Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 19

superposition of true time series values and a disturbance with upward or
downward trend. The presence of a disturbing component is detectable by pursuing
the changes in the mean values in certain successive time intervals across the time
series pattern.
Trend analysis is important in time series forecasting. In practice, it is
accomplished using linear and nonlinear regression technique that satisfactorily
helps in identifying non-monotonous trend component in the time series. For
instance, for identifying the character of the trend present in a time series, the
linear, exponential, or polynomial relation

x DEH

DEH
DEJH

t

t t

x exp( t )

t t

x t t

2

t t

is used for fitting the collected data.

2.2.1.4 Seasonality
The seasonality component of a time series is demonstrated through its periodically
fluctuating pattern. This feature is more common in economic time series and in
time series in which the observations are taken from real life, where the pattern
may repeat hourly, daily, weekly, monthly, yearly, etc . Thus, the main objective of
seasonal time series analysis is focused on the detection of the character of its
periodical fluctuations and on their interpretation. In engineering, seasonal time
series are found in the problems of power, gas, water, and other distribution
systems, where the prediction of consumer demands represents the basic problem.

2.2.1.5 Estimation and Elimination of Trend and Seasonality
When two or more time series with different features are superimposed, or when a
time series is superimposed by trend and/or seasonality component, decomposition
analysis is needed to discriminate and separate individual components involved.
More frequently, decomposition analysis is used for de- trending and de-
seasonalizing the time series data. A classical decomposition example is complex
decomposition, where a time series could be made up of various components, such
as trend, random, seasonal, and cycling components. In this context, the seasonal
component S ( t ) is viewed as a periodic component with a fixed cycling period
corresponding to the individual seasons. In practice, it is convenient to combine the
trend and the cyclical components into a trend-cycle component TC ( t ), so that the
observed resulting value of the time series X at time t can be written as

X ( t ) = S ( t ) + R ( t ) + TC ( t ),

where R ( t ) is the random component. This is the additive representation model of a
multi-component time series. The corresponding multiplicative representation
model is