Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 33

The components of the corresponding mean vector , made up of the mean of each
variable, are calculated as

1
5

3

x ¦ x , where i = 1, 2, 3, …

mi ij

j 1

The elements of the corresponding variant-covariant matrix , or dispersion , made
up of the variances of the variables along the main diagonal and the covariances
between each pair of variables in the remaining location, are given by

1

T

s ¦ ( x x )( x x ) , , where i , j = 1, 2, 3.

ij ij , i mi j

5

mj

Principal components, which are linear combinations of random variables with
some characteristic properties with respect to the variances, play a key role in the
analysis of multivariate time series. For instance, the first principal component is
the sum of squares of the coefficients having the maximal variance. Furthermore,
the principal components are in fact the characteristic vectors of the covariance
matrix, so that they help in the study of the characteristic vectors and characteristic
roots .

2.5.7 Linear Time Series Models

Linear models of time series are based on linear relationships between the observed
values. Typical examples of linear models are the AR, MA, ARMA, and ARIMA
models.

2.5.8 Nonlinear Time Series Models

The difficulty in testing for nonlinearity in a given set of observation values calls
for special approaches to building adequate time series models. The observation set
of nonlinear time series may contain various shocks of different form and of
different intensity. In financial engineering practice, it is common to check the time
series nonlinearity using first a linear time series model. If the linear model does
not fit the major part of observation data, then a nonlinear model is built and tested.
However, the problem then is what nonlinear model should be selected that will
best fit the collected data (Casdagli and Eubank, 1992). There are some traditional
examples of such models like STAR ( smooth transition autoregression model ),
ARCH ( autoregressive conditional heteroskedasticity ) and the bilinear model ,
widely used in econometrics and financial forecasting. Recently, the Markov
switching model , threshold autoregression model , and smooth transition
a utoregression model are also becoming popular.
For STAR models there have been some nonlinear alternatives like

x DD x f ( x )( MO H x ) ,

t 0 1 t 1 t d t 1 t