The generalized harmonic analysis, or spectral decomposition, of a time series results in its representation in terms of its harmonic, or sinusoidal, components. This paper, on the other hand, develops in an expository manner the generalized regression analysis, or predictive decomposition, of a time series. This decomposition results in the representation of the time series at any moment in terms of its own observable past history plus an unpredictable, random‐like innovation. For the purposes of this paper, it is assumed that a seismic trace (recorded with automatic volume control) is additively composed of many overlapping seismic wavelets which arrive as time progresses. It is assumed that each wavelet has the same stable, one‐sided, minimum‐phase shape and that the arrival times and strengths of these wavelets may be represented by a time sequence of uncorrelated random variables. By applying the predictive decomposition theorem, it is shown how the wavelet shape may be extracted from the trace, leaving as a residual the strengths of the wavelets at their respective arrival times.
A Quantum Version of Spectral Decomposition Theorem of dynamical systems, quantum chaos hierarchy: Ergodic, mixing and exact
