Robust electroencephalogram phase estimation with applications in brain-computer interface systems

The conventional procedure for extracting the instantaneous phase sequence of a signal consists of two main stages: (1) narrow-band filtering, and (2) estimating the phase of the narrow-band signal (Boashash 1992, Picinbono 1997).

For a unique and canonical definition, the instantaneous phase is extracted from very narrow frequency band signals (Chavez et al 2006). Moreover, the input signal's phase contents should not be affected by the filtering procedure. For the first stage, almost all previous studies on EEG phase extraction have used finite impulse response (FIR) filters to make the signal narrow-band in its frequency spectrum (Lachaux et al 1999, Mormann et al 2000, Le Van Quyen et al 2001, Freeman 2004, Fell and Axmacher 2011, Krusienski et al 2012). However, in order to have a reliable phase sequence, there are important considerations regarding this procedure, including: the filter's band-width, its phase response and the convolution process.

The second stage requires choosing a phase estimation method to extract the phase sequence from the narrow-band signal. The most common method for phase estimation is based on the analytic signal representation of the narrow-band signal (Chavez et al 2006). As shown in Sameni and Seraj (2016), the calculation of the instantaneous phase from the analytical representations becomes challenging and highly susceptible to noise in low analytical signal amplitudes; resulting in fake jumps and spikes in the extracted phase signal. In the following sections, this issue is further studied and partially solved by applying perturbations in the phase extraction procedure followed by ensemble averaging, as proposed in Sameni and Seraj (2016).

For the signal x(t), its analytical form is defined as follows (Picinbono 1997):

$$\begin{eqnarray}&&{{z}_{x}}(t)=x(t)+jH\left\{x(t)\right\}\end{eqnarray} \tag{ 1 }$$

where $H\left\{\centerdot \right\}$ denotes the Hilbert transform. Using the analytical form, the instantaneous envelope (IE), the instantaneous phase (IP), and the instantaneous frequency (IF) are uniquely defined as follows:

$$\begin{eqnarray}&&\text{IE}:\quad {{a}_{x}}(t)=\,|{{z}_{x}}(t)|=\sqrt{x{{(t)}^{2}}+H{{\left\{x(t)\right\}}^{2}}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\text{IP}:\quad {{\phi}_{x}}(t)=\arctan \left(\frac{H\left\{x(t)\right\}}{x(t)}\right)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\text{IF}:\quad {{f}_{x}}(t)=\frac{1}{2\pi}\left[\frac{\text{d}{{\phi}_{x}}(t)}{\text{d}t}~\text{mod}~2\pi \right]\end{eqnarray} \tag{ 4 }$$

For discrete-time implementations, (4) is replaced by a finite-difference approximation of the derivative operator (e.g. the first-order difference in the simplest case).

Unless the signal has a narrow-band spectral support, the pair $\left({{a}_{x}}(t),{{\phi}_{x}}(t)\right)$ does not convey significant information regarding the instantaneous phase (Boashash 1992). This is due to the fact that only for narrow-band signals, the relative variations of the amplitude a_x(t) are rather slow (and negligible) as compared with the variations of the phase ${{\phi}_{x}}(t)$ (Picinbono 1997), i.e.

$$\begin{eqnarray}&&\left|\frac{\text{d}\phi (t)}{\text{d}t}\right|\gg \left|\frac{1}{a(t)}\frac{\text{d}a(t)}{\text{d}t}\right|\end{eqnarray} \tag{ 5 }$$

For cerebral signals, it is known that the EEG has a wide frequency range (0 Hz–150 Hz in the extreme case), which makes narrow bandpass filtering (of the order of Hertz) an essential prerequisite for extracting a meaningful instantaneous phase sequence.

Previous studies on EEG phase extraction have commonly employed linear-phase FIR filters to make the signal narrow-band in its frequency spectrum (Lachaux et al 1999, Mormann et al 2000, Le Van Quyen et al 2001, Freeman 2004, Fell and Axmacher 2011, Krusienski et al 2012). The advantage of linear phase filters is their constant group delay, which avoids phase distortions in the filtered signal. Nevertheless, in most FIR filter design techniques, the order of the filter proportionally increases with the inverse of their transition bandwidth, which means that narrow-band FIR filters have very long impulse responses and input-output delays (Oppenheim et al 1999). Moreover, highly narrow band filters are difficult to design and susceptible to design parameters.

