Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 18

Although for the majority of time series used in practice the stationarity is a
common assumption, forecasting of nonstationary time series is still of
considerable importance. For instance, in engineering, business, and economics the
collected observation data are better represented through nonstationary time series.
Also, nonstationary time series can be transformed into the equivalent stationary
time series by taking the differences between the successive data values along the
time series pattern, i.e. by simple or multiple differencing the given time series
data. This approach is generally recommended, because some stationary looking
time series can still be nonstationary. To resolve the stationarity problem
experimentally, the time series should first be partitioned into two or more “long
enough” segments that are apparently stationary, then the autocorrelation and
spectrum properties of each segment are checked and the results compared.

2.2.1.2 Linearity
Linearity of a time series indicates that the shape of the time series depends on it's
state, so that the current state determines the local time series pattern. If a time
series is linear, then it can be represented by a linear function of the present value
and the past values. Example of linear representations are the AR, MA, ARMA,
and ARIMA models (see Section 2.5), based on autoregression and/or on a moving
average technique. Nonlinear time series can be represented by the corresponding
nonlinear or bilinear models.
Time series represented by the linear model

f

X ¦ \ Z ,

t i t i

i f

generally describe a linear process, where

\ is a set of constants that satisfies the

condition

f

\ f ,

¦

i

i f

and is white noise with a zero mean value and variance

2

V
The multivariable form of a linear process is statistically defined by the relation

Z .

t

f

,

X ¦ CZ

t i t i

i f

where C represents a series of n u matrices with the absolutely summable
elements, and

Z

is the white noise with zero mean value and covariance matrix

t

Ȉ .

2.2.1.3 Trend
The trend component of a time series is its long-term feature that is manifested
through the local or global increase or decrease of data values as a consequence of