{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T06:14:03Z","timestamp":1778220843551,"version":"3.51.4"},"reference-count":50,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T00:00:00Z","timestamp":1766966400000},"content-version":"vor","delay-in-days":362,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>Stock price forecasting is complex due to the nonlinear and nonstationary nature of financial time series. This study proposes a hybrid variational mode decomposition (VMD)\u2013generalized autoregressive conditional heteroskedasticity (GARCH)\u2013long short\u2010term memory (LSTM) model to predict Airtel\u2019s stock prices, integrating VMD, GARCH, and LSTM networks. VMD decomposes the stock price series into intrinsic mode functions (IMFs), enabling frequency\u2010specific modeling. High\u2010frequency IMFs, which exhibit volatility, are processed with GARCH to capture time\u2010varying volatility and are then fed into LSTM to model residual nonlinear dynamics. Low\u2010frequency IMFs, which reflect smoother trends, are directly modeled by LSTM. Final forecasts are aggregated via an additive ensemble. Initially evaluated on an 80:20 train\u2013test split, the model underwent robustness checks with 70:30 and 90:10 splits, consistently outperforming benchmark models: transformer, GRU, LSTM, BiGRU, BiLSTM, VMD\u2010transformer, VMD\u2010GRU, VMD\u2010LSTM, VMD\u2010BiGRU, VMD\u2010BiLSTM, VMD\u2010GARCH\u2010transformer, VMD\u2010GARCH\u2010GRU, VMD\u2010GARCH\u2010BiGRU, and VMD\u2010GARCH\u2010BiLSTM. Diebold\u2013Mariano tests confirmed statistical superiority across MSE, MAE, and MAPE loss functions, validating the model\u2019s enhanced accuracy. These robustness checks demonstrate the model\u2019s stability across varied data partitions, offering investors a reliable tool for risk mitigation. By effectively addressing volatility and nonlinearity, this hybrid framework advances financial forecasting, enhances decision\u2010making in dynamic financial stocks, and contributes to robust time\u2010series prediction methodologies.<\/jats:p>","DOI":"10.1155\/ijmm\/2710277","type":"journal-article","created":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T10:42:24Z","timestamp":1767004944000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Forecasting Airtel Stock Prices Through Decomposition and Integration: A Novel VMD\u2010GARCH\u2010LSTM Framework"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-7919-2064","authenticated-orcid":false,"given":"John Kamwele","family":"Mutinda","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7813-6835","authenticated-orcid":false,"given":"Amos Kipkorir","family":"Langat","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9703-6514","authenticated-orcid":false,"given":"Samuel Musili","family":"Mwalili","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,12,29]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.sciaf.2024.e02374"},{"key":"e_1_2_11_2_2","doi-asserted-by":"publisher","DOI":"10.1155\/jama\/7706431"},{"key":"e_1_2_11_3_2","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1970.10481180"},{"key":"e_1_2_11_4_2","doi-asserted-by":"publisher","DOI":"10.2307\/1925546"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.procs.2017.01.175"},{"key":"e_1_2_11_6_2","doi-asserted-by":"publisher","DOI":"10.1080\/0740817x.2014.999180"},{"key":"e_1_2_11_7_2","doi-asserted-by":"publisher","DOI":"10.1186\/s13677-023-00576-7"},{"key":"e_1_2_11_8_2","first-page":"37","article-title":"Cancer Cases in Kenya; Forecasting Incidents Using Box & Jenkins Arima Model","volume":"2","author":"Langat A.","year":"2017","journal-title":"Biomedical Statistics and Informatics"},{"key":"e_1_2_11_9_2","doi-asserted-by":"publisher","DOI":"10.1155\/2024\/1591016"},{"key":"e_1_2_11_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2019.04.175"},{"key":"e_1_2_11_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2014.12.084"},{"key":"e_1_2_11_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.egypro.2019.02.027"},{"key":"e_1_2_11_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2023.119951"},{"key":"e_1_2_11_14_2","volume-title":"Emerging Trends in Expert Applications and Security: Proceedings of ICETEAS 2018","author":"Pawar