To avoid these issues, previous studies have kept a trade-off between the order of the FIR filter and its bandwidth (BW). The BW is commonly chosen relatively wide, e.g. between 4 and 12 Hz, to have a low-order and practically realizable filter (Mormann et al 2000, 2003, Freeman 2004). However, as discussed before, using a bandwidth in this range, the envelope-phase pair $\left\{a(t),\phi (t)\right\}$ obtained from the Hilbert transform fails to correctly (and uniquely) define the instantaneous envelope. Therefore, the extracted phase becomes unreliable. To overcome this issue, very narrow band filters with reasonably low orders are required in practice to be less sensitive to noise and filter parameter variations. In section 3, it is shown that for offline applications, zero-phase infinite impulse response (IIR) filters can be used as an alternative solution.

The instantaneous phase sequence derived from the analytic representation of a signal is prone to fake jumps (without cerebral source) in low-amplitude analytical signal (LAAS) time instants (Picinbono 1997, Freeman 2004, Chavez et al 2006). As depicted in figure 1, the instantaneous frequency tends to have big jumps at LAAS epochs. The problem was rigorously studied in Sameni and Seraj (2016). The main reason underlying this phenomenon could be linked to the $\arctan (\centerdot )$ operator used for phase calculation. According to (3), LAAS leads into very small numerator and denominator values. Consequently, any minor change in the real or imaginary parts of the analytic form (due to noise or background EEG fluctuations), results in a significant change in the estimated phase. This fact is illustrated in figure 2 for a sample EEG signal. It is seen that phase values corresponding to lower amplitudes tend to have bigger variations. More rigorously, it was shown in Sameni and Seraj (2016) that during LASS epochs the instantaneous phase tends to a uniform distribution over $\left[-\pi,\pi \right]$ and in discrete-time implementations, the instantaneous frequency becomes uniform over the entire Nyquist band [0,f_s].

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The findings of Sameni and Seraj (2016) are in accordance with previous research (Freeman 2004), which reported that LAAS occurs more frequently in low power time- frequency regions, such as the high frequency bands of the EEG, for which the EEG power decays more rapidly with increasing frequency. To illustrate this point, in figure 3 the instantaneous phase differences of frequency components in the range of DC to 50 Hz have been extracted alongside their corresponding instantaneous envelopes. Accordingly, the first 5 s of the results are due to the FIR filter's transient response; resulting in very low-magnitude instantaneous amplitudes during this period. As it can be seen, the corresponding regions of the phase difference plot contains many phase jumps and spikes. For the rest of the signal, in lower frequencies, where the analytic form has higher amplitudes, the phase sequences are less contaminated with jumps and spikes. However, as the frequency increases and the power in EEG signal decays, the analytical signal envelope decreases and the rate of phase jumps increases once more.

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Based on these findings, in the next section a robust method is proposed for the estimation of the instantaneous phase using perturbation of filter parameters and a Monte Carlo estimation process.

The challenges in narrow-band FIR filter design were noted in section 2.3. To overcome these issues, we propose to use forward–backward zero-phase IIR filters. Although the filter is performed offline in a non-causal manner (which is not a limiting issue for offline applications), the major advantage is that the order of a narrow band IIR filter is much lower than its FIR counterpart and by applying it in a forward–backward manner, the nonlinear phase response of the filter is compensated and zero-phase distortion—which is necessary for EEG phase analysis—is guaranteed.

Various types of IIR filters such as Chebyshev types 1 and 2, Butterworth and Elliptic filters were studied to determine the best filter for this application, and the Elliptic filter was chosen due to its steeper roll-off characteristics (as compared with Butterworth or Chebyshev filters) and its equi-ripple behavior in both the passband and stopband. In general, by allowing ripples in both passband and stopbands, Elliptic filters meet given performance specifications with the lowest order as compared with their counterparts (Lutovac et al 2001). In order to preserve the filter's frequency response over all frequency bands, instead of designing various bandpass filters in each band, a fixed narrow band lowpass filter prototype was designed and applied to the signal using a frequency domain shifting scheme illustrated in figure 4. Further details regarding this scheme is presented in section 3.2. The prototype Elliptic IIR lowpass filter used in this study has the following characteristics: 0.3 Hz pass-band frequency, 0.5 Hz stop-band cutoff frequency, 0.1 dB maximum pass-band ripple, and 70 dB minimum stop-band attenuation, designed at a sampling frequency of 160 Hz (the sampling rate of the sample EEG). The order of this prototype filter was 6, which is computationally far more effective than any FIR filter with the same specifications. This filter was performed in a forward backward manner, which doubles its pass-band ripple and stop band attenuation in dB.

