Optimal transformations for prediction in continuous-time stochastic processes: finite past and future

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 L[squared]-sense). In this paper we treat a variant of the prediction problem, whereby we seek the "most predictabl...

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Bibliographic Details
Published in:Probability theory and related fields 2005-04, Vol.131 (4), p.479-492
Main Authors: GIDAS, Basilis, MURUA, Alejandro
Format: Article
Language:English
Subjects:
Boundary conditions
Differential equations
Eigenvalues
Exact sciences and technology
Inference from stochastic processes
time series analysis
Linear prediction
Mathematical analysis
Mathematics
Optimization
Ordinary differential equations
Predictions
Probability
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Statistics
Stochastic models
Stochastic processes
Studies
Transformations (mathematics)
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Summary: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 L[squared]-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. [PUBLICATION ABSTRACT]
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-004-0371-x