Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 35

In reality, the output system signals are corrupted by noise, as well as by
chaotic signals generated by dynamic systems. Therefore, for modelling noise-
corrupted chaotic signals it should first be established whether the noise present
corrupts the systems state vector (like the system noise) in the form

y ( t +1) = F [ x ( t ) + n ( t )]

or, like the measurement disturbances, whether it only adds to the output signal

yt ˆ() . yt () nt ()

The generation of chaotic signals by dynamic systems is based on the
phenomena of initial-value sensitivity of the corresponding differential equations.
This was first pointed out by Poincare´. For instance, the sequence

x 4(1 x x )

n n 1 n 1

for any initial value and for any n = 0, 1, 2, … etc. , produces the solution 0 x 1

0

2 1

x

sin [sin ( x )] ,

0

which is highly sensitive to the initial value selected, because it determines the
value of the arcsin function. A small deviation

' contributes here the

2 n ' changes in ' . arcsin x x

0

0 0

2.6 Time-domain Models

Two typical time series modelling approaches in the time domain are to build the

x transfer function model
x state space model

Both models are of fundamental importance in traditional and modern control
theory.

2.6.1 Transfer-function Models

Transfer-function models are the extension of regression models in which the
transfer function of a dynamic system is integrated into the model. This is used in
systems theory for representation of relationships between the systems input and
output variables.
Building transfer-function models is based on experimental records of input
and output time series. In engineering practice, transfer-function estimation is
preferred because it does not require any system disturbance, say by step, pulse, or