Interpolation and Prediction
In this chapter we will take a closer look at certain important questions about stationary sequences. In the case of interpolation, we assume that a stationary process (discrete time) is being recorded continuously, but that one or more observations are missed (perhaps while the experimenter is out to lunch). It is then desired to reconstruct the missing observations as well as possible using all the others, both earlier and later than the ones which were omitted. For prediction, on the other hand, we assume that the entire history of the process is known up to a certain point in time, and on the basis of these observations one or more of the future values must be estimated as accurately as possible. Still another problem, which we won’t discuss here, involves filtering an observed process which consists of a “desired signal” plus “noise” in order to recover the signal alone from the combination. Once again, Wiener’s Cybernetics is a source of interesting historical background.
KeywordsSpectral Density Prediction Error Fourier Series Spectral Representation Spectral Decomposition
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- M. Riesz (1916): “Über die Randwerte einer analytischen Funktion,” Skandinaviske Mathematikerkongres 4, pp. 27–44.Google Scholar