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The procedure of zero-phase forward–backward smoothing (FBS) is shown in figure 5. This process can be implemented using the filtfilt function in major signal processing languages such as Matlab, Octave, or R. Accordingly, FBS uses the time-reversal property of the Fourier transform to perform zero-phase smoothing by processing the input signal in both the forward and reverse directions (Oppenheim and Schafer 2010). Considering $H(\,j\omega )$ as the frequency response of the forward path digital filter, the overall response of FBS is

$$\begin{eqnarray}&&{{H}_{\text{eff}}}(\,j\omega )=\,|H(\,j\omega ){{|}^{2}}\end{eqnarray} \tag{ 6 }$$

which is real-valued. Therefore, regardless of the nonlinear phase-response of the IIR filter, FBS has a zero-phase (and zero-group delay) response, which preserves the input signal's phase.

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The next step is to compute the phase sequence. For this, we use the analytic representation of the filtered signal. In order to reduce the processing complexity and avoid the direct calculation of the Hilbert transform, this stage can be merged with the bandpass filtering as follows: as illustrated in figure 4, the proposed bandpass filtering scheme uses a lowpass filter prototype. To filter the signal x(t) around the center frequency ${{\omega}_{0}}$, x(t) is shifted in the frequency by multiplying the pure phase signal $\exp \left(\,j{{\omega}_{0}}t\right)$, to obtain a complex valued signal x_f (t). Next, the real and imaginary parts of x_f(t) are given to the lowpass prototype ${{H}_{\text{LP}}}(\,j\omega )$ to obtain the narrow-band analytical signal y_f(t). Finally, y_f(t) is shifted back to the center frequency ${{\omega}_{0}}$, by multiplying the phase signal $\exp \left(-j{{\omega}_{0}}t\right)$. This procedure provides x_a(t), which is the narrow band analytical form of the original signal x(t) around the center frequency ${{\omega}_{0}}$.

After computing the analytic form of the filtered EEG, the instantaneous phase is calculated as follows:

$$\begin{eqnarray}&&\phi (t)=\arctan \left(\frac{\text{Im}\left({{x}_{a}}(t)\right)}{\text{Re}\left({{x}_{a}}(t)\right)}\right)\end{eqnarray} \tag{ 7 }$$

For discrete-time signals, the instantaneous frequency can be approximated by the first order difference of the instantaneous phase:

$$\begin{eqnarray}&&f(t)\approx {{f}_{s}}\frac{\phi (t)-\phi (t- \Delta )}{2\pi}\end{eqnarray} \tag{ 8 }$$

where $\Delta$ is the sampling time and ${{f}_{s}}=1/ \Delta$ is the sampling frequency.

In the proposed scheme, the previous two stages (narrow band filtering and phase calculation) are repeated several times, each time with very minor changes in the filter design parameters λ (and zero-pole positions). Here, the idea is to generate random ensembles of the signal's analytical form and EEG phase, using infinitesimal perturbations in the filter parameters. Apparently, clinically relevant EEG phase information should not be susceptible to minor filter design variations at the order of, e.g. 0.01 Hz. However, during LAAS epochs, even minor deviations in the filter parameters can significantly change the phase estimates, resulting into fake phase jumps.

The zero-pole plot of the utilized prototype IIR filter and the loci of its perturbed zeros and poles are illustrated in figure 6(c). It is known that all Elliptic IIR filter zeros are located on the unit-circle. Thus, in order to prevent any major change in the filter characteristics, these zeros are perturbed randomly only on the unit-circle. For this, the filter's zeros are taken into polar coordinates $(\rho,\theta )$, and the random perturbations are applied only to θ. Another important consideration is that any perturbation in the pole loci of a causal filter should not move its poles out of the unit-circle (to preserve its stability). Moreover, the conjugate symmetry of the zeros and poles should be preserved to guarantee the realness of the impulse response. These properties are achieved by applying random complex-valued perturbations $\delta _{i}^{p}$ (to the poles) and $\delta _{i}^{z}$ (to the zeros) to each pair of conjugate symmetric pair of poles/zeros. The complex values added to each conjugate symmetric pair should have equal real parts, and imaginary parts with equal magnitudes and opposite signs, to preserve the conjugate symmetry of the perturbed poles/zeros. The perturbed pole loci is finally checked to be inside the unit circle (figure 6(c)), to guarantee the filter's stability. The magnitude and phase response of 100 lowpass prototype filters obtained by the proposed zero-pole perturbation procedure are shown in figures 6(a) and (b), where it is seen that the zero-pole perturbations have had rather minor impact on the filters' response (irrelevant to most cerebral studies). However, it is later shown that even these minor changes can significantly change the EEG phase, especially during LAAS.

