Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 37

based on state vector

x and the system disturbance vector w and the other set

,

models the systems output

yCxv ,

t t t

based on the observability vector

y at system output and on the system output

disturbances vector A B and C represent the

system matrix , control matrix , and observation matrix respectively, all at the
instant t . In addition, the covariance matrices of the disturbance vectors are
supposed to be

In the two sets of equations,

v

. , ,

t t

cov{ wQ }

t t

cov{ vR }

t t

The objective of state-space modelling (Aoki, 1990) is to estimate the values of
the state vector and to forecast its future values based on observations

y

.

2.7 Frequency-domain Models

In analogy with the signal representation in the frequency domain, also a time
series can be represented in the same domain. This is because the time series
values are generated by equidistant sampling of signals. Therefore, a time series
made up of sampled values can also be presented as a collection of sine and cosine
waves with different frequencies and be processed using spectral analysis
(Warner, 1998).
A device that was designed to represent the given time series visually in the
frequency domain is the periodogram . It is a simple spectral analysis facility made
up of mixtures of sine and cosine components within a frequency spectrum.
To illustrate the calculation of the periodogram, suppose that the number of
observations is odd, say N = 2 q +1, and the Fourier series model to be fitted using
the observation data is given as

q

x a ( a Cb S ) e ,

¦

t 0 i it i it t

i 1

where

C cos(2 S

S

f t ),

it i

S sin(2 f t ),

it i

i

f .

i

N