Advertisement

Prediction and Filtering of Processes

  • M. M. Rao
Part of the Mathematics and Its Applications book series (MAIA, volume 508)

Abstract

This chapter is devoted to a different class of applications complementing the preceding work. The first section contains a comparative analysis of general prediction operations relative to a convex loss function, and its relation to projection operators. This is refined in the next section, for least squares prediction with the Cramér-Hida method. Then Section 3 treats linear filters as formulated by Bochner [2]. The results are specialized and sharpened in Section 4 for linear Kalman-Bucy filters of interest in many applications. Then in Section 5, we consider nonlinear filtering, which is a counter part of the preceding showing that there are many new possibilities, as well as illustrating the essential use of the general theory of SDEs in this subject. Thus Sections 3–5 contain mathematical glimpses of some of the vast filter technology. Finally some related complements are included as exercises, often with sketches of proof.

Keywords

Hilbert Space Conditional Expectation Generalize Inverse Orlicz Space Borel Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical notes

  1. [1]
    Yaglom, A. M. “ Second order homogeneous random fields,” Proc. 4th Berkeley Symp. Math. Statist. and Prob., 2 (1960), 593–622.Google Scholar
  2. [1]
    Bru, B., and Heinich, H. “Meilleurs approximations et médeines conditionnelles,” Annales Inst. Henri Poincaré, 21 (1985), 197–224.MathSciNetMATHGoogle Scholar
  3. [1]
    Urbanik, K. “Some prediction problems for strictly stationary processes,” Proc. 5th Berkeley Symp. Math. Statist. and Prob., 2 -I (1966), 235–258. Vakhania, N. N., and Tarieladze, V. I.Google Scholar
  4. [1]
    Rao, C. R., and Mitra, S. K. Generalized Inverse of Matrices and Its Applications,Wiley, New York, 1971.Google Scholar
  5. [1]
    Hannan, E. J. The concept of a filter,“ Proc. Carob. Phil. Soc.,63 (1967), 221–227.Google Scholar
  6. [1]
    Kalman, R. E. A new approach to linear filtering and prediction problems,“ J. Basic Eng.,82 (1960), 35–45.CrossRefGoogle Scholar
  7. [1]
    Kalman, R. E., and Bucy, R. S. New results in linear filtering and prediction theory,“ J. Basic Eng.,83 (1961), 95–108.Google Scholar
  8. [1]
    Leonov, V. P., and Shiryayev, A. N. On the technique of computing semi-invariants,“ Theor. Prob. Appl.,4 (1959), 319–329.Google Scholar
  9. [1]
    Shald, S. The continuous Kalman filter as the limit of the discrete Kalman filter,“ Stoch. Anal. Appl., 17 (1999), 841–856.MathSciNetMATHCrossRefGoogle Scholar
  10. [1]
    Elliot, R. J., and Glowinski, R. “Approximations to solutions of the Zakai filtering equation,” Stoch. Anal. Appl. 7 (1989) 145–168.Google Scholar
  11. [1]
    Lototsky, S., and Rozovskii, B. L. Recursive multiple Wiener integral expansion for nonlinear filtering of diffusion processes,“ In Stochastic Processes and Functional Analysis,Lect. Notes in Pure and Appl. Math., 186, Marcel Dekker, New York, (1997), 199–208.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.University of CaliforniaRiversideUSA

Personalised recommendations