"Constructing a stock-price forecast CNN model with gold and crude oil indicators","volume":"112","author":"Chen","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_16","first-page":"439","article-title":"Stock Market Price Forecasting Using the Arima Model: An Application to Istanbul, Turkiye","volume":"9","author":"Mashadihasanli","year":"2022","journal-title":"J. Econ. Policy Res.\/\u0130ktisat Polit. Ara\u015ft\u0131rmalar\u0131 Derg."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ferdinand, F.V., Santoso, T.H., and Saputra, K.V.I. (, January 18\u201321). Performance comparison between Facebook Prophet and SARIMA on Indonesian stock. Proceedings of the 2023 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), Singapore.","DOI":"10.1109\/IEEM58616.2023.10406940"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.jfds.2018.04.003","article-title":"Stock price prediction using support vector regression on daily and up to the minute prices","volume":"4","author":"Henrique","year":"2018","journal-title":"J. Financ. Data Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gorde, P.S., and Borkar, S.N. (2024, January 23\u201324). Comparative analysis of linear regression, random forest regressor and LSTM for stock price prediction. Proceedings of the 2024 8th International Conference on Computing, Communication, Control and Automation (ICCUBEA), Pune, India.","DOI":"10.1109\/ICCUBEA61740.2024.10775094"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Raudys, A., and Goldstein, E. (2022). Forecasting detrended volatility risk and financial price series using LSTM neural networks and XGBOOST Regressor. J. Risk Financ. Manag., 15.","DOI":"10.3390\/jrfm15120602"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ecer, F., Ardabili, S., Band, S.S., and Mosavi, A. (2020). Training Multilayer Perceptron with Genetic Algorithms and Particle Swarm Optimization for Modeling Stock Price Index Prediction. Entropy, 22.","DOI":"10.3390\/e22111239"},{"key":"ref_22","first-page":"40","article-title":"An application of artificial neural networks and fuzzy logic on the stock price prediction problem","volume":"1","author":"Khuat","year":"2017","journal-title":"JOIV Int. J. Inform. Vis."},{"key":"ref_23","unstructured":"Sunny, M.a.I., Maswood, M.M.S., and Alharbi, A.G. (2020, January 24\u201326). Deep Learning-Based Stock Price Prediction using LSTM and Bi-Directional LSTM model. Proceedings of the 2020 2nd Novel Intelligent and Leading Emerging Sciences Conference (NILES), Giza, Egypt."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Bhavani, A., Ramana, A.V., and Chakravarthy, A.S.N. (2022, January 13\u201315). Comparative Analysis between LSTM and GRU in Stock Price Prediction. Proceedings of the 2022 International Conference on Edge Computing and Applications (ICECAA), Tamilnadu, India.","DOI":"10.1109\/ICECAA55415.2022.9936434"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"6622927","DOI":"10.1155\/2020\/6622927","article-title":"A CNN-LSTM-Based model to forecast stock prices","volume":"2020","author":"Lu","year":"2020","journal-title":"Complexity"},{"key":"ref_26","first-page":"2446543","article-title":"Stock price forecast based on CNN-BILSTM-ECA model","volume":"2021","author":"Chen","year":"2021","journal-title":"Sci. Program."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Xu, C., Li, J., Feng, B., and Lu, B. (2023). A financial time-series prediction model based on multiplex attention and linear transformer structure. Appl. Sci., 13.","DOI":"10.3390\/app13085175"},{"key":"ref_28","first-page":"1","article-title":"Multi-perspective Learning Based on Transformer for Stock Price Trend","volume":"18","author":"Li","year":"2025","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_29","first-page":"100320","article-title":"Predicting stock market index using LSTM","volume":"9","author":"Bhandari","year":"2022","journal-title":"Mach. Learn. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.eswa.2017.04.030","article-title":"Deep learning networks for stock market analysis and prediction: Methodology, data representations, and case studies","volume":"83","author":"Chong","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"117986","DOI":"10.1016\/j.eswa.2022.117986","article-title":"StockNet\u2014GRU based stock index prediction","volume":"207","author":"Gupta","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1016\/j.asoc.2016.08.029","article-title":"Evaluation of co-evolutionary neural network architectures for time series prediction with mobile application in finance","volume":"49","author":"Chandra","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"86230","DOI":"10.1109\/ACCESS.2021.3088999","article-title":"Novel Stock Crisis Prediction Technique\u2014A Study on Indian Stock Market","volume":"9","author":"Naik","year":"2021","journal-title":"IEEE Access"},{"key":"ref_34","first-page":"89","article-title":"Analysis of price volatility in BIST 100 index with time series: Comparison of Fbprophet and LSTM model","volume":"35","author":"Aker","year":"2022","journal-title":"Avrupa Bilim Ve Teknol. Derg."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"117600","DOI":"10.1016\/j.eswa.2022.117600","article-title":"Prediction of SSE Shanghai Enterprises index based on bidirectional LSTM model of air pollutants","volume":"204","author":"Liu","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"71","DOI":"10.54097\/hset.v34i.5380","article-title":"Long Short-term Memory Applied on Amazon\u2019s Stock Prediction","volume":"34","author":"Zhou","year":"2023","journal-title":"Highlights Sci. Eng. Technol."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Saini, A., Singh, N., Singh, R.K., and Sachan, M.K. (2024, January 25\u201327). Financial Time Series Prediction on Apple stocks using machine and deep learning models. Proceedings of the 2024 International Conference on Electrical, Computer and Energy Technologies (ICECET), Sydney, Australia.","DOI":"10.1109\/ICECET61485.2024.10698514"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"120346","DOI":"10.1016\/j.eswa.2023.120346","article-title":"Stock price prediction with optimized deep LSTM network with artificial rabbits optimization algorithm","volume":"227","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Zhang, J., Ye, L., and Lai, Y. (2023). Stock price prediction using CNN-BILSTM-Attention model. Mathematics, 11.","DOI":"10.3390\/math11091985"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Ali, M., Khan, D.M., Alshanbari, H.M., and El-Bagoury, A.A.