{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T21:45:47Z","timestamp":1774647947200,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T00:00:00Z","timestamp":1745884800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T00:00:00Z","timestamp":1745884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2025,8]]},"DOI":"10.1007\/s11222-025-10619-5","type":"journal-article","created":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T15:30:28Z","timestamp":1745940628000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Nonlinear principal component analysis for two-dimensional functional data using neural networks"],"prefix":"10.1007","volume":"35","author":[{"given":"Xinran","family":"Chen","sequence":"first","affiliation":[]},{"given":"Jingxiao","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Rou","family":"Zhong","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,29]]},"reference":[{"issue":"4","key":"10619_CR1","first-page":"1571","volume":"24","author":"J-M Chiou","year":"2014","unstructured":"Chiou, J.-M., Chen, Y.-T., Yang, Y.-F.: Multivariate functional principal component analysis: a normalization approach. Stat. Sin. 24(4), 1571\u20131596 (2014)","journal-title":"Stat. Sin."},{"key":"10619_CR2","doi-asserted-by":"publisher","DOI":"10.1080\/10618600.2024.2374584","author":"L Cai","year":"2024","unstructured":"Cai, L., Hu, Q.: Global inference and test for eigensystems of imaging data over complicated domains. J. Comput. Graph. Stat. (2024). https:\/\/doi.org\/10.1080\/10618600.2024.2374584","journal-title":"J. Comput. Graph. Stat."},{"issue":"522","key":"10619_CR3","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1080\/01621459.2016.1273115","volume":"113","author":"C Happ","year":"2018","unstructured":"Happ, C., Greven, S., Initi, A.D.N.: Multivariate functional principal component analysis for data observed on different (dimensional) domains. J. Am. Stat. Assoc. 113(522), 649\u2013659 (2018)","journal-title":"J. Am. Stat. Assoc."},{"issue":"2","key":"10619_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11222-025-10578-x","volume":"35","author":"S Jia","year":"2025","unstructured":"Jia, S., Shi, H., Guan, T.: Function-on-function regression models with nonlinear dynamic effect and linear concurrent effect. Stat. Computing 35(2), 1\u201315 (2025). https:\/\/doi.org\/10.1007\/s11222-025-10578-x","journal-title":"Stat. Computing"},{"key":"10619_CR5","unstructured":"Kingma, D.P., Ba, J. (2017): Adam: A Method for Stochastic Optimization . https:\/\/arxiv.org\/abs\/1412.6980"},{"issue":"539","key":"10619_CR6","doi-asserted-by":"publisher","first-page":"1228","DOI":"10.1080\/01621459.2020.1844211","volume":"117","author":"T Li","year":"2022","unstructured":"Li, T., Li, T., Zhu, Z., Zhu, H.: Regression analysis of asynchronous longitudinal functional and scalar data. J. Am. Stat. Assoc. 117(539), 1228\u20131242 (2022)","journal-title":"J. Am. Stat. Assoc."},{"issue":"3","key":"10619_CR7","doi-asserted-by":"publisher","first-page":"846","DOI":"10.1111\/biom.12457","volume":"72","author":"Z Lin","year":"2016","unstructured":"Lin, Z., Wang, L., Cao, J.: Interpretable functional principal component analysis. Biometrics 72(3), 846\u2013854 (2016)","journal-title":"Biometrics"},{"key":"10619_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2020.107016","volume":"152","author":"Y Nie","year":"2020","unstructured":"Nie, Y., Cao, J.: Sparse functional principal component analysis in a new regression framework. Comput. Stat. Data Anal. 152, 107016 (2020)","journal-title":"Comput. Stat. Data Anal."},{"issue":"3","key":"10619_CR9","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s11222-017-9758-2","volume":"28","author":"Y Nie","year":"2018","unstructured":"Nie, Y., Wang, L., Liu, B., Cao, J.: Supervised functional principal component analysis. Stat. Computing 28(3), 713\u2013723 (2018)","journal-title":"Stat. Computing"},{"issue":"6088","key":"10619_CR10","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1016\/B978-1-4832-1446-7.50035-2","volume":"323","author":"DE Rumelhart","year":"1988","unstructured":"Rumelhart, D.E., Hinton, G.E., Williams, R.J.: Learning internal representations by error propagation. Read. Cognit. Sci. 323(6088), 399\u2013421 (1988)","journal-title":"Read. Cognit. Sci."},{"issue":"4","key":"10619_CR11","doi-asserted-by":"publisher","first-page":"1248","DOI":"10.1080\/10618600.2023.2165498","volume":"32","author":"AR Rao","year":"2023","unstructured":"Rao, A.R., Reimherr, M.: Nonlinear functional modeling using neural networks. J. Computational Graph. Stat. 32(4), 1248\u20131257 (2023)","journal-title":"J. Computational Graph. Stat."},{"key":"10619_CR12","first-page":"44","volume":"18","author":"C Stoehr","year":"2021","unstructured":"Stoehr, C., Aston, J.A.D., Kirch, C.: Detecting changes in the covariance structure of functional time series with application to fmri data. Econom. Stat. 18, 44\u201362 (2021)","journal-title":"Econom. Stat."