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Calculating the phase using the four quadrant arc-tangent causes phase-wrapping (Itoh 1982). The amplitude of the phase sequence can take any value and even exceed the range $\left[-\pi,\pi \right]$. In cases where the phase exceeds this range, it is wrapped so that it stays within the principal range (Itoh 1982, Ghiglia and Pritt 1998). In such cases, the wrapped phase sequence will contain some phase-jumps greater than $\pm \pi$. Therefore, for EEG phase analysis (either from the phase itself or from its time difference), an unwrapping procedure is required to obtain the phase sequence in its original continuous form. An unwrapped phase sequence typically diverges from zero over time (similar to a random-walk process). For better illustration of the phase fluctuations, one may either subtract the constant linear phase signal ${{\omega}_{0}}t$ from the instantaneous phase to obtain its temporal fluctuations, or alternatively use the time difference of the phase signal, which is proportional to the instantaneous frequency.

The final stage of the algorithm is to average over the randomized ensembles of the phase sequences obtained from different filter responses, to obtain an average phase estimate.

For illustration, figure 7 shows 100 ensembles of the instantaneous frequency of a sample EEG segment obtained by the aforementioned randomization scheme (zero-pole perturbation) for a center frequency of 8 Hz, and the average of the randomized ensembles. This figure illustrates the importance of the proposed scheme and the significance of the zero-pole perturbation. Accordingly, without this procedure (by simply calculating the phase from a single filtered signal as in conventional methods), the obtained phase sequence is unreliable, since each set of filter parameters (even with minor differences) would lead to significantly different results; especially during low analytical signal envelope epochs in which the standard deviation of the instantaneous frequency is significantly high (figure 7).

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In this section, the robustness of the proposed phase/frequency calculation method is verified versus conventional methods (without random ensemble generation and averaging) for a sample EEG signal. The sample signal, represented in figure 9(a), consists of 24 s of an ongoing EEG acquired with a sampling frequency of f_s  =  173.61 Hz, recorded during a BCI study. The complete description of the data is presented in Andrzejak et al (2001). In the following, the phase robustness of this sample data is studied from three aspects: (1) impact of filter parameter variations, (2) robustness to non-stationary background noise and (3) low-amplitude analytic signal and phase jumps. The power spectral density (PSD) of this sample data is shown in figure 8. Accordingly, f₀  =  8 Hz (the alpha-band peak) is chosen as the center frequency in the following evaluations.

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3.6.1. Filter parameter variations.

3.6.2. Low-amplitude analytic signal and phase jumps.

Figure 10 shows the instantaneous phase difference and the instantaneous analytical signal envelope calculated for the sample EEG in figure 9(a) using the conventional method, for frequency components in the range of DC to 30 Hz, using the three filtering schemes stated in section 3.6.1. The same procedure is performed using the proposed method and the results are shown in figure 11 for comparison.

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The comparison of figures 10 and 11, clearly shows the effects of LAAS on the EEG phase jumps using the classical and proposed methods. It is seen that the phase sequences estimated by the conventional method are prone to fake jumps at points where the corresponding analytic signal have lower amplitudes; while this issue has been significantly improved using the proposed method.

