×

zbMATH — the first resource for mathematics

An RKHS formulation of the inverse regression dimension-reduction problem. (English) Zbl 1162.62053
Summary: Suppose that \(Y\) is a scalar and \(X\) is a second-order stochastic process, where \(Y\) and \(X\) are conditionally independent given the random variables \(\xi _{1}, \dots , \xi _p\) which belong to the closed span \(L_X^{2}\) of \(X\). This paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of \(L_X^{2}\) with the reproducing kernel Hilbert space of \(X\) provides a platform for a seamless extension from the finite- to infinite-dimensional settings. It also facilitates convenient computational algorithms that can be applied to a variety of models.

MSC:
62H12 Estimation in multivariate analysis
62M99 Inference from stochastic processes
46N30 Applications of functional analysis in probability theory and statistics
62H99 Multivariate analysis
62J99 Linear inference, regression
65C60 Computational problems in statistics (MSC2010)
Software:
fda (R); gss
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Amato, U., Antoniadis, A. and De Feis, I. (2006). Dimension reduction in functional regression with applications. Cumput. Statist. Data Anal. 50 2422-2446. · Zbl 1445.62078
[2] Aronszajn, N. (1950). Theory of reproducing kernel. Trans. Amer. Math. Soc. 68 337-404. JSTOR: · Zbl 0037.20701
[3] Ash, R. B. and Gardner, M. N. (1975). Topics in Stochastic Processes. Probability and Mathematical Statistics 27 . Academic Press, New York. · Zbl 0317.60014
[4] Berlinet, A. and Thomas-Agnan, C. (2004). Reproducing Kernel Hilbert Spaces in Probability . Kluwer, Dordrecht. · Zbl 1145.62002
[5] Chen, C.-H. and Li, K. C. (1998). Can SIR be as popular as multiple linear regression? Statist. Sinica 8 289-316. · Zbl 0897.62069
[6] Chiaromonte, F. and Martinelli, J. (2002). Dimension reduction strategies for analyzing global gene expression data with a response. Math. Biosci. 176 123-144. · Zbl 0999.62090
[7] Cook, R. D. (1998). Regression Graphics . Wiley, New York. · Zbl 0903.62001
[8] Cook, R. D. and Li, B. (2002). Dimension reduction for conditional mean in regression. Ann. Statist. 30 455-474. · Zbl 1012.62035
[9] Dauxois, J., Ferré, L. and Yao, A. F. (2001). Un modèle semi-paramétrique pour variables aléatoires hilbertiennes. C. R. Acad. Sci. Paris Sér. I Math. 333 947-952. · Zbl 0996.62035
[10] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136-154. · Zbl 0539.62064
[11] Driscoll, M. F. (1973). The reproducing kernel Hilbert space structure of the sample paths of a Gaussian process. Z. Wahrsch. Verw. Gebiete 26 309-316. · Zbl 0251.60033
[12] Duan, N. and Li, K. C. (1991). Slicing regression: A link-free regression method. Ann. Statist. 19 505-530. · Zbl 0738.62070
[13] Dunford, N. and Schwarz, J. T. (1988). Linear Operators . Wiley, New York.
[14] Eubank, R. and Hsing, T. (2007). Canonical correlation for stochastic processes. Stochastic Process. Appl. 118 1634-1661. · Zbl 1145.62048
[15] Ferré, L. and Yao, A. F. (2003). Functional sliced inverse regression analysis. Statistics 37 475-488. · Zbl 1032.62052
[16] Ferré, L. and Yao, A. F. (2005). Smooth function inverse regression. Statist. Sinica 15 665-683. · Zbl 1086.62054
[17] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice . Springer, New York. · Zbl 1119.62046
[18] Fortet, R. M. (1973). Espaces à noyau reproduisant et lois de probabilités des fonctions alèatoires. Ann. Inst. H. Poincaré Ser. B (N.S.) 9 41-48. · Zbl 0264.60028
[19] Gohberg, I. C. and Kreĭn, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators . Amer. Math. Soc., Providence, RI. · Zbl 0181.13504
[20] Gu, C. (2002). Smoothing Spline ANOVA Models . Springer, New York. · Zbl 1051.62034
[21] Hall, P. and Li, K. C. (1993). On almost linearity of low-dimensional projections from high-dimensional data. Ann. Statist. 21 867-889. · Zbl 0782.62065
[22] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis . Cambridge Univ. Press., Cambridge. · Zbl 0704.15002
[23] James, G., Hastie, T. and Sugar, C. (2000). Principal component models for sparse functional data. Biometrika 87 587-602. JSTOR: · Zbl 0962.62056
[24] Li, K. C. (1991). Sliced inverse regression for dimension-reduction. J. Amer. Statist. Assoc. 86 316-342. JSTOR: · Zbl 0742.62044
[25] Li, K. C. (1992). On Principal Hessian directions for data visualization and dimension-reduction: Another application of Stein’s lemma. J. Amer. Statist. Assoc. 87 1025-1039. JSTOR: · Zbl 0765.62003
[26] Loève, M. (1948). Fonctions Aléatoires du Second Ordre. Supplement to P. Lévy. Processus Stochastiques et Mouvement Brownien . Gauthier-Villars, Paris.
[27] Lukić, M. N. and Beder, J. H. (2001). Stochastic process with sample paths in reproducing kernel Hilbert spaces. Trans. Amer. Math. Soc. 353 3945-3969. JSTOR: · Zbl 0973.60036
[28] Parzen, E. (1959). Statistical inference on time series by Hilbert space methods. I. Technical Report, No. 23, Dept. Statistics, Stanford Univ.
[29] Parzen, E. (1961a). An approach to time series analysis. Ann. Math. Statist. 32 951-989. · Zbl 0107.13801
[30] Parzen, E. (1961b). Regression analysis of continuous parameter time series. In Proc. 4th Berkeley Sympos. Math. Statist. Probab. 1 469-489. Univ. California Press, Berkeley. · Zbl 0107.13802
[31] Parzen, E. (1963). Probability density functionals and reproducing kernel Hilbert spaces. Proc. Sympos. Time Series Analysis (M. Rosenblatt, ed.) 155-169. Wiley, New York. · Zbl 0168.18101
[32] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer. · Zbl 1079.62006
[33] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233-243. JSTOR: · Zbl 0800.62214
[34] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance . Chapman and Hall, New York. · Zbl 0925.60027
[35] Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1-24. · Zbl 0853.62044
[36] Wahba, G. (1990). Spline Models for Observational Data . CBMS 59, SIAM, Philadelphia, PA. · Zbl 0813.62001
[37] Wu, W. B. and Pourahmadi, M. (2003). Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90 831-844. · Zbl 1436.62347
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.