-A.H. (2023). Prediction of complex stock market data using an improved hybrid EMD-LSTM model. Appl. Sci., 13.","DOI":"10.3390\/app13031429"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Jaiswal, R., and Singh, B. (2022, January 23\u201324). A hybrid convolutional recurrent (CNN-GRU) model for stock price prediction. Proceedings of the 2022 IEEE 11th International Conference on Communication Systems and Network Technologies (CSNT), Indore, India.","DOI":"10.1109\/CSNT54456.2022.9787651"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Song, H., and Choi, H. (2023). Forecasting stock market indices using the recurrent neural network based hybrid models: CNN-LSTM, GRU-CNN, and ensemble models. Appl. Sci., 13.","DOI":"10.3390\/app13074644"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Singh, U., Tamrakar, S., Saurabh, K., Vyas, R., and Vyas, O. (2024, January 24\u201328). Optimizing hyperparameters of deep learning models for stock price prediction. Proceedings of the 2024 15th International Conference on Computing Communication and Networking Technologies (ICCCNT), Kamand, India.","DOI":"10.1109\/ICCCNT61001.2024.10724813"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"110835","DOI":"10.1016\/j.asoc.2023.110835","article-title":"Stock price forecasting using PSO hypertuned neural nets and ensembling","volume":"147","author":"Chauhan","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Chung, H., and Shin, K. (2018). Genetic Algorithm-Optimized Long Short-Term Memory Network for stock market prediction. Sustainability, 10.","DOI":"10.3390\/su10103765"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Chen, X., Yang, F., Sun, Q., and Yi, W. (2024). Research on stock prediction based on CED-PSO-StockNet time series model. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-78984-1"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Yan, J., and Huang, Y. (2025). MambaLLM: Integrating Macro-Index and Micro-Stock Data for Enhanced Stock Price Prediction. Mathematics, 13.","DOI":"10.3390\/math13101599"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1089\/big.2020.0159","article-title":"Deep learning for Time Series Forecasting: A survey","volume":"9","author":"Torres","year":"2020","journal-title":"Big Data"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"119951","DOI":"10.1016\/j.ins.2023.119951","article-title":"TRNN: An efficient time-series recurrent neural network for stock price prediction","volume":"657","author":"Lu","year":"2023","journal-title":"Inf. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.neucom.2018.09.082","article-title":"Time series forecasting of petroleum production using deep LSTM recurrent networks","volume":"323","author":"Sagheer","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long Short-Term memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.physa.2018.11.061","article-title":"Financial time series forecasting model based on CEEMDAN and LSTM","volume":"519","author":"Cao","year":"2018","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Mahjoub, S., Chrifi-Alaoui, L., Marhic, B., and Delahoche, L. (2022). Predicting energy consumption using LSTM, Multi-Layer GRU and Drop-GRU neural networks. Sensors, 22.","DOI":"10.3390\/s22114062"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Pirani, M., Thakkar, P., Jivrani, P., Bohara, M.H., and Garg, D. (2022, January 23\u201324). A comparative analysis of ARIMA, GRU, LSTM and BILSTM on financial Time Series forecasting. Proceedings of the 2022 IEEE International Conference on Distributed Computing and Electrical Circuits and Electronics (ICDCECE), Ballari, India.","DOI":"10.1109\/ICDCECE53908.2022.9793213"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"106393","DOI":"10.1016\/j.jastp.2024.106393","article-title":"Comparative Time series analysis of SARIMA, LSTM, and GRU models for global SF6 emission Management System","volume":"265","year":"2024","journal-title":"J. Atmos. Sol. -Terr. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Jiang, X., and Xu, C. (2022). Deep Learning and Machine Learning with Grid Search to Predict Later Occurrence of Breast Cancer Metastasis Using Clinical Data. J. Clin. Med., 11.","DOI":"10.3390\/jcm11195772"},{"key":"ref_57","first-page":"281","article-title":"Random search for hyper-parameter optimization","volume":"13","author":"Bergstra","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"ref_58","unstructured":"Snoek, J., Larochelle, H., and Adams, R.P. (2012, January 3\u20136). Practical bayesian optimization of machine learning algorithms. Proceedings of the Neural Information Processing Systems, Red Hook, NY, USA."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Lee, S., Kim, J., Kang, H., Kang, D., and Park, J. (2021). Genetic algorithm based deep learning neural network structure and hyperparameter optimization. Appl. Sci., 11.","DOI":"10.3390\/app11020744"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Teixeira, D.M., and Barbosa, R.S. (2024). Stock Price Prediction in the Financial Market Using Machine Learning Models. Computation, 13.","DOI":"10.3390\/computation13010003"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"21229","DOI":"10.1007\/s00521-024-10303-1","article-title":"A deep fusion model for stock market prediction with news headlines and time series data","volume":"36","author":"Chen","year":"2024","journal-title":"Neural Comput. Appl."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.neucom.2020.07.061","article-title":"On hyperparameter optimization of machine learning algorithms: Theory and practice","volume":"415","author":"Yang","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_63","first-page":"26","article-title":"Hyperparameter optimization for machine learning models based on Bayesian optimization","volume":"17","author":"Wu","year":"2019","journal-title":"J. Electron. Sci. Technol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1905\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T11:18:55Z","timestamp":1762514335000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1905"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,7]]},"references-count":63,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111905"],"URL":"https:\/\/doi.org\/10.3390\/sym17111905","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,7]]}}}