19 Complexity-Based Analysis of Microvascular Blood …
303
two different skin sites, also in healthy individuals, demonstrated both local and
central mechanisms regulating low-frequency blood flow oscillations.
19.6.2 Attractor Reconstruction
Attractor reconstruction analysis has been used previously to determine changes
in the shape and variability of quasi periodic signals, achieving a two-dimensional
attractor and providing features such as density and symmetry. Attractor reconstruction has been previously applied to various physiological data including blood pressure [5, 60, 61], plethysmographic [87], electroencephalographic (EEG) [88] and
blood flow signals [7, 12, 32, 64]. The attractors reconstructed from these times
series, using a suitable time delay and embedding dimension, contain properties that
can be used to define the dynamics of the system and provide a visual representation
of the system’s stability [44].
In this approach, the signals are first reconstructed in three-dimensional space
with a time delay, τ , computed using the mutual information analysis where the
average mutual information between two instances i and i + τ reaches its first local
minimum [44, 81] and the first local maximum is the average period,T , of the signal.
Then, the ideal time delay will be either τ = T /3 or τ = 2T /3. So, for a time series
x(t), the two additional variables will be:
y(t) = x(t − τ ) and z(t) = x(t − 2τ ).
The reconstructed phase space can be now plotted as (x, y, z). Then, the variation
of the time series of the (x, y, z) attractor is removed by projecting the attractor in
two-dimensional space, referred as plane (v, w) perpendicular to the vector (1, 1, 1)
forming two new variables:
v =
1
√
6
(x + y − 2z) and w =
1
√
2
(x − y).
The two-dimensional plane (v, w) will be defined as periodic, with period T = 3τ
when a symmetric triangular shape is observed.
Recently, Aston et al. [5] have applied a new approach, attractor reconstruction
analysis (ARA), which quantifies the changes in the morphology and variability of a
quasi-periodic signal without affecting the signal information, to arterial blood pressure signals, photoplethysmogram signals and electrocardiogram signals captured
from animals and humans. ARA provides a two-dimensional colour-scaled representation of the signal producing features like density and symmetry by which Aston
and colleagues [4, 5, 16, 60] were able to identify changes in the shape and variability
of the signal associated with cardiovascular function. Similarly, González et al. [32]
investigating the attractors of rheoencephalographic signals in human volunteers,
303
two different skin sites, also in healthy individuals, demonstrated both local and
central mechanisms regulating low-frequency blood flow oscillations.
19.6.2 Attractor Reconstruction
Attractor reconstruction analysis has been used previously to determine changes
in the shape and variability of quasi periodic signals, achieving a two-dimensional
attractor and providing features such as density and symmetry. Attractor reconstruction has been previously applied to various physiological data including blood pressure [5, 60, 61], plethysmographic [87], electroencephalographic (EEG) [88] and
blood flow signals [7, 12, 32, 64]. The attractors reconstructed from these times
series, using a suitable time delay and embedding dimension, contain properties that
can be used to define the dynamics of the system and provide a visual representation
of the system’s stability [44].
In this approach, the signals are first reconstructed in three-dimensional space
with a time delay, τ , computed using the mutual information analysis where the
average mutual information between two instances i and i + τ reaches its first local
minimum [44, 81] and the first local maximum is the average period,T , of the signal.
Then, the ideal time delay will be either τ = T /3 or τ = 2T /3. So, for a time series
x(t), the two additional variables will be:
y(t) = x(t − τ ) and z(t) = x(t − 2τ ).
The reconstructed phase space can be now plotted as (x, y, z). Then, the variation
of the time series of the (x, y, z) attractor is removed by projecting the attractor in
two-dimensional space, referred as plane (v, w) perpendicular to the vector (1, 1, 1)
forming two new variables:
v =
1
√
6
(x + y − 2z) and w =
1
√
2
(x − y).
The two-dimensional plane (v, w) will be defined as periodic, with period T = 3τ
when a symmetric triangular shape is observed.
Recently, Aston et al. [5] have applied a new approach, attractor reconstruction
analysis (ARA), which quantifies the changes in the morphology and variability of a
quasi-periodic signal without affecting the signal information, to arterial blood pressure signals, photoplethysmogram signals and electrocardiogram signals captured
from animals and humans. ARA provides a two-dimensional colour-scaled representation of the signal producing features like density and symmetry by which Aston
and colleagues [4, 5, 16, 60] were able to identify changes in the shape and variability
of the signal associated with cardiovascular function. Similarly, González et al. [32]
investigating the attractors of rheoencephalographic signals in human volunteers,