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Deciding the dimension of effective dimension reduction space for functional and high-dimensional data. (English) Zbl 1200.62115
Summary: We consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is called the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential \(\chi ^{2}\) testing procedures under the assumption that the predictor has an elliptically contoured distribution. We further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores. The proposed procedures are supported by theory, validated by simulation studies, and illustrated by a real-data example. Our methods and theory are applicable to functional data and high-dimensional multivariate data.

MSC:
62M20 Inference from stochastic processes and prediction
62J05 Linear regression; mixed models
62H25 Factor analysis and principal components; correspondence analysis
62L10 Sequential statistical analysis
62G20 Asymptotic properties of nonparametric inference
46N30 Applications of functional analysis in probability theory and statistics
65C60 Computational problems in statistics (MSC2010)
Software:
fda (R); gss; RSIR
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References:
[1] Amato, U., Antoniadis, A. and De Feis, I. (2006). Dimension reduction in functional regression with applications. Comput. Statist. Data Anal. 50 2422-2446. · Zbl 1445.62078
[2] Ash, R. B. and Gardner, M. F. (1975). Topics in Stochastic Processes . Academic Press, New York. · Zbl 0317.60014
[3] Cai, T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159-2179. · Zbl 1106.62036 · doi:10.1214/009053606000000830
[4] Cambanis, S., Huang, S. and Simons, G. (1981). On the theory of elliptically contoured distributions. J. Multivariate Anal. 11 368-385. · Zbl 0469.60019 · doi:10.1016/0047-259X(81)90082-8
[5] Cardot, H., Ferraty, F. and Sarda, P. (2003). Spline estimators for the functional linear model. Statist. Sinica 13 571-591. · Zbl 1050.62041
[6] Cardot, H. and Sarda, P. (2005). Estimation in generalized linear models for functional data via penalized likelihood. J. Multivariate Anal. 92 24-41. · Zbl 1065.62127 · doi:10.1016/j.jmva.2003.08.008
[7] Carroll, R. J. and Li, K. C. (1992). Errors in variables for nonlinear regression: Dimension reduction and data visualization. J. Amer. Statist. Assoc. 87 1040-1050. · Zbl 0765.62002
[8] Cook, D. R. and Weisberg, S. (1991). Comments on “Sliced Inverse Regression for Dimension Reduction,” by K. C. Li. J. Amer. Statist. Assoc. 86 328-332. JSTOR: · Zbl 0742.62044 · doi:10.2307/2290563 · links.jstor.org
[9] Cook, D. R. (1998). Regression Graphics . Wiley, New York. · Zbl 0903.62001
[10] Crambes, C., Kneip, A. and Sarda, P. (2009). Smoothing spline estimators for functional linear regression. Ann. Statist. 37 35-72. · Zbl 1169.62027 · doi:10.1214/07-AOS563
[11] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector of random function: Some application to statistical inference. J. Multivariate Anal. 12 136-154. · Zbl 0539.62064 · doi:10.1016/0047-259X(82)90088-4
[12] Eaton, M. L. and Tyler, D. (1994). The asymptotic distribution of singular values with application to canonical correlations and correspondence analysis. J. Multivariate Anal. 50 238-264. · Zbl 0805.62020 · doi:10.1006/jmva.1994.1041
[13] Eubank, R. and Hsing, T. (2010). The Essentials of Functional Data Analysis . Unpublished manuscript. Dept. Statistics, Univ. Michigan.