},{"issue":"20","key":"10619_CR13","doi-asserted-by":"publisher","first-page":"3887","DOI":"10.1093\/bioinformatics\/bti634","volume":"21","author":"M Scholz","year":"2005","unstructured":"Scholz, M., Kaplan, F., Guy, C., Kopka, J., Selbig, J.: Non-linear pca: a missing data approach. Bioinformatics 21(20), 3887\u20133895 (2005)","journal-title":"Bioinformatics"},{"key":"10619_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2020.104675","volume":"181","author":"J Song","year":"2021","unstructured":"Song, J., Li, B.: Nonlinear and additive principal component analysis for functional data. J. Multivar. Anal. 181, 104675 (2021)","journal-title":"J. Multivar. Anal."},{"issue":"4","key":"10619_CR15","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1080\/10618600.2022.2035738","volume":"31","author":"H Shi","year":"2022","unstructured":"Shi, H., Yang, Y., Wang, L., Ma, D., Beg, M.F., Pei, J., Cao, J.: Two-dimensional functional principal component analysis for image feature extraction. J. Computational Graph. Stat. 31(4), 1127\u20131140 (2022)","journal-title":"J. Computational Graph. Stat."},{"key":"10619_CR16","unstructured":"Thind, B., Multani, K., Cao, J. (2020): Neural networks as functional classifiers. ArXiv abs\/2010.04305"},{"issue":"1","key":"10619_CR17","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1080\/10618600.2022.2097914","volume":"32","author":"B Thind","year":"2023","unstructured":"Thind, B., Multani, K., Cao, J.: Deep learning with functional inputs. J. Computational Graph. Stat. 32(1), 171\u2013180 (2023)","journal-title":"J. Computational Graph. Stat."},{"issue":"5","key":"10619_CR18","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1007\/s11222-023-10287-3","volume":"33","author":"S Wu","year":"2023","unstructured":"Wu, S., Beaulac, C., Cao, J.: Neural networks for scalar input and functional output. Stat. Computing 33(5), 118 (2023)","journal-title":"Stat. Computing"},{"issue":"6","key":"10619_CR19","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/s11222-024-10501-w","volume":"34","author":"S Wu","year":"2024","unstructured":"Wu, S., Beaulac, C., Cao, J.: Functional autoencoder for smoothing and representation learning. Stat. Computing 34(6), 203 (2024)","journal-title":"Stat. Computing"},{"issue":"5","key":"10619_CR20","doi-asserted-by":"publisher","DOI":"10.1002\/env.2792","volume":"34","author":"H Wang","year":"2023","unstructured":"Wang, H., Cao, J.: Nonlinear prediction of functional time series. Environmetrics 34(5), e2792 (2023)","journal-title":"Environmetrics"},{"issue":"4","key":"10619_CR21","doi-asserted-by":"publisher","first-page":"1667","DOI":"10.1111\/sjos.12660","volume":"50","author":"S Wang","year":"2023","unstructured":"Wang, S., Cao, G., Shang, Z.: Alzheimer\u2019s disease neuroimaging initiative: deep neural network classifier for multidimensional functional data. Scand. J. Stat. 50(4), 1667\u20131686 (2023)","journal-title":"Scand. J. Stat."},{"issue":"538","key":"10619_CR22","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1080\/01621459.2020.1820344","volume":"117","author":"J Wang","year":"2022","unstructured":"Wang, J., Wong, R.K.W., Zhang, X.: Low-rank covariance function estimation for multidimensional functional data. J. Am. Stat. Assoc. 117(538), 809\u2013822 (2022)","journal-title":"J. Am. Stat. Assoc."},{"key":"10619_CR23","doi-asserted-by":"crossref","unstructured":"Zhou, L., Pan, H.: Principal component analysis of two-dimensional functional data. J. Computational Gr. Stat. 23(3), 779\u2013801 (2014)","DOI":"10.1080\/10618600.2013.827986"},{"key":"10619_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2025.108169","volume":"209","author":"Q Zhong","year":"2025","unstructured":"Zhong, Q., Song, X.: Functional nonlinear principal component analysis. Computational Stat. Data Anal. 209, 108169 (2025)","journal-title":"Computational Stat. Data Anal."},{"issue":"1\u20133","key":"10619_CR25","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1016\/j.neucom.2005.06.004","volume":"69","author":"D Zhang","year":"2005","unstructured":"Zhang, D., Zhou, Z.: (2d)2pca: two-directional two-dimensional pca for efficient face representation and recognition. Neurocomputing 69(1\u20133), 224\u2013231 (2005)","journal-title":"Neurocomputing"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10619-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-025-10619-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10619-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T02:03:51Z","timestamp":1750212231000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-025-10619-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,29]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["10619"],"URL":"https:\/\/doi.org\/10.1007\/s11222-025-10619-5","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,29]]},"assertion":[{"value":"20 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2025","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 no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"86"}}