{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T23:57:55Z","timestamp":1769731075862,"version":"3.49.0"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T00:00:00Z","timestamp":1723075200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T00:00:00Z","timestamp":1723075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"PRIN 2022 Project: Numerical Optimization with Adaptive Accuracy and Applications to Machine Learning","award":["2022N3ZNAX"],"award-info":[{"award-number":["2022N3ZNAX"]}]},{"name":"PNRR PRIN 2022 Pro ject: A multidisciplinary approach to evaluate ecosystems resilience under climate change","award":["P2022WC2ZZ"],"award-info":[{"award-number":["P2022WC2ZZ"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the era of big data, an ever-growing volume of information is recorded, either continuously over time or sporadically, at distinct time intervals. Functional Data Analysis (FDA) stands at the cutting edge of this data revolution, offering a powerful framework for handling and extracting meaningful insights from such complex datasets. The currently proposed FDA methods can often encounter challenges, especially when dealing with curves of varying shapes. This can largely be attributed to the method\u2019s strong dependence on data approximation as a key aspect of the analysis process. In this work, we propose a free knots spline estimation method for functional data with two penalty terms and demonstrate its performance by comparing the results of several clustering methods on simulated and real data.<\/jats:p>","DOI":"10.1007\/s11222-024-10474-w","type":"journal-article","created":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T04:02:05Z","timestamp":1723089725000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Roughness regularization for functional data analysis with free knots spline estimation"],"prefix":"10.1007","volume":"34","author":[{"given":"Anna","family":"De Magistris","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Valentina","family":"De Simone","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elvira","family":"Romano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gerardo","family":"Toraldo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,8]]},"reference":[{"key":"10474_CR1","doi-asserted-by":"crossref","unstructured":"Basna, R., Nassar, H., Podgski, K.: Data driven orthogonal basis selection for functional data analysis. J. Multivar. Anal. 189, 1048\u20131068 (2022). (ISSN 0047-259X)","DOI":"10.1016\/j.jmva.2021.104868"},{"key":"10474_CR2","doi-asserted-by":"crossref","unstructured":"Bouveyron, C., C\u00c3\u2019me, E., Jacques, J.: The discriminative functional mixture model for the analysis of bike sharing systems. HAL n.01024186, (2014)","DOI":"10.1214\/15-AOAS861"},{"key":"10474_CR3","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1111\/biom.12161","volume":"70","author":"H Chen","year":"2014","unstructured":"Chen, H., Reiss, P.T., Tarpey, T.: Optimally weighted l2 distance for functional data. Biometrics 70, 516\u201325 (2014)","journal-title":"Biometrics"},{"issue":"2","key":"10474_CR4","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1111\/1467-9868.00128","volume":"60","author":"DGT Denison","year":"2002","unstructured":"Denison, D.G.T., Mallick, B.K., Smith, A.F.M.: Automatic Bayesian curve fitting. J. R. Stat. Soc. Ser. B Stat Methodol. 60(2), 333\u2013350 (2002). https:\/\/doi.org\/10.1111\/1467-9868.00128","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"issue":"4","key":"10474_CR5","doi-asserted-by":"publisher","first-page":"1055","DOI":"10.1093\/biomet\/88.4.1055","volume":"88","author":"I Dimatteo","year":"2001","unstructured":"Dimatteo, I., Genovese, C.R., Kass, R.E.: Bayesian curve-fitting with free-knot splines. Biometrika 88(4), 1055\u20131071 (2001). https:\/\/doi.org\/10.1093\/biomet\/88.4.1055","journal-title":"Biometrika"},{"key":"10474_CR6","unstructured":"Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis: Theory and Practice. Springer, 2006. ISBN: 978-0-387-30369-7"},{"issue":"4","key":"10474_CR7","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1111\/j.1467-9868.2006.00561.x","volume":"68","author":"D Gervini","year":"2006","unstructured":"Gervini, D.: Free-knot spline smoothing for functional data. J. R. Stat. Soc. 68(4), 671\u2013687 (2006)","journal-title":"J. R. Stat. Soc."