[14] Fan, J. and Lin, S.-K. (1998). Test of significance when data are curves. J. Amer. Statist. Assoc. 93 1007-1021. JSTOR: · Zbl 1064.62525 · doi:10.2307/2669845 · links.jstor.org
[15] Ferré, L. and Yao, A. (2003). Functional sliced inverse regression analysis. Statistics 37 475-488. · Zbl 1032.62052 · doi:10.1080/0233188031000112845
[16] Ferré, L. and Yao, A. (2005). Smoothed functional sliced inverse regression. Statist. Sinica 15 665-685. · Zbl 1086.62054
[17] Ferré, L. and Yao, A. (2007). Reply to the paper “A note on smoothed functional inverse regression,” by L. Forzani and R. D. Cook. Statist. Sinica 17 1683-1687. · Zbl 1133.62028
[18] Forzani, L. and Cook, R. D. (2007). A note on smoothed functional inverse regression. Statist. Sinica 17 1677-1681. · Zbl 1132.62023
[19] Gu, C. (2002). Smoothing Spline ANOVA Models . Springer, New York. · Zbl 1051.62034
[20] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109-126. · Zbl 1141.62048 · doi:10.1111/j.1467-9868.2005.00535.x
[21] Hall, P., Müller, H. and Wang, J. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493-1517. · Zbl 1113.62073 · doi:10.1214/009053606000000272
[22] Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models . Chapman and Hall, New York. · Zbl 0747.62061
[23] Hsing, T. and Ren, H. (2009). An RKHS formulation of the inverse regression dimension reduction problem. Ann. Statist. 37 726-755. · Zbl 1162.62053 · doi:10.1214/07-AOS589
[24] James, G. A. and Silverman, B. W. (2005). Functional adaptive model estimation. J. Amer. Statist. Assoc. 100 565-576. · Zbl 1117.62364 · doi:10.1198/016214504000001556 · miranda.asa.catchword.org
[25] Li, K. C. (1991). Sliced inverse regression for dimension reduction. J. Amer. Statist. Assoc. 86 316-327. JSTOR: · Zbl 0742.62044 · doi:10.2307/2290563 · links.jstor.org
[26] Li, Y. (2007). A note on Hilbertian elliptically contoured distribution. Unpublished manuscript, Dept. Statistics, Univ. Georgia.
[27] Müller, H. G. and Stadtmüller, U. (2005). Generalized functional linear models. Ann. Statist. 33 774-805. · Zbl 1068.62048 · doi:10.1214/009053604000001156
[28] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006
[29] Schoenberg, I. J. (1938). Metric spaces and completely monotone functions. Ann. Math. 39 811-841. · Zbl 0019.41503 · doi:10.2307/1968466
[30] Schott, J. R. (1994). Determining the dimensionality in sliced inverse regression. J. Amer. Statist. Assoc. 89 141-148. JSTOR: · Zbl 0791.62069 · doi:10.2307/2291210 · links.jstor.org
[31] Spruill, M. C. (2007). Asymptotic distribution of coordinates on high dimensional spheres. Electron. Comm. Probab. 12 234-247. · Zbl 1132.62012 · eudml:128572
[32] Thodberg, H. H. (1996). A review of Bayesian neural networks with an application to near infrared spectroscopy. IEEE Transactions on Neural Network 7 56-72.
[33] Xia, Y., Tong, H., Li, W. K. and Zhu, L.-X. (2002). An adaptive estimation of dimension reduction space (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 64 363-410. JSTOR: · Zbl 1091.62028 · doi:10.1111/1467-9868.03411 · links.jstor.org
[34] Zhu, Y. and Zeng, P. (2006). Fourier methods for estimating the central subspace and the central mean subspace in regression. J. Amer. Statist. Assoc. 101 1638-1651. · Zbl 1171.62325 · doi:10.1198/016214506000000140
[35] Zhu, Y. and Zeng, P. (2008). An integral transform method for estimating the central mean and central subspace. J. Multivariate Anal. 101 271-290. · Zbl 1177.62054 · doi:10.1016/j.jmva.2009.08.004
[36] Zhang, J.-T. and Chen, J. (2007). Statistical inferences for functional data. Ann. Statist. 35 1052-1079. · Zbl 1129.62029 · doi:10.1214/009053606000001505
[37] Zhong, W., Zeng, P., Ma, P., Liu, J. and Zhu, Y. (2005). RSIR: Regularized sliced inverse regression for motif discovery. Bioinformatics 21 4169-4175.
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