3.6.3. Noise susceptibility.

For evaluating the robustness of the proposed phase estimation method to noise, the results of phase estimations using both the proposed and conventional methods are tested in presence of additive non-stationary noise. Non-stationary noise with a time-variant variance and EEG-like spectra is used to model background non-stationary EEG activity. For this purpose, a time-varying auto-regressive (TVAR) model of order 20 is trained over sliding windows of length 4 s using a sample 20 s EEG. The resulting TVAR model is next fed by white Gaussian noise, to generate non-stationary noises of the same length with a spectra similar to a real EEG and added to the EEG understudy at different signal-to-noise ratios (SNR). For this scenario, an Elliptic IIR filter with f₀  =  8 Hz, BW  =  0.5 Hz, TB  =  0.5 Hz, ${{\delta}_{p}}=0.1$ dB and ${{\delta}_{s}}=70$ dB is used as the bandpass filter.

The corresponding results are shown in figure 12 for both the conventional and presented methods in SNR  =  15 dB. It can be seen that while the conventional phase estimation procedure (gray line) becomes unreliable during noisy epochs, the proposed method (thick black line) yields a robust estimation of phase information despite the non-stationary background noise (which is statistically independent from the underlying mental task). Therefore, the proposed method significantly improves the reliability of the extracted instantaneous phase/frequency estimates. A rigorous discussion on the effect of noise and SNR level on the probability of correct and false phase detections has been presented in Sameni and Seraj (2016).

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Figure 13 shows the instantaneous phase difference and the instantaneous analytical signal envelope calculated for the sample EEG in figure 9(a), using the conventional method for frequency components in the range of DC to 30 Hz, at three SNR: infinity (no noise), SNR  =  10 dB, and SNR  =  0 dB. In figure 14, the same procedure is repeated using the proposed method for comparison; where it is seen that the proposed method has been considerably less susceptible to additive noise.

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The dataset used for this study is adopted from the Neuroelectric and Myoelectric Databases, which are online available on Physionet (Goldberger et al 2000). This dataset includes one and two-minute recordings of 109 volunteers, performing a series of motor and motor-imagery tasks. Each record contains sixty four channels of EEG recorded using the BCI2000 System, during a set of annotated mental tasks (Schalk et al 2004). The complete description of the dataset is available at Goldberger et al (2000). Each subject has performed a series of mental tasks: two one-minute baseline runs, with open and closed eyes and three two-minute runs of the four following tasks:

• (i)  
  A target appears on the left or right side of a screen in front of the subject. The subject opens and closes the corresponding fist until the target disappears. Then the subject relaxes.
• (ii)  
  A target appears on the left or right side of the screen. The subject imagines opening and closing the corresponding fist until the target disappears. Then the subject relaxes.
• (iii)  
  A target appears on the top or bottom of the screen. The subject opens and closes both fists (if the target is on top) or both feet (if the target is on the bottom) until the target disappears. Then the subject relaxes.
• (iv)  
  A target appears on the top or bottom of the screen. The subject imagines opening and closing both fists (if the target is on top) or both feet (if the target is on the bottom) until the target disappears. Then the subject relaxes.

Figure 15 shows the placement of the electrodes used for recording EEG signals in this dataset. Since the primary cortical regions involved in the task of motor imagery are the supplementary motor area (SMA) and the primary motor cortex area (M1), electrodes FCz, C3, and C4 are chosen for this study (Wang et al 2006, Wei et al 2007, Krusienski et al 2012).

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The annotations provided in the dataset consist of three classes for identifying rest versus left/up or right/down side activities: (1) T₀ corresponding to rest condition, (2) T₁ corresponding to motion (real or imagined) onset of either the left fist or both fists, and (3) T₂ corresponding to motion (real or imagined) onset of either the right fist or both feet. We therefore have a three-class classification problem.

The targets appeared on the screen every four seconds, resulting in thirty 4 s annotated EEG segments for each two minute records (per subject), each corresponding to a mental task trial.

A survey of previous studies on VEP-based BCI systems reveals that EEG phase-related features are among the most discriminative and informative features for BCI applications (Pan et al 2011, He et al 2012, Krusienski et al 2012). During the feature extraction phase, a broad range of features are commonly extracted from the frequency band of interest and passed to the feature selection and classification stages. However, in this study, a single phase-related feature, namely the PLV, is used to evaluate the robustness and feasibility of the proposed instantaneous phase estimation procedure.

