Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 64

is regarded as such a system. It follows that the integration of forecasts is more
than their sum, because the performance of the integrated system is more than the
sum of the performances of its subsystems. So, the trustworthiness of the linear
forecast combination is quite questionable. Rather, more trust should be paid to a
nonlinear interrelation between the individual forecasts, such as

f \ [ F ( I ), F ( I ),..., F ( )]

kk

I

c 1 1 2 2

where \ is a nonlinear function. While the given information is processed by
individual forecasting models, it is likely that the parts of the entire information
can be lost. For instance, it could happen that the information set I i is not used
efficiently, or different forecasts may have different parts of information lost. This
is why as many different forecasts should be present in the combination as
possible, even when the individual forecast depends on the same set of
information. What still remains is how to determine the form of the nonlinear
relationship

\

2.10 Application Examples

In the following, some examples are given of practical applications of time series
analysis and forecasting in business and industry.

2.10.1 Forecasting Nonstationary Processes

As the first example, forecasting of a nonstationary non-seasonal time series is
taken, based on collected equidistantly spaced temperature values of an
uncontrolled chemical plant (Box and Jenkins, 1976). For forecasting, the ARMA
process model and the Holt-Winter exponential smoothing technique are used. It is
an experiment based on 226 time series data, approximately fitted by the model

z 0.8 z 1.8 z a

t 1 t 1 2 t 1 1 t 1

or by

ˆ (1)

z 0.8 z 1 .8 z

t t 1 t

where the time t is the origin at which the forecast

ˆ t zl is made and l is the lead
time of forecast, representing the number of time steps ahead the forecast should be
made with respect to origin, and a l

t is the random shock. Based on the above
model, the forecast has been made with the lead time l = 1 at different origins t = 2,
3, 4, ..., 225. Consequently, a total of m = 224 data have been generated as a Box
and Jenkins forecast series.