Abstract
Frequency domain properties of the operators to decompose a time series into the multi-components along the Akaike's Bayesian model (Akaike (1980, Bayesian Statistics, 143–165, University Press, Valencia, Spain)) are shown. In that analysis a normal disturbance-linear-stochastic regression prior model is applied to the time series. A prior distribution, characterized by a small number of hyperparameters, is specified for model parameters. The posterior distribution is a linear function (filter) of observations. Here we use frequency domain analysis or filter characteristics of several prior models parametrically as a function of the hyperparameters.
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References
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, 2nd International Symposium on Information Theory (eds. B. N. Petrov and F. Csáki), 267–281, Akadémiai Kiadó, Budapest.
Akaike, H. (1980). Likelihood and the Bayes procedure (with discussion), Bayesian Statistics, 143–165, University Press, Valencia, Spain.
Akaike, H. and Ishiguro, M. (1983). Comparative study of the X-11 and BAYSEA procedures of seasonal adjustment, Applied Time Series Analysis of Economic Data (ed. A. Zellner), 17–53, U.S. Department of Commerce, Bureau of the Census, Washington, D.C.
Fukao, Y. and Suda, N. (1989). Core modes of the earth's free oscillations and structure of the inner core, Geophysical Research Letter, 16, 401–404.
Gersch, W. and Kitagawa, G. (1983). The prediction of time series with trends and seasonalities, J. Business & Economic Statistics, 1, 253–264.
Gersch, W. and Kitagawa, G. (1988). Smoothness priors in time series, Bayesian Analysis of Time Series and Dynamic Models (ed. J. C. Spall), 431–476, Dekker, New York and Basel.
Gersch, W. and Kitagawa, G. (1989). Smoothness priors transfer function estimation, Automatica—J. IFAC, 25, 603–608.
Good, I. J. (1964). The Estimation of Probabilities, MIT Press, Cambridge, Massachusetts.
Good, I. J. and Gaskin, R. A. (1971). Non-parametric roughness penalties for probability densities, Biometrika, 58, 255–277.
Good, I. J. and Gaskin, R. A. (1980). Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data (with discussion), J. Amer. Statist. Assoc., 75, 42–73.
Green, P. J. (1987). Penalized likelihood for general semi-parametric regression models, Internat. Statist. Rev., 55, 245–259.
Higuchi, T., Kita, K. and Ogawa, T. (1988). Bayesian statistical inference to remove periodic noise in the optical observations aboard a spacecraft, Applied Optics, 27, 4514–4519.
Ishiguro, M. (1984). Computationally efficient implementation of a Bayesian seasonal adjustment procedure, J. Time Ser. Anal., 5, 245–253.
Ishiguro, M. and Arahata, E. (1982). A Bayesian spline regression, Proc. Inst. Statist. Math., 30, 29–36 (in Japanese).
Ishiguro, M. and Sakamoto, Y. (1983). A Bayesian approach to binary response curve estimation, Ann. Inst. Statist. Math., 35, 115–137.
Ishiguro, M. and Sakamoto, Y. (1984). A Bayesian approach to the probability density estimation, Ann. Inst. Statist. Math., 36, 523–538.
Karl, J. H. (1989). An Introduction of Digital Signal Processing, Academic Press, San Diego.
Kashiwagi, N. (1982). Estimation of fertilities in field experiments, Proc. Inst. Statist. Math., 30, 1–10 (in Japanese).
Kay, S. M. and Marple, S. L. Jr. (1989). Spectrum analysis—A modern perspective, Proc. IEEE, 69, 1380–1419.
Kita, K., Higuchi, T. and Ogawa, T. (1989). Bayesian statistical inference of airglow profiles from rocket observational data: comparison with conventional methods, Planet Space Science, 37, 1327–1331.
Kitagawa, G. (1981). A nonstationary time series model and its fitting by a recursive filter, J. Time Ser. Anal., 2, 103–116.
Kitagawa, G. (1983). Changing spectrum estimation, J. Sound Vibration, 89, 433–445.
Kitagawa, G. (1986). Decomposition of a nonstationary time series—An introduction of the program DECOMP, Proc. Inst. Statist. Math., 34, 255–271 (in Japanese).
Kitagawa, G. (1987). Non-Gaussian state-space modeling of nonstationary time series (with discussion), J. Amer. Statist. Assoc., 79, 1032–1063.
Kitagawa, G. and Gersch, W. (1984). A smoothness priors-state space modeling of time series with trend and seasonality, J. Amer. Statist. Assoc., 79, 378–389.
Kitagawa, G. and Gersch, W. (1985a). A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series, IEEE Trans. Automat. Control, AC 30, 57–65.
Kitagawa, G. and Gersch, W. (1985b). A smoothness priors long AR model method for spectral estimation, IEEE Trans. Automat. Control, AC 30, 57–65.
Koike, M. (1990). A study on stratospheric ozone distribution with the backscattered ultraviolet spectrometer aboard the satellite OZORA, Doctor Thesis, University of Tokyo.
Leonard, T. (1978). Density estimation, stochastic process and prior information (with discussion), J. Roy. Statist. Soc. Ser. B, 40, 113–146.
Lindley, D. V. and Smith, A. F. M. (1972). Bayes estimates of the linear model (with discussion), J. Roy. Statist. Soc. Ser. B, 34, 1–41.
Nakamura, T. (1986). Bayesian cohort models for general cohort table analysis, Ann. Inst. Statist. Math., 38, 353–370.
Sakamoto, Y. and Ishiguro, M. (1988). A Bayesian approach to nonparametric test problems, Ann. Inst. Statist. Math., 40, 587–602.
Sakamoto, Y., Ishiguro, M. and Kitagawa, G. (1986). Akaike Information Criterion Statistics, Reidel, Dordrecht.
Silverman, B. W. (1984a). Spline smoothing: the equivalent variable kernel method, Ann. Statist., 12, 898–916.
Silverman, B. W. (1984b). A fast and efficient cross-validation method for smoothing parameter choice in spline regression, J. Amer. Statist. Assoc., 79, 584–589.
Silverman, B. W. (1985). Some aspects of the spline smoothing approach to non-parametric regression curve fitting (with discussion), J. Roy. Statist. Soc. Ser. B, 47, 1–52.
Silverman, B. W. (1986). Density estimation for statistics and data analysis, Monographs on Statist. Appl. Probab., Chapman and Hall, London.
Tamura, Y. H. (1987). An approach to the nonstationary process analysis, Ann. Inst. Statist. Math., 39, 227–241.
Tanabe, K. and Tanaka, T. (1983). Fitting curves and surface by Bayesian modeling, Gekkan Chikyu, 5, 179–186 (in Japanese).
Titterington, D. M. (1985). Common structure of smoothing techniques in statisties, Internat. Statist. Rev., 53, 141–170.
Wahba, G. (1975). Smoothing noisy data with spline functions, Numer. Math., 24, 383–393.
Wahba, G. (1990). Spline Models for Observational Data, Society for industrial and Applied Mathematics, Philadelphia.
Wahba, G. and Wold, S. (1975). A completely automatic French curve—Fitting spline functions by cross validation, Comm. Statist., 4, 1–17.
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Higuchi, T. Frequency domain characteristics of linear operator to decompose a time series into the multi-components. Ann Inst Stat Math 43, 469–492 (1991). https://doi.org/10.1007/BF00053367
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DOI: https://doi.org/10.1007/BF00053367