{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:40:23Z","timestamp":1760240423923,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,17]],"date-time":"2019-06-17T00:00:00Z","timestamp":1560729600000},"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 online sequential extreme learning machine with persistent regularization and forgetting factor (OSELM-PRFF) can avoid potential singularities or ill-posed problems of online sequential regularized extreme learning machines with forgetting factors (FR-OSELM), and is particularly suitable for modelling in non-stationary environments. However, existing algorithms for OSELM-PRFF are time-consuming or unstable in certain paradigms or parameters setups. This paper presents a novel algorithm for OSELM-PRFF, named \u201cCholesky factorization based\u201d OSELM-PRFF (CF-OSELM-PRFF), which recurrently constructs an equation for extreme learning machine and efficiently solves the equation via Cholesky factorization during every cycle. CF-OSELM-PRFF deals with timeliness of samples by forgetting factor, and the regularization term in its cost function works persistently. CF-OSELM-PRFF can learn data one-by-one or chunk-by-chunk with a fixed or varying chunk size. Detailed performance comparisons between CF-OSELM-PRFF and relevant approaches are carried out on several regression problems. The numerical simulation results show that CF-OSELM-PRFF demonstrates higher computational efficiency than its counterparts, and can yield stable predictions.<\/jats:p>","DOI":"10.3390\/sym11060801","type":"journal-article","created":{"date-parts":[[2019,6,17]],"date-time":"2019-06-17T11:24:45Z","timestamp":1560770685000},"page":"801","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Cholesky Factorization Based Online Sequential Extreme Learning Machines with Persistent Regularization and Forgetting Factor"],"prefix":"10.3390","volume":"11","author":[{"given":"Xinran","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Computer Science and Engineering, Central South University, Changsha 410083, China"}]},{"given":"Xiaoyan","family":"Kui","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Central South University, Changsha 410083, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1162\/neco.1991.3.2.246","article-title":"Universal approximation using radial basis function networks","volume":"3","author":"Park","year":"1991","journal-title":"Neural Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1109\/72.846750","article-title":"Classification ability of single hidden layer feedforward neural networks","volume":"11","author":"Huang","year":"2000","journal-title":"IEEE Trans. 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