Ferré, Louis; Villa, Nathalie Multilayer perceptron with functional inputs: an inverse regression approach. (English) Zbl 1164.62339 Scand. J. Stat. 33, No. 4, 807-823 (2006). A functional sliced inverse regression (FSIR) technique is considered for fitting the model \[ Y=f(\langle X,a_1\rangle,\dots,\langle X,a_p\rangle,\varepsilon), \] where \(Y\) is the response, \(X\) is a functional regressor (a random function in \(L_2[a,b]\)), \(f\) is an unknown regression function, \(a_1\) are unknown functions, and \(\varepsilon\) is an error term. Consistency of regularized FSIR estimates for \(a_i\) is demonstrated. It is proposed to estimate the function \(f\) by a multilayer perceptron technique. Consistency of the proposed training algorithm for such perceptrons is shown. Applications to real data are considered. Reviewer: R. E. Maiboroda (Kyïv) Cited in 12 Documents MSC: 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62G99 Nonparametric inference 62J12 Generalized linear models (logistic models) Keywords:functional sliced inverse regression; dimension reduction; functional data analysis Software:fda (R) PDFBibTeX XMLCite \textit{L. Ferré} and \textit{N. Villa}, Scand. J. Stat. 33, No. 4, 807--823 (2006; Zbl 1164.62339) Full Text: DOI arXiv References: [1] DOI: 10.1109/TIT.2005.847705 · Zbl 1285.94015 · doi:10.1109/TIT.2005.847705 [2] Bishop C., Neural networks for pattern recognition (1995) · Zbl 0868.68096 [3] DOI: 10.1021/ac00029a018 · doi:10.1021/ac00029a018 [4] Bosq D., Nonparametric functional estimation and related topics, Nato ASI Series C pp 509– (1991) · Zbl 0737.62032 · doi:10.1007/978-94-011-3222-0_38 [5] DOI: 10.1016/S0167-7152(99)00036-X · Zbl 0962.62081 · doi:10.1016/S0167-7152(99)00036-X [6] DOI: 10.1109/72.392253 · doi:10.1109/72.392253 [7] DOI: 10.2307/2290564 · doi:10.2307/2290564 [8] Dauxois J., C. R. Math. Acad. Sci. Paris 327 pp 947– (2001) · Zbl 0996.62035 · doi:10.1016/S0764-4442(01)02163-2 [9] DOI: 10.1016/S0167-9473(03)00032-X · Zbl 1429.62241 · doi:10.1016/S0167-9473(03)00032-X [10] Ferre L., Rev. Statist. Appliquee pp 39– (2005) [11] Ferre L., Statistics 37 pp 475– (2003) [12] Ferre L., Statist. Sinica 15 pp 665– (2005) [13] DOI: 10.2307/2289860 · doi:10.2307/2289860 [14] Hastie T., Ann. Statist. 23 pp 73– (1995) [15] DOI: 10.2307/2290989 · Zbl 0812.62067 · doi:10.2307/2290989 [16] DOI: 10.1016/S0893-6080(05)80067-X · doi:10.1016/S0893-6080(05)80067-X [17] Hsing T., Ann. Statist. 20 pp 1040– (1992) [18] DOI: 10.1198/016214503000189 · Zbl 1041.62052 · doi:10.1198/016214503000189 [19] Leurgans S., J. Roy. Statist. Soc. Ser. B 55 pp 725– (1993) [20] DOI: 10.2307/2290563 · Zbl 0742.62044 · doi:10.2307/2290563 [21] Li K., Ann. Statist. 87 pp 1025– (1992) [22] Ramsay J., Functional data analysis (1997) · Zbl 0882.62002 · doi:10.1007/978-1-4757-7107-7 [23] DOI: 10.1016/j.neunet.2004.07.001 · Zbl 1085.68134 · doi:10.1016/j.neunet.2004.07.001 [24] Rossi F., ESANN’2004 Proceedings pp 305– (2004) [25] DOI: 10.1007/BF01190124 · Zbl 0876.94055 · doi:10.1007/BF01190124 [26] DOI: 10.1016/S0893-6080(98)00108-7 · doi:10.1016/S0893-6080(98)00108-7 [27] DOI: 10.1109/72.478392 · doi:10.1109/72.478392 [28] White H., Neural Comput. 1 pp 425– (1989) [29] DOI: 10.1111/1467-9868.03411 · Zbl 1091.62028 · doi:10.1111/1467-9868.03411 [30] A. Yao (2001 ). Un modele semi-paramerique pour variables fonctionnelles: la regression inverse fonctionnelle . PhD thesis , Universite Toulouse III, Toulouse, France. [31] DOI: 10.1214/aos/1032526955 · Zbl 0864.62027 · doi:10.1214/aos/1032526955 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.