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Nonlinear principal components and long-run implications of multivariate diffusions. (English) Zbl 1191.62107
Summary: We investigate a method for extracting nonlinear principal components (NPCs). These NPCs maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and multivariate probability densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these NPCs. By exploiting the theory of continuous-time, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the NPCs maximize the long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the NPCs behave as scalar autoregressions with heteroskedastic innovations; this supports semiparametric identification and estimation of a multivariate reversible diffusion process and tests of the overidentifying restrictions implied by such a process from low-frequency data. We also explore implications for stationary, possibly nonreversible diffusion processes. Finally, we suggest a sieve method to estimate the NPCs from discretely-sampled data.

MSC:
62H25 Factor analysis and principal components; correspondence analysis
60J60 Diffusion processes
47N30 Applications of operator theory in probability theory and statistics
47D07 Markov semigroups and applications to diffusion processes
35P05 General topics in linear spectral theory for PDEs
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[1] Agmon, S. (1965). Lectures on Elliptic Boundary Problems . Van Nostrand, Princeton, NJ. · Zbl 0142.37401
[2] Azencott, R. (1974). Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France 102 193-240. · Zbl 0293.60071
[3] Banon, G. (1978). Nonparametric identification of diffusions. SIAM J. Control Optim. 16 380-395. · Zbl 0404.93045
[4] Benko, M., Hardle, W. and Kneip, A. (2009). Common functional principal components. Ann. Statist. 37 1-34. · Zbl 1169.62057
[5] Beurling, A. and Deny, J. (1958). Espaces de Dirichlet i , le cas elementaire. Acta Math. 99 203-224. · Zbl 0089.08106
[6] Bhattacharya, R. N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete 60 185-201. · Zbl 0468.60034
[7] Box, G. E. P. and Tiao, G. C. (1977). Canonical analysis of multiple time series. Biometrika 64 355-365. JSTOR: · Zbl 0362.62091
[8] Brezis, H. (1983). Analyse Fonctionelle . Masson, Paris.
[9] Chen, X., Hansen, L. P. and Scheinkman, J. (1998). Shape-preserving estimation of diffusions. Working paper, Univ. Chicago.
[10] Cobb, L. P., Kopstein, P. and Chen, N. (1983). Estimation and moment recursion relations for multimodal distributions in the exponential family. J. Amer. Statist. Assoc. 78 124-130. · Zbl 0508.62015
[11] Darolles, S., Florens, J. P. and Gourieroux, C. (2004). Kernel based nonlinear canonical analysis and time reversibility. J. Econometrics 119 323-353. · Zbl 1282.91265
[12] Dauxois, J. and Nkiet, G. M. (1998). Nonlinear canonical analysis and independence tests. Ann. Statist. 26 1254-1278. · Zbl 0934.62061
[13] Davies, E. B. (1985). l 1 properties of second order operators. Bull. London Math. Soc. 17 417-436. · Zbl 0583.35032
[14] Davies, E. B. (1989). Heat Kernels and Spectral Theory . Cambridge Univ. Press, Cambridge. · Zbl 0699.35006
[15] Demoura, S. (1998). The nonparametric estimation of the expected value operator. Workshop Presentation, Univ. Chicago.
[16] Fan, J. (2005). A selective overview of nonparametric methods in financial econometrics (with discussion). Statist. Science 20 317-357. · Zbl 1130.62364
[17] Florens, J. P., Renault, E. and Touzi, N. (1998). Testing for embeddability by stationary reversible continuous-time Markov processes. Econometric Theory 69 744-69. JSTOR: · Zbl 04549530
[18] Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes . Walter de Gruyter, Berlin. · Zbl 0838.31001
[19] Gobet, E., Hoffmann, M. and Reib, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Statist. 26 2223-2253. · Zbl 1056.62091
[20] Hall, P., Muller, H. G. and Wang, J. L. (2006). Properties of principal components methods for functional and longitudinal data analysis. Ann. Statist. 34 1493-1517. · Zbl 1113.62073
[21] Hansen, L. P. and Scheinkman, J. (1995). Back to the future. Econometrica 63 767-804. JSTOR: · Zbl 0834.60083
[22] Hansen, L. P., Scheinkman, J. and Touzi, N. (1998). Spectral methods for identifying scalar diffusions. J. Econometrics 86 1-32. · Zbl 0962.62094
[23] Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology 17 417-441. · JFM 59.1182.04
[24] Kessler, M. and Sorensen, M. (1999). Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli 5 299-314. · Zbl 0980.62074
[25] Nelson, E. (1958). The adjoint Markov process. Duke Math. J. 25 671-690. · Zbl 0084.13402
[26] Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors. Biometrika 95 365-379. Available at http://biomet.oxfordjournals.org/cgi/reprint/95/2/365.pdf, http://biomet.oxfordjournals.org/cgi/content/abstract/95/2/365. · Zbl 1437.62574
[27] Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. Philosophical Magazine 2 559-572. · JFM 32.0710.04
[28] Ramsay, J. and Silverman, B. W. (2005). Functional Data Analysis . Springer, New York. · Zbl 1079.62006
[29] Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics IV: Analysis of Operators . Academic Press, San Diego. · Zbl 0401.47001
[30] Rudin, W. (1973). Functional Analysis . McGraw-Hill, New York. · Zbl 0253.46001
[31] Salinelli, E. (1998). Nonlinear principal components i : Absolutely continuous variables. Ann. Statist. 26 596-616. · Zbl 0929.62067
[32] Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1-24. · Zbl 0853.62044
[33] Wong, E. (1964). The construction of a class of stationary Markov processes. In Stochastic Processes in Mathematical Physics and Engineering (R. E. Bellman, ed.). Proceedings of Symposia in Applied Mathematics 17 264-276. Amer. Math. Soc., Providence, RI. · Zbl 0139.34406
[34] Zhou, J. and He, X. (2008). Dimension reduction based on constrained cononical correlation and variable filtering. Ann. Statist. 36 1649-1668. · Zbl 1142.62045
[35] Zhou, L., Huang, J. Z. and Carroll, R. J. (2008). Joint modelling of paried sparse functional data using principal components. Biometrika 95 601-619. · Zbl 1437.62676
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