Abstract
In the classical Wiener-Kolmogorov prediction problem, one fixes a functional of the “future” and seeks its best predictor (in the L 2-sense). In this paper we treat a variant of this problem, whereby we seek the “most predictable” non-trivial functional of the future and its best predictor. In contrast to the Wiener-Kolmogorov problem, our problem may not have solutions, and if solutions exist, they might not be unique. We prove the existence of solutions for linear functionals under appropriate conditions on the spectral function of weakly stationary, continuous-time processes.
This Research was partially supported by ARO Grant DAAH04-95-1-0101, ONR Grant NOOO14-19-J-1021, and ARPA via ARLMDA972-93-1-0012. The manuscript was prepared using computer facilities supported in part by The University of Chicago Block Fund.
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References
A. Böottcher and B. Silbermann. Analysis of Toeplitz Operators. Springer-Verlag, 1990.
L. Breiman and J. H. Friedman. Estimating optimal transformations for multiple regression and correlation (with discussion). Journal of the American Statistical Association, 80:580–619, 1985.
L. Breiman and R. Ihaka. Nonlinear discriminant analysis via scaling and ACE. Technical report, University of California, Berkeley, Dept. of Statistics, 1988.
H. Dym and H. P. McKean. Gaussian Processes, Function Theory, and the Inverse Spectral Theorem. Academic Press, Inc., New York, 1976.
J. Garnett. Bounded Analytic Functions. Academic Press, New York, 1981.
B. Gidas and A. Murua. Estimation and consistency for Linear functional of continuous-time processes from a finite data set, II: Optimal transformations for prediction. Preprint.
B. Gidas and A. Murua. Optimal transformations for prediction in continuous-time stochastic processes with rational spectral densities. In preparation.
B. Gidas and A. Murua. Classification and clustering of stop consonants via nonparametric transformations and wavelets. In ICASSP-95, volume 1, pages 872–875, 1995.
B. Gidas and A. Murua. Stop consonants discrimination and clustering using nonlinear transformations and wavelets. In S. E. Levinson and L. Shepp, editors, Image Models (And Their Speech Model Cousins), volume 80 of IMA, pages 13–62. Springer-Verlag, 1996.
H. Helson and G. Szego. A problem in prediction theory. Ann. Mat. Pura Appl, 51:107–138, 1960.
I. A. Ibragimov and Y. A. Rozanov. Gaussian Random Processes. Springer-Verlag, New York, 1978.
P. Koosis. Introduction to Hp Spaces. Cambridge University Press, 1980.
P. Koosis. The Logarithmic Integral, volume 2. Cambridge University Press, 1988.
N. Levinson and H. P. McKean. Weighted trigonometrical approximation on the line with application to the germ field of a stationary gaussian noise. Acta. Math., 112:99–143, 1964.
W. Magnus and F. Oberthettinger. Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, 1966.
J. R. Partington. An Introduction to Hankel Operators. Cambridge University Press, 1988.
M. Reed and B. Simon. Methods of Modern Mathematical Physics, volume 1. Academic Press, New York, 1972.
Y. A. Rozanov. Stationary Random Processes. Holden-Day, San Francisco, 1967.
D. Sarason. Function theory on the unit disc. Virginia Polytechnique Institute and State University, Blacksburg, Virginia, 1978.
B. Simon. The P(ϕ)2 Euclidean (Quantum) Field Theory. Princeton University Press, Princeton, New Jersey, 197
B. Simon. Functional Integration and Quantum Physics. Academic Press, New York, 1979.
J. Weidman. Linear Operators in Hilbert Spaces. Springer-Verlag, New York, 1980.
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This paper is dedicated to the memory of Stamatis Cambanis-a good friend and a wonderful person. In the early stages of this paper (when we barely understood the problem), Stamatis provided a driving impetus through long and joyful conversations that would go on till 2:00 or 3:00 am (during a four day visit to Brown). He seems to have enjoyed these conversations himself: “I enjoyed my visit thoroughly, specially the long late night discussions! Perhaps something may come out” (e-mail to B. G.).
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Gidas, B., Murua, A. (1998). Optimal Transformations for Prediction in Continuous-Time Stochastic Processes. In: Karatzas, I., Rajput, B.S., Taqqu, M.S. (eds) Stochastic Processes and Related Topics. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2030-5_9
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DOI: https://doi.org/10.1007/978-1-4612-2030-5_9
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