Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-17T12:44:19.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Alternating projections and interpolation of stationary processes

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Mohsen Pourahmadi
Mohsen Pourahmadi*
Affiliation:
Northern Illinois University
*
∗ Postal address: Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

Keywords

MISSING VALUESPREDICTION AND INTERPOLATION

MSC classification

Primary: 60G25: Prediction theory
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 921 - 931
Copyright
Copyright © Applied Probability Trust 1992 

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.)

Footnotes

Research supported by AFOSR-88–0284.

References

Abraham, B. (1981) Missing observations in time series. Commun. Statist. Theory Methods A10, 1643–1653.Google Scholar
Adamyan, V. M. and Arov, D. Z. (1968) A general solution of a problem in linear prediction of stationary processes. Theory Prob. Appl. 13, 394–407.CrossRefGoogle Scholar
Aronszajn, N. (1950) Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404.CrossRefGoogle Scholar
Bhansali, R. J. (1990) On a relationship between the inverse of a stationary covariance matrix and the linear interpolator. J. Appl. Prob. 27, 156–170.Google Scholar
Brieman, L. and Friedman, J. H. (1985) Estimating optimal transformation for multiple regression and correlation. J. Amer. Statist. Assoc. 80, 580–598.Google Scholar
Damsleth, E. (1980) Interpolating missing values in time series. Scand. J. Statist. 7, 33–39.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Gaffke, N. and Mathar, R. (1989) A cyclic projection algorithm via duality. Metrika 36, 29–54.CrossRefGoogle Scholar
Kolmogorov, A. N. (1941) Stationary sequences in a Hilbert space. Bull. Moscow State Univ. 2, 1–40.Google Scholar
Miamee, A. G. and Pourahmadi, M. (1988) Best approximation in Lp(dµ) and prediction problems of Szego, Kolmogorov, Yaglom and Nakazi. J. London Math. Soc. (2) 38, 133–145.Google Scholar
Nakazi, T. (1984) Two problems in prediction theory. Studia Math. 78, 7–14.Google Scholar
Pourahmadi, M. (1989) Estimation and interpolation of missing values of a stationary time series. J. Time Series Analysis 10, 149–169.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and its Applications. Wiley, New York.CrossRefGoogle Scholar
Salehi, H. (1974) On the bilateral linear predictor for minimal stationary stochastic processes. SIAM J. Appl. Math. 26, 502–507.Google Scholar
Wiener, N. and Masani, P. R. (1958) The prediction theory of multivariate stationary processes, II. Acta Math. 99, 93–137.CrossRefGoogle Scholar