14
Overview
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 re-
ferred to as a deterministic time series. Alternately, the value may be ob-
tained 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 dis-
tributions 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.^1 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 rig-
orous 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 pre-
scription to the process of model building not only had great intuitive appeal
but also managed to transform what was considered an esoteric science into
a robust engineering discipline. The approach could now be readily applied
to forecasting problems. The methodology gained instant popularity with
time series analysts and has become the staple by far for the analysis of sto-
CHAPTER
2
Time Series
(^1) Nobert Weiner is also credited with coining the word cybernetics, the shortened
version of which is the ubiquitous cyber, which has by usage become a prefix for a
lot of terms associated with the Internet.