Linear least squares prediction in non-stochastic time series

Abstract

Many problems arising in the physical and social sciences relate to processes which happen sequentially. Such processes are usually investigated by means of the theory of stationary stochastic processes, but there have been some attempts to develop techniques which are not subject to the conceptual difficulties inherent in the probabilistic approach. These difficulties stem from the fact that in practice one is often restricted to a single record which, from the probabilistic point of view, is only one sample from an ensemble of possible records. In some instances such a viewpoint seems artificial, and for some time series it is questionable whether any objective reality corresponds to the idea of an ensemble of possible time series. For example, as noted in Feller (1967), a theory of probability based on a frequency interpretation cannot meaningfully attach a probability to a statement such as “the sun will rise tomorrow”, because to do so one would have to set up a conceptual universe of possible worlds.