Information Technology Reference
In-Depth Information
time series can only be represented in terms of a probability distribution function ,
then the time series is said to be non-deterministic or stochastic .
2.3.4 Multivariate Time Series
Multivariate time series are generated by simultaneous observation of two or more
processes. The observation values collected are represented here as vector values.
These kinds of observation are very common in engineering, where two or more
physical variables (temperature, pressure, flow, etc .) have to be simultaneously
sampled for building the model of a dynamic system.
Multivariate time series are best understood as being a set of simultaneously
built time series, the value of each series - apart from their internal dependency
within the series itself - also have an interdependency with the values of other
component series. Multivariate analysis , a branch of mathematical statistics
qualified for processing of multidimensional sampled data, is used for their
processing (Dillon and Goldstein, 1985; Johnson and Wichern, 1988).
2.3.5 Chaotic Time Series
Random components of a time series mainly fall into one of two categories:
x They are truly random , i.e . the observations are drawn from the underlying
probability distribution characterized by a statistical distribution function or
by statistical moments of data, such as mean, variance, skew, etc .
x They are chaotic , characterized by values that appear to be randomly
distributed and non-periodic, but are actually resulting from a completely
deterministic process.
The main feature of chaotic time series is that they have no definite periodicity, i.e .
they can be represented by the values that may be randomly repeated several times
without maintaining any definite periodicity. A typical example of a chaotic signal
generator is the nonlinear dynamic oscillating system
^
2
dt
2
xd
2
x
0.5
dxdt
,
which is sensitive to its initial conditions. This can be presented geometrically by
the trajectory of the system in the phase plane, in which the trajectory of non-
dissipative systems make up a set of nested closed curves, whereas those of
dissipative nonlinear systems for all initial conditions lead to trajectories that
either lie on a single surface or converge to individual points in phase space. The
set of surfaces and points in the phase space to which all trajectories of the system
converge is called the attractor of the system. The attractors of a chaotic system
can have a non-integral, i.e . fractal, dimensions and are called strange attractors .
Such attractors are very important for forecasting of chaotic time series.
Search WWH ::




Custom Search