Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 21

x chaotic.

Time series encountered in practice can have two or more of the properties listed
above. For instance, linear time series can be stationary, seasonal, and can have the
trend component incorporated. In the following we will mainly focus on linear,
nonlinear, univariate, multivariate, and chaotic time series.

2.3.1 Linear Time Series

Linear time series are generated through observation of linear processes ,
mathematically defined by linear models of the form

f

yt () D xt ( j ) ,

¦

j

j f

where the coefficients Į are subjected to the restriction

f

D f

¦

i

i f

Linear time series could be generated by second-order stationary processes that
are generally linear processes or they can be transformed to linear processes using
World's decomposition (Brockwell and Davis, 2002) technique for elimination of
its deterministic component.

2.3.2 Nonlinear Time Series

Many time series in engineering and macroeconometrics require nonlinear
modelling (see Section 2.5.8). Some of them are represented as bilinear time
series , modeled as

p q rs

x z a x b z c x z .

¦ ¦ ¦ ¦

t t iti jtj jtitj

i 1 j 1 i 1 j 1

2.3.3 Univariate Time Series

The term univariate time series refers to time series obtained by sampling a single
observation pattern, for instance the values of a single physical variable or of a
single time-dependent signal at equal time intervals. Thus, in univariate time series
the time is an implicit variable that is usually replaced by an index variable . If the
data sampling is equispaced then the index variable can be omitted.
Time series presented here in the majority of cases are univariate time series. In
the case where a univariate time series can be exactly represented by a
mathematical model, the time series is said to be deterministic . Otherwise, if the