{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T07:52:17Z","timestamp":1762329137138,"version":"build-2065373602"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"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":[[2025,12]]},"DOI":"10.1007\/s11222-025-10727-2","type":"journal-article","created":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T11:14:21Z","timestamp":1758539661000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Measuring dependence between functional data via Projection Hilbert-Schmidt Covariance"],"prefix":"10.1007","volume":"35","author":[{"given":"Zhentao","family":"Tian","sequence":"first","affiliation":[]},{"given":"Darong","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Zhongzhan","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,22]]},"reference":[{"issue":"2","key":"10727_CR1","doi-asserted-by":"publisher","first-page":"858","DOI":"10.1214\/21-AOS2129","volume":"50","author":"M Albert","year":"2022","unstructured":"Albert, M., Laurent, B., Marrel, A., Meynaoui, A.: Adaptive test of independence based on hsic measures. Ann. Stat. 50(2), 858\u2013879 (2022)","journal-title":"Ann. Stat."},{"key":"10727_CR2","first-page":"126","volume-title":"International Conference on Artificial Intelligence and Statistics","author":"K Balasubramanian","year":"2013","unstructured":"Balasubramanian, K., Sriperumbudur, B., Lebanon, G.: Ultrahigh dimensional feature screening via rkhs embeddings. In: International Conference on Artificial Intelligence and Statistics, vol. 31, pp. 126\u2013134. PMLR, Scottsdale, Arizona, USA (2013)"},{"issue":"14","key":"10727_CR3","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1093\/bioinformatics\/btz333","volume":"35","author":"H Climente-Gonz\u00e1lez","year":"2019","unstructured":"Climente-Gonz\u00e1lez, H., Azencott, C.A., Kaski, S., Yamada, M.: Block hsic lasso: model-free biomarker detection for ultra-high dimensional data. Bioinformatics 35(14), 427\u2013435 (2019)","journal-title":"Bioinformatics"},{"issue":"1","key":"10727_CR4","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1016\/j.acha.2006.04.006","volume":"21","author":"RR Coifman","year":"2006","unstructured":"Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5\u201330 (2006)","journal-title":"Appl. Comput. Harmon. Anal."},{"issue":"4","key":"10727_CR5","doi-asserted-by":"publisher","first-page":"2572","DOI":"10.3150\/23-BEJ1685","volume":"30","author":"X Cheng","year":"2024","unstructured":"Cheng, X., Xie, Y.: Kernel two-sample tests for manifold data. Bernoulli 30(4), 2572\u20132597 (2024)","journal-title":"Bernoulli"},{"key":"10727_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/b97287","volume-title":"Statistical Methods for the Analysis of Repeated Measurements","author":"CS Davis","year":"2002","unstructured":"Davis, C.S.: Statistical Methods for the Analysis of Repeated Measurements. Springer, New York (2002)"},{"key":"10727_CR7","first-page":"473","volume-title":"Advances in Neural Information Processing Systems","author":"K Fukumizu","year":"2009","unstructured":"Fukumizu, K., Gretton, A., Sch\u00f6lkopf, B.: Characteristic kernels on groups and semigroups. In: Advances in Neural Information Processing Systems, vol. 21, pp. 473\u2013480. Curran Associates, Vancouver, BC, Canada (2009)"},{"key":"10727_CR8","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/11564089_7","volume-title":"International Conference on Algorithmic Learning Theory","author":"A Gretton","year":"2005","unstructured":"Gretton, A., Bousquet, O., Smola, A., Sch\u00f6lkopf, B.: Measuring statistical dependence with hilbert-schmidt norms. In: International Conference on Algorithmic Learning Theory, pp. 63\u201377. Springer, Berlin, Heidelberg (2005)"},{"key":"10727_CR9","first-page":"585","volume-title":"Advances in Neural Information Processing Systems","author":"A Gretton","year":"2007","unstructured":"Gretton, A., Fukumizu, K., Teo, C., Song, L., Sch\u00f6lkopf, B., Smola, A.: A kernel statistical test of independence. In: Advances in Neural Information Processing Systems, pp. 585\u2013592. Curran Associates, Vancouver British Columbia Canada (2007)"},{"key":"10727_CR10","first-page":"2075","volume":"6","author":"A Gretton","year":"2005","unstructured":"Gretton, A., Herbrich, R., Smola, A., Bousquet, O., Sch\u00f6lkopf, B.: Kernel methods for measuring independence. J. Mach. Learn. Res. 6, 2075\u20132129 (2005)","journal-title":"J. Mach. Learn. Res."},{"key":"10727_CR11","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1007\/s10462-018-9666-7","volume":"53","author":"T Go\u0155ecki","year":"2020","unstructured":"Go\u0155ecki, T., Krzysko, M., Woly\u0144ski, W.: Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data. Artif. Intell. Rev. 53, 475\u2013499 (2020)","journal-title":"Artif. Intell. Rev."},{"issue":"3","key":"10727_CR12","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1111\/rssc.12408","volume":"69","author":"R Ghosal","year":"2020","unstructured":"Ghosal, R., Maity, A., Clark, T.: Variable selection in functional linear concurrent regression. J. R. Stat. Soc.: Ser. C: Appl. Stat. 69(3), 565\u2013587 (2020)","journal-title":"J. R. Stat. Soc.: Ser. C: Appl. Stat."},{"key":"10727_CR13","first-page":"3759","volume-title":"International Conference on Machine Learning","author":"D Greenfeld","year":"2020","unstructured":"Greenfeld, D., Shalit, U.: Robust learning with the hilbert-schmidt independence criterion. In: International Conference on Machine Learning, pp. 3759\u20133768. PMLR, Red Hook, NY (2020)"},{"key":"10727_CR14","volume-title":"Measure Theory","author":"PR Halmos","year":"1974","unstructured":"Halmos, P.R.: Measure Theory. Springer, New York (1974)"},{"key":"10727_CR15","doi-asserted-by":"publisher","first-page":"3284","DOI":"10.1214\/12-AOP803","volume":"41","author":"R Lyons","year":"2013","unstructured":"Lyons, R.: Distance covariance in metric spaces. Ann. Probab. 