},{"issue":"1","key":"10474_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41597-022-01245-1","volume":"9","author":"E Guidotti","year":"2022","unstructured":"Guidotti, E.: A worldwide epidemiological database for COVID-19 at fine-grained spatial resolution. Sci. Data 9(1), 1\u20137 (2022)","journal-title":"Sci. Data"},{"issue":"51","key":"10474_CR9","first-page":"2376","volume":"5","author":"E Guidotti","year":"2020","unstructured":"Guidotti, E., Ardia, D.: COVID-19 data hub. Open J. 5(51), 2376 (2020)","journal-title":"Open J."},{"key":"10474_CR10","doi-asserted-by":"publisher","first-page":"100","DOI":"10.2307\/2346830","volume":"28","author":"JA Hartigan","year":"1979","unstructured":"Hartigan, J.A., Wong, M.A.: A k-means clustering algorithm. Appl. Stat. 28, 100\u2013108 (1979)","journal-title":"Appl. Stat."},{"key":"10474_CR11","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1137\/0715022","volume":"15","author":"DLB Jupp","year":"1978","unstructured":"Jupp, D.L.B.: Approximation to data by splines with free knots. SIAM J. Numer. Anal. 15, 328\u2013343 (1978)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"10474_CR12","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.dsm.2023.03.003","volume":"6","author":"Z Luo","year":"2023","unstructured":"Luo, Z., Liu, N., Zhang, L., Wu, Y.: Time series clustering of COVID-19 pandemic-related data. Data Sci. Manage. 6(2), 79\u201387 (2023). https:\/\/doi.org\/10.1016\/j.dsm.2023.03.003","journal-title":"Data Sci. Manage."},{"key":"10474_CR13","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1007\/978-3-030-55874-1_76","volume-title":"Numerical Mathematics and Advanced Applications, ENUMATH 2019\u2014European Conference, Lecture Notes in Computational Science and Engineering","author":"H Nassar","year":"2021","unstructured":"Nassar, H., Podg\u00f3rski, K.: Empirically driven orthonormal bases for functional data analysis. In: Vermolen, F.J., Vuik, C. (eds.) Numerical Mathematics and Advanced Applications, ENUMATH 2019\u2014European Conference, Lecture Notes in Computational Science and Engineering, pp. 773\u2013783. Springer Science and Business Media B.V., United States (2021). https:\/\/doi.org\/10.1007\/978-3-030-55874-1_76"},{"issue":"4","key":"10474_CR14","first-page":"502","volume":"1","author":"F O\u2019Sullivan","year":"1986","unstructured":"O\u2019Sullivan, F.: A statistical perspective on ill-posed inverse problems. Stat. Sci. 1(4), 502\u2013518 (1986)","journal-title":"Stat. Sci."},{"key":"10474_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-7107-7","volume-title":"Functional Data Analysis","author":"JO Ramsay","year":"1997","unstructured":"Ramsay, J.O., Silverman, B.: Functional Data Analysis. Springer, New York (1997)"},{"key":"10474_CR16","unstructured":"Ramsay, J.O., Marron, J.S., Sangalli, L.M., Srivastava, A.: Functional data analysis of amplitude and phase variation. Stat. Sci. 468\u201384 (2015)"},{"issue":"1","key":"10474_CR17","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1111\/j.2517-6161.1991.tb01821.x","volume":"53","author":"JA Rice","year":"1991","unstructured":"Rice, J.A., Silverman, B.W.: Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. 53(1), 233\u2013243 (1991)","journal-title":"J. R. Stat. Soc."},{"key":"10474_CR18","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.jco.2016.05.003","volume":"36","author":"S Sivananthan","year":"2016","unstructured":"Sivananthan, S.: Multi-penalty regularization in learning theory. J. Complex. 36, 141\u2013165 (2016)","journal-title":"J. Complex."},{"key":"10474_CR19","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.csda.2012.12.001","volume":"61","author":"JD Tucker","year":"2013","unstructured":"Tucker, J.D., Wu, W., Srivastava, A.: Generative models for functional data using phase and amplitude separation. Comput. Stat. Data Anal. 61, 50\u201366 (2013)","journal-title":"Comput. Stat. Data Anal."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10474-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-024-10474-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10474-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,3]],"date-time":"2024-10-03T15:07:06Z","timestamp":1727968026000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-024-10474-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,8]]},"references-count":19,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["10474"],"URL":"https:\/\/doi.org\/10.1007\/s11222-024-10474-w","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,8]]},"assertion":[{"value":"7 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"165"}}