K.","year":"2018"},{"key":"e_1_2_11_15_2","doi-asserted-by":"publisher","DOI":"10.3390\/app13158813"},{"key":"e_1_2_11_16_2","first-page":"5246","article-title":"Stock Price Prediction Using Lstm","volume":"83","author":"Pramod B. S.","year":"2020","journal-title":"Test Engineering and Management"},{"key":"e_1_2_11_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/PDGC50313.2020.9315800"},{"key":"e_1_2_11_18_2","doi-asserted-by":"publisher","DOI":"10.3390\/electronics12122718"},{"key":"e_1_2_11_19_2","doi-asserted-by":"publisher","DOI":"10.11591\/ijeecs.v35.i1.pp263-273"},{"key":"e_1_2_11_20_2","doi-asserted-by":"publisher","DOI":"10.1142\/s146902682350013x"},{"key":"e_1_2_11_21_2","doi-asserted-by":"publisher","DOI":"10.3390\/app13010222"},{"key":"e_1_2_11_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2022.118128"},{"key":"e_1_2_11_23_2","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1998.0193"},{"key":"e_1_2_11_24_2","doi-asserted-by":"publisher","DOI":"10.1142\/s1793536909000047"},{"key":"e_1_2_11_25_2","doi-asserted-by":"publisher","DOI":"10.1109\/tsp.2013.2288675"},{"key":"e_1_2_11_26_2","doi-asserted-by":"publisher","DOI":"10.1109\/ICASSP.2011.5947265"},{"key":"e_1_2_11_27_2","doi-asserted-by":"publisher","DOI":"10.9734\/ajeba\/2024\/v24i61356"},{"key":"e_1_2_11_28_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2018.11.061"},{"key":"e_1_2_11_29_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2020.114332"},{"key":"e_1_2_11_30_2","doi-asserted-by":"publisher","DOI":"10.1002\/for.3201"},{"key":"e_1_2_11_31_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.najef.2023.101915"},{"key":"e_1_2_11_32_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-023-08441-0"},{"key":"e_1_2_11_33_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2025.128473"},{"key":"e_1_2_11_34_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2024.126222"},{"key":"e_1_2_11_35_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2023.110356"},{"key":"e_1_2_11_36_2","first-page":"1","article-title":"Application of the Deep Learning Integrated Framework ceemdan-gru-informer in Financial Time Series Prediction","author":"Liu F.","year":"2025","journal-title":"Computational Economics"},{"key":"e_1_2_11_37_2","doi-asserted-by":"publisher","DOI":"10.1186\/s40537-025-01127-4"},{"key":"e_1_2_11_38_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2022.109324"},{"key":"e_1_2_11_39_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10614-025-11020-0"},{"key":"e_1_2_11_40_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.sciaf.2024.e02423"},{"key":"e_1_2_11_41_2","doi-asserted-by":"publisher","DOI":"10.1080\/07474938608800095"},{"key":"e_1_2_11_42_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.eneco.2017.12.030"},{"key":"e_1_2_11_43_2","doi-asserted-by":"publisher","DOI":"10.1155\/ijmm\/4795841"},{"key":"e_1_2_11_44_2","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1997.9.8.1735"},{"key":"e_1_2_11_45_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10614-025-11030-y"},{"key":"e_1_2_11_46_2","doi-asserted-by":"publisher","DOI":"10.9734\/ajrcos\/2024\/v17i5445"},{"key":"e_1_2_11_47_2","doi-asserted-by":"publisher","DOI":"10.3115\/v1\/w14-4012"},{"key":"e_1_2_11_48_2","doi-asserted-by":"publisher","DOI":"10.1002\/for.70010"},{"key":"e_1_2_11_49_2","doi-asserted-by":"publisher","DOI":"10.1515\/cmb-2025-0026"},{"key":"e_1_2_11_50_2","doi-asserted-by":"publisher","DOI":"10.1080\/07350015.2014.983236"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/ijmm\/2710277","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/ijmm\/2710277","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/ijmm\/2710277","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T12:16:05Z","timestamp":1772972165000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/ijmm\/2710277"}},"subtitle":[],"editor":[{"given":"Andrzej","family":"Skowron","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":50,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1155\/ijmm\/2710277"],"URL":"https:\/\/doi.org\/10.1155\/ijmm\/2710277","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2025-03-19","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-10-10","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-12-29","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"2710277"}}