A discrete-time signal or time series [1] is a set of
observations taken sequentially in time, space or some other
independent variable. Many sets of data appear as time series:
a monthly sequence of the quantity of goods shipped from a
factory, a weekly series of the number of road accidents,
hourly observations made on the yield of a chemical process
and so on. Examples of time series abound in such fields as
economics, business, engineering, natural sciences, medicine
and social sciences.
An intrinsic feature of a time series is that, typically,
adjacent observations are related or dependent. The nature of
this dependence among observations of a time series is of
considerable practical interest. Time Series Analysis is
concerned with techniques for the analysis of this dependence
[2]. This requires the development of models for time series
data and the use of such models in important areas of
application.
A discrete-time signal x (n) is basically a sequence of real
or complex numbers called samples. Discrete-time signals can
arise in various ways. Very often, a discrete-time signal is
obtained by periodically sampling a continuous-time signal,
that is x (n) = xc (nT), where T = 1 / Fs is the sampling period
and Fs is the sampling frequency. At other times, the samples
of a discrete-time signal are obtained by accumulating some
quantity over equal intervals of time, for example, the number
of cars per day traveling on a certain road. Financial signals,
like daily stock market prices are inherently discrete-time.
When successive observations of the series are dependent,
the past observations may be used to predict future values. If
the prediction is exact, the series is said to be deterministic.
We cannot predict a time series exactly in most practical
situations. Such time series are called random or stochastic,
and the degree of their predictability is determined by the
dependence between consecutive observations. The ultimate
case of randomness occurs when every sample of a random
signal is independent of all other samples. Such a signal,
which is completely unpredictable, is known as White noise
and is used as a building block to simulate random signals
with different types of dependence. To properly model and
predict a time series, it becomes important to fundamentally
and thoroughly analyze the signal it self, and hence there is a
strong need for signal analysis.