284
A. V. Tankanag et al.
18.2.3 Data Analysis
The correlation degree between the phases of respiratory-related skin blood flow
oscillations of left (SBF left ) and right (SBF right ) fingers was estimated from the value
of wavelet phase coherence [3, 14, 17–19, 21, 22]. We chose the complex valued
Morlet wavelet (t) = e
iω 0 t e
(−t
2 /2) . For each signal a spectral function X (ω k , t n ) =
a k,n + ib k,n was determined, describing the spectral properties of the x(t) signal in
the time–frequency domain. The phase difference for a pair of signals x 1 (t) and x 2 (t)
at any time point t n and frequency ω k was calculated, where x 1 (t) and x 2 (t) were
SBF left and SBF right , respectively. The coefficients were determined as
cos((ϕ k,n ) =
a 1k,n a 2k,n + b 1k,n b 2k,n
a 2
1k,n + b 2
1k,n
a 2
2k,n + b 2
2k,n
, sin((ϕ k,n ) =
b 1k,n a 2k,n − a 1k,n b 2k,n
a 2
1k,n + b 2
1k,n
a 2
2k,n + b 2
2k,n
,
and averaged over the whole signal length (N)
cos((ϕ k,n )
=
1
N
N
n=1
cos((ϕ k,n ),
sin((ϕ k,n )
=
1
N
N
n=1
sin((ϕ k,n ).
Subsequently an averaged over time wavelet phase coherence was calculated for each
analyzed frequency ω k
C ϕ (ω k ) =
cos((ϕ k,n )
2 +
sin((ϕ k,n )
2 .
The C ϕ (ω k ) function takes values from 0 to 1, containing the information on the
degree of phase coherence of the two signals x 1 (t) and x 2 (t) at frequency ω k .
The statistical analysis of the results obtained was carried out with SigmaPlot
13.0 software (Systat Software, Inc., 2014). Nonparametric statistical tests were
performed to avoid any assumption of normality. The Wilcoxon test was used to
estimate the significance of differences between the wavelet phase coherence values
obtained in spontaneous breathing and controlled regimes. The differences were
considered statistically significant at p < 0.05.
18.3 Results
Firstly, we tested reversibility of the breathing practices. The Fig. 18.2 presented
the phase coherence values for SBF left –SBF right pair under spontaneous breathing
before and after all controlled breathing tests. There are no differences between
phase coherence calculated for oscillations registered initially and after all breathing
practices.
A. V. Tankanag et al.
18.2.3 Data Analysis
The correlation degree between the phases of respiratory-related skin blood flow
oscillations of left (SBF left ) and right (SBF right ) fingers was estimated from the value
of wavelet phase coherence [3, 14, 17–19, 21, 22]. We chose the complex valued
Morlet wavelet (t) = e
iω 0 t e
(−t
2 /2) . For each signal a spectral function X (ω k , t n ) =
a k,n + ib k,n was determined, describing the spectral properties of the x(t) signal in
the time–frequency domain. The phase difference for a pair of signals x 1 (t) and x 2 (t)
at any time point t n and frequency ω k was calculated, where x 1 (t) and x 2 (t) were
SBF left and SBF right , respectively. The coefficients were determined as
cos((ϕ k,n ) =
a 1k,n a 2k,n + b 1k,n b 2k,n
a 2
1k,n + b 2
1k,n
a 2
2k,n + b 2
2k,n
, sin((ϕ k,n ) =
b 1k,n a 2k,n − a 1k,n b 2k,n
a 2
1k,n + b 2
1k,n
a 2
2k,n + b 2
2k,n
,
and averaged over the whole signal length (N)
cos((ϕ k,n )
=
1
N
N
n=1
cos((ϕ k,n ),
sin((ϕ k,n )
=
1
N
N
n=1
sin((ϕ k,n ).
Subsequently an averaged over time wavelet phase coherence was calculated for each
analyzed frequency ω k
C ϕ (ω k ) =
cos((ϕ k,n )
2 +
sin((ϕ k,n )
2 .
The C ϕ (ω k ) function takes values from 0 to 1, containing the information on the
degree of phase coherence of the two signals x 1 (t) and x 2 (t) at frequency ω k .
The statistical analysis of the results obtained was carried out with SigmaPlot
13.0 software (Systat Software, Inc., 2014). Nonparametric statistical tests were
performed to avoid any assumption of normality. The Wilcoxon test was used to
estimate the significance of differences between the wavelet phase coherence values
obtained in spontaneous breathing and controlled regimes. The differences were
considered statistically significant at p < 0.05.
18.3 Results
Firstly, we tested reversibility of the breathing practices. The Fig. 18.2 presented
the phase coherence values for SBF left –SBF right pair under spontaneous breathing
before and after all controlled breathing tests. There are no differences between
phase coherence calculated for oscillations registered initially and after all breathing
practices.