Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 40

(FFT), and then by computing the power spectral density function as a measure of
the energy at various frequencies

yy PYY

*

where “*” represents the one-to-one multiplication of vector components or of
matrix elements, and Y represents the complex conjugate of Y . The calculated
values y P are then plotted against the frequencies. From the resulting plot, the
major frequency components present in the residual time series signal can be
identified.
The FFT is a computational technique that substantially reduces the time
required to perform Fourier transformations on a digital computer. Introduced in
the 1960s by Cooley and Tukey (1965), the transformation has steadily increased
its popularity.

2.8 Model Building

The Box and Jenkins methodology (Box and Jenkins, 1976) for building time
series models includes a

x model identification phase , in which - apart from some preliminary
statistical calculations - the number of model parameters is determined that
are needed to ensure that the mathematical model to be built matches the
collected time series data with the desired accuracy
x model estimation phase , in which the values of model parameters are
estimated by minimizing the sum of squares of residuals
x model validation phase , in which the model accuracy is checked and the
possible model improvement is established
x model forecasting phase , in which the model is used to establish the
confidence limits of the forecast.
The above methodology, however, is not a straight-forward process, rather it is a
chain of iterative actions that Box and Jenkins described using the flow chart
shown in Figure 2.3.