PLV is an index for quantifying how constant the phase difference between two signals is. In order to calculate the PLV between two signals (or channels) x(t) and y(t), the following steps are applied (Rosenblum et al 1996, Lachaux et al 1999):

• Channelize the signals using narrow-band filters centered at f
• Calculate the instantaneous frequency-specific phase values ${{\phi}_{x}}(t,f)$ and ${{\phi}_{y}}(t,f)$.
• Calculate the instantaneous phase-difference between x(t) and y(t) and quantify the local stability of this phase-difference over time:
  $$\begin{eqnarray}&&\text{PL}{{\text{V}}_{xy}}(\,f)=\,\left|\frac{1}{T}\underset{t=1}{\overset{T}{\sum}}\,\exp \left(\,j\left[{{\phi}_{x}}(t,f)-{{\phi}_{y}}(t,f)\right]\right)\right|\end{eqnarray} \tag{ 9 }$$
  where T is the signal length and the summation is taken over all temporal samples of the instantaneous phases.

PLV varies between 0 and 1, corresponding to completely non-synchronized signals and complete synchronization, respectively (Rosenblum et al 1996, Lachaux et al 1999). The notion of phase-locking using PLV has been previously studied in BCI applications (Hamner et al 2011, Carreiras et al 2012). For illustration, the envelope and phase sequences (phase derivatives), and the corresponding PLV index extracted from two 10 s segments of EEG acquired from channels C3 and C1 of the first subject in the dataset are shown in figures 16(a) and (b). During this segment, the subject is performing a right-hand task (starting between t  =  55 and t  =  56 s) for four seconds and then relaxes. The PLV index obtained by averaging over 250 ms sliding windows (controlled by the window length T, in (9)) is shown in figure 16(c), using phase values obtained from the conventional and proposed methods. Accordingly, the PLV computed using the conventional phase estimation method is reporting some de-synchronization around t  =  50.2 s, 50.6 s, 52.4 s, 57.7 s and 59 s, which have coincided with the envelope drops at the same instants (figure 16(a)). This shows that phase variations during low analytical signal epochs are unreliable in conventional methods, which is not the case for the PLV reported by the proposed method.

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In order to evaluate the overall performance, the EEG phase and the corresponding PLV features were extracted using the conventional and proposed procedures, from 105 out of 109 subjects (four subjects were excluded due to data deficiency). The PLV was calculated for a single frequency band f  =  10 Hz with an effective bandwidth of 1 Hz, which was identified as the dominant alpha-band peak by visual inspection of the EEG spectra. The index was calculated between the three possible combinations of the selected electrodes, i.e. FCz–C3, FCz–C4 and C3–C4 (as shown in figure 15), resulting in PLV feature vectors of length three. The feature vectors were computed from each of the thirty 4 s annotated temporal windows over all two-minutes records of each subject.

The PLV feature vectors calculated by both the conventional and proposed methods together with the annotations provided in the database were used for training and testing the classifiers. K-nearest neighbors (KNN) with K  =  30 (the number of nearest neighbors used in the classification), and random forest (RF) with number of trees equal to 10, were used as classifiers in a leave-one-out cross-validation process, in which the feature-set of one subject is considered as test data and the rest of the feature sets are used for training the classifiers.

The comparison has been made both in absence and presence of an additive white Gaussian noise with SNR  =  5 dB, to investigate the robustness of the conventional and proposed procedures to noise. The average results of the noiseless and noisy cases over all subjects are reported in figure 17. Accordingly, the proposed method for extracting PLV features has improved the mean RF classification rates 7%, as compared with conventional methods of EEG phase extraction (in absence of additive noise) and 12% (in presence of additive noise). Using the KNN classifier, the mean classification rates have been improved 4% (in absence of additive noise) and 8% (in presence of additive noise). The significance of these results was tested by a paired sample t-test between the conventional and proposed methods, under the null hypothesis of equal average performance of PLV features obtained by both methods (conventional versus proposed). The null hypothesis was rejected with p-values p  =  0.01 and p  =  0.03, for the noiseless and noisy scenarios, respectively, using 6442 degrees of freedom in both cases.

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The per-subject three-class accuracies using PLV as feature are shown and compared for the proposed and conventional methods in figure 18; where the outperformance of the proposed method is seen, both in terms of higher classification accuracies and smaller inter-subject variances. The higher performance can be associated to the robustness of the proposed EEG phase extraction method. The average and standard deviation of the per-subject results shown in figure 18 are reported in table 1, which show that KNN has slightly outperformed RF.

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In order to make the results reproducible, all source codes related to this study are online available in the open-source electrophysiological toolbox (OSET) (Sameni 2014).