State space models on special manifolds. (English) Zbl 1099.62098

Summary: This paper concerns modeling time series observations in state space form considered on Stiefel and Grassmann manifolds. We develop a state space model relating the time series observations to a sequence of unobserved state or parameter matrices assuming the matrix Langevin noise processes on the Stiefel manifolds. We show a Bayes method for estimating the state matrices by the posterior modes. We consider a further extended state space model where two sequences of unobserved state matrices are involved. A simple state space model on the Grassmann manifolds with matrix Langevin noise processes is also investigated.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H12 Estimation in multivariate analysis
62F15 Bayesian inference
62H11 Directional data; spatial statistics
60G20 Generalized stochastic processes
Full Text: DOI


[1] Anderson, T.W., The statistical analysis of time series, (1971), Wiley New York · Zbl 0225.62108
[2] J. Breckling, The analysis of directional time series: applications to wind speed and direction, Lecture Notes in Statistics, vol. 61, Springer, New York, 1989. · Zbl 0698.62093
[3] Brockwell, P.J.; Davis, R.A., Time series: theory and methods, (1991), Springer New York · Zbl 0673.62085
[4] R.S. Bucy, Recent results in linear and non-linear filtering, in: 1968 Stochastic Problems in Control, Joint Automatic Control Conference, University of Michigan, Ann Arbor, MI, 1968, The American Society of Mechanical Engineers, New York, pp. 87-105.
[5] Y. Chikuse, Statistics on special manifolds, Lecture Notes in Statistics, vol. 174, Springer, New York, 2003. · Zbl 1026.62051
[6] Y. Chikuse, Multivariate time series models on Stiefel manifolds and related results, (2004), submitted for publication.
[7] Chikuse, Y.; Watson, G.S., Large sample asymptotic theory of tests for uniformity on the Grassmann manifold, J. multivariate anal., 54, 18-31, (1995) · Zbl 0863.62058
[8] Constantine, A.G., Some non-central distribution problems in multivariate analysis, Ann. math. statist., 34, 1270-1285, (1963) · Zbl 0123.36801
[9] Downs, T.D., Orientation statistics, Biometrika, 59, 665-676, (1972) · Zbl 0269.62027
[10] Durbin, J.; Koopman, S.J., Time series analysis by state space methods, (2001), Oxford University Press New York · Zbl 0995.62504
[11] Fahrmeir, L., Posterior mode estimation by extended Kalman filtering for multivariate dynamic generalized linear models, J. amer. statist. assoc., 87, 501-509, (1992) · Zbl 0781.62147
[12] Farrell, R.H., Multivariate calculation, (1985), Springer New York · Zbl 0575.62009
[13] Fisher, N.I.; Lewis, T.; Embleton, B.J.J., Statistical analysis of spherical data, (1987), Cambridge University Press London, New York · Zbl 0651.62045
[14] Hannan, E.J., Multiple time series, (1970), Wiley New York · Zbl 0211.49804
[15] Herz, C.S., Bessel functions of matrix argument, Ann. math., 61, 474-523, (1955) · Zbl 0066.32002
[16] James, A.T., Normal multivariate analysis and the orthogonal group, Ann. math. statist., 25, 40-75, (1954) · Zbl 0055.13203
[17] James, A.T., Distributions of matrix variates and latent roots derived from normal samples, Ann. math. statist., 35, 475-501, (1964) · Zbl 0121.36605
[18] Jupp, P.E.; Mardia, K.V., Maximum likelihood estimators for the matrix von mises – fisher and Bingham distributions, Ann. statist., 7, 599-606, (1979) · Zbl 0406.62012
[19] Kitagawa, G., Non-Gaussian state-space modeling of nonstationary time series, J. amer. statist. assoc., 82, 1032-1041, (1987) · Zbl 0644.62088
[20] Koopman, L.H., The spectral analysis of time series, (1974), Academic Press New York
[21] Mardia, K.V.; Jupp, P.E., Directional statistics, (2000), Wiley New York · Zbl 0707.62095
[22] Meinhold, R.J.; Singpurwalla, N.D., Understanding the Kalman filter, Amer. statist., 37, 123-127, (1983)
[23] Muirhead, R.J., Aspects of multivariate statistical theory, (1982), Wiley New York · Zbl 0556.62028
[24] Naik-Nimbalkar, U.V.; Rajarshi, M.B., Filtering and smoothing via estimating functions, J. amer. statist. assoc., 90, 301-306, (1995) · Zbl 0818.62082
[25] Watson, G.S., Orientation statistics in the Earth sciences, Bull. geol. inst. univ. Uppsala, 2, 73-89, (1970)
[26] G.S. Watson, Statistics on spheres, Lecture Notes in Mathematics, vol. 6, University of Arkansas, Wiley, New York, 1983. · Zbl 0646.62045
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