Abstract.
In the classical Wiener-Kolmogorov linear prediction problem, one fixes a linear functional in the ‘‘future’’ of a stochastic process, and seeks its best predictor (in the L2-sense). In this paper we treat a variant of the prediction problem, whereby we seek the ‘‘most predictable’‘ non-trivial functional of the future and its best predictor; we refer to such a pair (if it exists) as an optimal transformation for prediction. In contrast to the Wiener-Kolmogorov problem, an optimal transformation for prediction may not exist, and if it exists, it may not be unique. We prove the existence of optimal transformations for finite ‘‘past’’ and ‘‘future’’ intervals, under appropriate conditions on the spectral density of a weakly stationary, continuous-time stochastic process. For rational spectral densities, we provide an explicit construction of the transformations via differential equations with boundary conditions and an associated eigenvalue problem of a finite matrix.
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Breiman, L., Friedman, J. H.: Estimating optimal transformations for multiple regression and correlation (with discussion). J. Am. Statis. Association 80, 580–619 (1985)
Butzer, P. L., Nessel, R. J.: Fourier Analysis and Approximation. New York: Academic Press, 1971
Dym, H., McKean, H. P.: Gaussian Processes, Function Theory, and the Inverse Spectral Theorem. Academic Press, Inc., New York, 1976
Garnett, J.: Bounded Analytic Functions. Academic Press, New York, 1981
Gelfand, I. M., Shilov, G. E.: Generalized Functions. Volume I. (Translated by Eugene Saletan). Academic Press, New York, 1964
Gelfand, I. M., Yaglom, A. M.: Calculation of the amount of information about a random function contained in another such function. Uspekhi Mat. Nauk 12 (73), 3–52 (1957)
Gidas, B., Murua, A.: Estimation and consistency for certain linear functionals of continuous-time processes from discrete data. Technical Report 473, Department of Statistics, University of Washington, Seattle, Washington, USA, 2004
Gidas, B., Murua, A.: Classification and clustering of stop consonants via nonparametric transformations and wavelets. In Int. Conf. Acoustics, Speech, and Signal Processing 1, 872–875 (1995)
Gidas, B., Murua, A.: Stop consonants discrimination and clustering using nonlinear transformations and wavelets. In: (eds). Levinson, S. E., Shepp, L. editors. Image Models (And Their Speech Model Cousins), v. 80 of IMA, pp. 13–62. Springer-Verlag, 1996
Gidas, B., Murua, A.: Optimal transformations for prediction in continuous time stochastic processes. In: (eds). Rajput, B., Karatzas, I., Taqqu, M. Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis, pp. 167–183. Birkhäuser, 1998
Helson, H., Szegö, G.: A problem in prediction theory. Ann. Mat. Pura Appl. 51, 107–138 (1960)
Ibragimov, I. A., Rozanov, Y. A.: Gaussian Random Processes. Springer-Verlag, New York, 1978
Koosis, P.: The Logarithmic Integral I. Cambridge University Press, 1988
Koosis, P.: The Logarithmic Integral II. Cambridge University Press, 1992
Partington, J. R.: An Introduction to Hankel Operators. Cambridge University Press, 1988
Peller, V. V., Krushchev, S. V.: Hankel Operators, Best Approximations, and Stationary Gaussian Processes. Russian Math. Surveys 37 (1), 61–144 (1982)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. 2, Academic Press, New York, 1974
Rozanov, Y. A.: Stationary Random Processes. Holden-Day, San Francisco, 1967
Rudin, W.: Real and Complex Analysis. McGraw-Hill, Inc., third edition, 1987
Yaglom, A. M.: Stationary Gaussian Processes Satisfying the Strong Mixing Condition and Best Predictable Functionals. Bernoulli-Bayes-Laplace Anniversary Vol., Proceedings of International Research Seminar, Stat. Lab., Univ. of California, Berkeley, Springer-Verlag 1965, pp. 241–252
Acknowledgments.
We thank one of the referees for bringing Yaglom’s paper to our attention, and the other referee for his constructive remarks. We also thank the referee whose question on equation (2.5) led to the Appendix.
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This research was partially supported by ARO (MURI grant) DAAH04-96-1-0445, NSF grant DMS-0074276, and CNPq grant 301179/00-0.
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Gidas, B., Murua, A. Optimal transformations for prediction in continuous-time stochastic processes: finite past and future. Probab. Theory Relat. Fields 131, 479–492 (2005). https://doi.org/10.1007/s00440-004-0371-x
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DOI: https://doi.org/10.1007/s00440-004-0371-x