Biomedical Engineering Reference
In-Depth Information
D1
×
10
4
D2
1000
1000
1
0
0
0
−
1000
−
1000
−
1
0
1000
2000
3000
0
500
1000
1500
D3
D4
×
10
4
×
10
4
2
2
0
0
−
2
−
2
0
200
400
600
0
100
200
300
D5
A5
×
10
4
×
10
4
2
2
0
0
−
2
−
2
0
50
100
150
0
50
100
150
Figure 3.20
A five-level MRWA for a 4,096-point EEG data segment using the Daubechies 4
wavelet.
motivation is that a successful study of such complex systems may have a significant
impact on our ability to forecast their future behavior and intervene in time to con-
trol catastrophic crises.
In principle, the dynamics of complex nonlinear systems can be studied both by
analytical and numerical techniques. In the majority of these systems, analytical
solutions cannot be found following mathematical modeling, because either exact
nonlinear equations are difficult to derive from the data or to subsequently solve in
closed form. Given our inadequate knowledge of their initial conditions, individual
components, and intercomponent connections, mathematical modeling seems to be
a formidable task. Therefore, it appears that time-series analysis of such systems is a
viable alternative.
Although traditional linear time-series techniques appeared to enjoy initial suc-
cess in the study of several problems [57], it has progressively become clear that
additional information provided by employment of techniques from nonlinear
dynamics may be crucial to satisfactorily address these problems. Theoretically,
even simple nonlinear systems can exhibit extremely rich (complicated) behavior
(e.g., chaotic dynamics). Furthermore, standard linear methods, such as power spec-
trum analyses, Fourier transforms, and parametric linear modeling, may fail to cap-
ture and, in fact, may lead to erroneous conclusions about those systems' behavior
[58]. Thus, employing existing and developing new methods within the framework
of nonlinear dynamics and higher order statistics for the study of complex nonlinear
systems is of practical significance, and could also be of theoretical significance for
the fields of signal processing and time-series analysis.
Nonlinear dynamics has opened a new window for understanding the behavior
of the brain. Nonlinear dynamic measures of complexity (e.g., the correlation
dimension) and stability (e.g., the Lyapunov exponent and Kolmogorov entropy)
Search WWH ::
Custom Search