Biomedical Engineering Reference
In-Depth Information
system by the minimal number of bits required to transfer the data. Mathematically,
the information quantity of a random event
A
is the logarithm of its occurrence
probability (
P
A
), that is, log
2
P
A
. Therefore, the number of bits needed for transfer-
ring
N-
symbol data (
A
i
) with probability distribution {
P
i
,
i
=
1, ...,
N
} is the averaged
information of each symbol:
SE
=−
P
log
2
P
(3.61)
i
i
A straight conclusion from (3.61) is that
SE
reaches its global maximum under
uniform distribution, that is,
SE
max
=
P
N
. Therefore,
SE
measures the extent to which the probability distribution of a random variable
diverges from a uniform one, and can be implemented to analyze the variation dis-
tribution of physiological signals, such as EEG and electromyogram (EMG).
log
2
(
N
) when
P
1
=
P
2
=
...
=
3.3.1.1 Formality of Entropy Implementation in EEG Signal Processing
Entropy has been used in EEG signal analysis in different formalities, including: (1)
approximate entropy (
ApEn
), a descriptor of the changing complexity in embed-
ding space [92, 93]; (2) Kolmogorov entropy (
K
2
), another nonlinear measure cap-
turing the dynamic properties of the system orbiting within the EEG attractor [94];
(3) spectral entropy, evaluating the energy distribution in wavelet subspace [95] or
uniformity of spectral components [96]; and (4) amplitude entropy, a direct uncer-
tainty measure of the EEG signals in the time domain [97-99]. In applications,
entropy has also been used to analyze spontaneous regular EEG [95, 96], epileptic
seizures [100], and EEG from people with Alzheimer's disease [101] and Parkin-
son's disease [102]. Compared with other nonlinear methods, such as fractal dimen-
sion and Lyapunov components, entropy does not require a huge dataset and, more
importantly, it can be used to investigate the interdependence across the cerebral
cortex [103, 104].
3.3.1.2 Beyond the Formalism of Shannon Entropy
The classic formalism in (3.61) has been shown to be restricted to the domain of
validity of Boltzmann-Gibbs statistics (BGS), which describes a system in which the
effective microscopic interactions and the microscopic memory are of short range.
Such a BGS-based entropy is generally applicable to extensive or additive systems.
For two independent subsystems
A
and
B
, their joint probability distribution is
equal to the product of their individual probability, that is,
(
)
( ) ( )
PAB PAPB
ij
∪=
(3.62)
,
i
j
where
P
i,j
(
A B
) is the probability of the combined system
A B
, and
P
i
(
A
) and
P
j
(
B
) are the probability distribution of systems
A
and
B
, respectively. Combining
(3.62)and (3.61), we can easily conclude additivity in such a combined system:
(
)
( )
( )
SE A
∪=
B
SE A
+
SE B
(3.63)
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