A time series is a sequence of values measured over time. These values may be derived from a fixed deterministic formula, in which case they are referred to as a deterministic time series. Alternately, the value may be obtained by drawing a sample from a probability distribution, in which case they may be termed as probabilistic or stochastic time series. In this chapter, we will focus on stochastic time series.
Now, if the value at each instance in a stochastic time series is drawn from a probability distribution, how is it different from repeated drawings from a probability distribution? The added twist is that the probability distributions used for the drawings can themselves vary with time. The formal specification prescribing ways in which the distributions could vary with time and the discipline of analyzing stochastic time series was pioneered and popularized by Nobert Weiner.2
For this reason, the subject area is also referred to at times as Weiner filtering.
In the early days of Weiner filtering, the ideas were in theorem form, and to use them in practical applications one had to work through the rigorous mathematical definitions and theorems. Along came George Box and Gwilym Jenkins in the early 1970s, who formulated the application of Weiner filtering concepts into a recipe-like format. Their step-by-step prescription to the process of model building not only had great intuitive appeal but also managed to transform what was considered ...