Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T04:55:41.325Z Has data issue: false hasContentIssue false
  • English
  • Français

A comparison of linear versus non-linear prediction for polynomial functions of the Ornstein-Uhlenbeck process

Published online by Cambridge University Press:  14 July 2016

John Donelson III  and
Fred Maltz
Fred Maltz
Affiliation:
Lockheed Palo Alto Research Laboratory, Palo Alto, California
Get access Rights & Permissions [Opens in a new window]

Abstract

The general minimum mean square error prediction problem is studied for the class of processes defined by polynomial functions of the Ornstein-Uhlenbeck process. In particular, the relative error difference for non-linear prediction compared with optimal linear prediction is calculated. Limiting values are determined, including an upper bound for the maximal relative error difference. These results show no difference over the entire class for both long and short prediction lead times. For a single Hermite polynomial, the relative error difference is zero. An analytical solution for the maximal difference for lead times below a given threshold is derived. This latter result has been verified numerically on a digital computer for polynomials up to 17th degree.

Keywords

NON-LINEAR PREDICTIONLINEAR PREDICTIONHERMITE POLYNOMIAL; ORNSTEIN-UHLENBECK PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972 , pp. 725 - 744
Copyright
Copyright © Applied Probability Trust 1972 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Doob, J. L. (1953) Stochastic Processes. John Wiley, New York.Google Scholar
[2] Erdelyi, A. (editor) (1953) Higher Transcendental Functions. Bateman Manuscript Project, Vol. II. McGraw-Hill, New York.Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley, New York.Google Scholar
[4] Gantmacher, F. R. (1959) The Theory of Matrices. Vol. I. Chelsea, New York.Google Scholar
[5] Gradshteyn, I. S. and Ryzhik, I. M. (1965) Table of Integrals, Series and Products. Academic Press, New York.Google Scholar
[6] Grenander, U. and Szegö, G. (1958) Toeplitz Forms and their Applications. Univ. of Calif. Press, Berkeley.CrossRefGoogle Scholar
[7] Jaglom, A. M. (1971) Optimal non-linear extrapolation. Selected Translations in Mathematical Statistics and Probability, 273–298. Amer. Math. Soc., Providence, R. I. Google Scholar
[8] Wiener, N. (1949) Time Series. M. I. T. Press, Cambridge, Mass.Google Scholar