41, 3284\u20133305 (2013)","journal-title":"Ann. Probab."},{"key":"10727_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2020.104711","volume":"182","author":"T Lai","year":"2021","unstructured":"Lai, T., Zhang, Z., Wang, Y., Kong, L.: Testing independence of functional variables by angle covariance. J. Multivar. Anal. 182, 104711 (2021)","journal-title":"J. Multivar. Anal."},{"issue":"543","key":"10727_CR17","doi-asserted-by":"publisher","first-page":"1876","DOI":"10.1080\/01621459.2021.2020126","volume":"118","author":"R Miao","year":"2022","unstructured":"Miao, R., Zhang, X., Wong, R.K.W.: A wavelet-based independence test for functional data with an application to meg functional connectivity. J. Am. Stat. Assoc. 118(543), 1876\u20131889 (2022)","journal-title":"J. Am. Stat. Assoc."},{"key":"10727_CR18","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1111\/rssb.12235","volume":"80","author":"N Pfister","year":"2018","unstructured":"Pfister, N., B\u00fchlmann, P., Sch\u00f6lkopf, B., Peters, J.: Kernel-based tests for joint independence. J. R. Stat. Soc. Ser. B Stat Methodol. 80, 5\u201331 (2018)","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"key":"10727_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-88-7642-499-1","volume-title":"Introduction to Stochastic Analysis and Malliavin Calculus","author":"GD Prato","year":"2014","unstructured":"Prato, G.D.: Introduction to Stochastic Analysis and Malliavin Calculus. Springer, New York (2014)"},{"key":"10727_CR20","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1080\/01621459.2018.1543600","volume":"115","author":"W Pan","year":"2020","unstructured":"Pan, W., Wang, X., Zhang, H., Zhu, H., Zhu, J.: Ball covariance: a generic measure of dependence in banach space. J. Am. Stat. Assoc. 115, 307\u2013317 (2020)","journal-title":"J. Am. Stat. Assoc."},{"key":"10727_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1214\/aos\/1176342996","volume":"3","author":"M Rosenblatt","year":"1975","unstructured":"Rosenblatt, M.: A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Ann. Stat. 3, 1\u201314 (1975)","journal-title":"Ann. Stat."},{"key":"10727_CR22","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316481","volume-title":"Approximation Theorems of Mathematical Statistics","author":"RJ Serfling","year":"1980","unstructured":"Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)"},{"key":"10727_CR23","first-page":"1393","volume":"13","author":"L Song","year":"2012","unstructured":"Song, L., Smola, A., Gretton, A., Borgwardt, K., Bedo, J.: Feature selection via dependence maximization. J. Mach. Learn. Res. 13, 1393\u20131434 (2012)","journal-title":"J. Mach. Learn. Res."},{"key":"10727_CR24","doi-asserted-by":"crossref","unstructured":"Vershynin, R.: High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, Cambridge (2018)","DOI":"10.1017\/9781108231596"},{"issue":"3","key":"10727_CR25","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1093\/biomet\/asy021","volume":"105","author":"L Weihs","year":"2018","unstructured":"Weihs, L., Drton, M., Meinshausen, N.: Symmetric rank covariances: a generalized framework for nonparametric measures of dependence. Biometrika 105(3), 547\u2013562 (2018)","journal-title":"Biometrika"},{"issue":"1","key":"10727_CR26","first-page":"269","volume":"31","author":"G Wang","year":"2021","unstructured":"Wang, G., Li, W.K., Zhu, K.: New hsic-based tests for independence between two stationary multivariate time series. Stat. Sin. 31(1), 269\u2013300 (2021)","journal-title":"Stat. Sin."},{"key":"10727_CR27","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1093\/biomet\/asx043","volume":"104","author":"L Zhu","year":"2017","unstructured":"Zhu, L., Xu, K., Li, R., Zhong, W.: Projection correlation between two random vectors. Biometrika 104, 829\u2013843 (2017)","journal-title":"Biometrika"},{"issue":"6","key":"10727_CR28","first-page":"3366","volume":"48","author":"C Zhu","year":"2019","unstructured":"Zhu, C., Yao, S., Zhang, X., Shao, X.: Distance-based and rkhs-based dependence metrics in high dimension. Ann. Stat. 48(6), 3366\u20133394 (2019)","journal-title":"Ann. Stat."},{"key":"10727_CR29","first-page":"19145","volume-title":"Advances in Neural Information Processing Systems","author":"T Zhang","year":"2024","unstructured":"Zhang, T., Zhang, Y., Zhou, T.: Statistical insights into hsic in high dimensions. In: Advances in Neural Information Processing Systems, pp. 19145\u201319156. Curran Associates, New Orleans LA USA (2024)"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10727-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-025-10727-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10727-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T07:17:11Z","timestamp":1762327031000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-025-10727-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,22]]},"references-count":29,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["10727"],"URL":"https:\/\/doi.org\/10.1007\/s11222-025-10727-2","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"type":"print","value":"0960-3174"},{"type":"electronic","value":"1573-1375"}],"subject":[],"published":{"date-parts":[[2025,9,22]]},"assertion":[{"value":"26 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 September 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 competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"198"}}