Spectral Analysis and Filtering

Abstract

The notion that a time series exhibits repetitive or regular behavior over time is of fundamental importance because it distinguishes time series analysis from classical statistics which assumes complete independence over time. We have seen in Chapters 1 and 2 how dependence over time can be introduced through models that describe in detail the way certain empirical data behaves, even to the extent of producing forecasts based on the models. It is natural that models based on predicting the present as a regression on the past such as are provided by the celebrated ARIMA or state-space forms, will be attractive to statisticians, who are trained to view nature in terms of linear models. In fact, the difference equations used to represent these kinds of models are simply the discrete versions of linear differential equations that may, in some instances, provide the ideal physical model for a certain phenomenon. An alternate version of the way nature behaves exists, however, based on a decomposition of an empirical series into its regular components.