408
Y. A. Abdulhameed et al.
Δφ k n = φ 2k , n −φ 1k , n .
(26.4)
The phase coherence function C φ ( f k ) is obtained by calculating and averaging in
time the components of the sine and cosine of the phase differences for the whole
signal, effectively defining the time-averaged WPC as
C φ ( f k ) =
cos Δφ k n 2 + +sin Δφ k n 2 .
(26.5)
The idea behind Eq. (26.5) is that, while we are considering individual times and
frequencies, these come from a discrete set (since all the signals in the present study
are discrete and finite-time), and so the subscripts k and n just reflect this discreteness.
The phase coherence function C φ ( f k ) as defined in Eq. (26.5) is exactly the discrete
version of the phase coherence formula Eq. (26.3), where φ is the phase difference
between the signals in question.
The function C φ ( f k ) characterises the tendency of Δφ k n to remain constant, or
not, at a certain frequency. Its value lies between 1 implying perfect coherence, and
0 implying total incoherence.
26.3.3.1 Effective (or Significant) Coherence
Note that the coherence computed in the first instance does not necessarily reflect
a genuine phase relationship and requires careful evaluation. The problem arises
because some of the coherence values obtained can be less than zero (although
formally coherence values range between 0 and 1). These negative coherence values
are then subtracted. Following this procedure, the very low frequency oscillations
may appear to have a coherence values close to 1, because of bias resulting from the
use of recordings that are too short to encompass the content at low frequencies.
Even in the case of two noisy signals, there is a tendency for there to be some
apparent coherence in the sense that C φ ( f k ) rarely approaches 0 at very low frequencies. The degree of apparent phase coherence depends on frequency. So the coherence
baseline will not be the same for all scales. The low-frequency components, particularly for signals of finite length (like those recorded in this study) are evaluated
using fewer periods than for the higher frequency components [5]. The result can
be an artificially increased coherence 1 even where, in reality, the dynamics of the
signals are completely unrelated.
To minimise random effects giving rise to apparent (but spurious) coherence,
whether at low or high frequency, we checked/tested the significance of the computed
coherence using the method of surrogates [24, 31]—by setting as a null hypothesis
that, for all frequencies, the phases in the signals are independent. We used iterative
amplitude-adjusted Fourier transform (IAAFT) surrogates to estimate the significance level of the apparent coherence, thereby avoiding the bias associated with the
power spectrum of the more commonly used amplitude-adjusted Fourier transform
(AAFT) surrogates. First, the IAAFT surrogates are constructed by randomizing all
the properties of the signals in question, whilst keeping only the phases unshuffled.
Y. A. Abdulhameed et al.
Δφ k n = φ 2k , n −φ 1k , n .
(26.4)
The phase coherence function C φ ( f k ) is obtained by calculating and averaging in
time the components of the sine and cosine of the phase differences for the whole
signal, effectively defining the time-averaged WPC as
C φ ( f k ) =
cos Δφ k n 2 + +sin Δφ k n 2 .
(26.5)
The idea behind Eq. (26.5) is that, while we are considering individual times and
frequencies, these come from a discrete set (since all the signals in the present study
are discrete and finite-time), and so the subscripts k and n just reflect this discreteness.
The phase coherence function C φ ( f k ) as defined in Eq. (26.5) is exactly the discrete
version of the phase coherence formula Eq. (26.3), where φ is the phase difference
between the signals in question.
The function C φ ( f k ) characterises the tendency of Δφ k n to remain constant, or
not, at a certain frequency. Its value lies between 1 implying perfect coherence, and
0 implying total incoherence.
26.3.3.1 Effective (or Significant) Coherence
Note that the coherence computed in the first instance does not necessarily reflect
a genuine phase relationship and requires careful evaluation. The problem arises
because some of the coherence values obtained can be less than zero (although
formally coherence values range between 0 and 1). These negative coherence values
are then subtracted. Following this procedure, the very low frequency oscillations
may appear to have a coherence values close to 1, because of bias resulting from the
use of recordings that are too short to encompass the content at low frequencies.
Even in the case of two noisy signals, there is a tendency for there to be some
apparent coherence in the sense that C φ ( f k ) rarely approaches 0 at very low frequencies. The degree of apparent phase coherence depends on frequency. So the coherence
baseline will not be the same for all scales. The low-frequency components, particularly for signals of finite length (like those recorded in this study) are evaluated
using fewer periods than for the higher frequency components [5]. The result can
be an artificially increased coherence 1 even where, in reality, the dynamics of the
signals are completely unrelated.
To minimise random effects giving rise to apparent (but spurious) coherence,
whether at low or high frequency, we checked/tested the significance of the computed
coherence using the method of surrogates [24, 31]—by setting as a null hypothesis
that, for all frequencies, the phases in the signals are independent. We used iterative
amplitude-adjusted Fourier transform (IAAFT) surrogates to estimate the significance level of the apparent coherence, thereby avoiding the bias associated with the
power spectrum of the more commonly used amplitude-adjusted Fourier transform
(AAFT) surrogates. First, the IAAFT surrogates are constructed by randomizing all
the properties of the signals in question, whilst keeping only the phases unshuffled.