The EEG data set was originally provided for classifying EEG data with small training sets and is widely employed in BCI studies to compare different classification algorithms. It consists of five data subsets derived from five healthy subjects (aa, al, av, aw and ay). Each subject participated in a MI-based BCI experiment, in which they were required to conduct mental tasks of imagining left hand, right hand or right foot movements, following a given visual cue denoted by a letter (L, R, or F). Starting from the visual cue, the subjects carried out the corresponding MI task for 3.5 s. These visual cues were presented intermittently with random lengths ranging from 1.75 s to 2.25 s, during which the subject could relax.

EEG signals were collected using a BrainAmp amplifier and a 128-channel Ag/AgCl electrode cap. 118 electrodes were used for recording experimental data, according to the extended international 10/20 system. The EEG data were digitalized at 1000 Hz by the amplifier and re-sampled to 100 Hz by the competition organizers for offline analysis. A total of 280 trials per class were performed by each subject. Only the data from MI tasks of right hand (R) and right foot (F) were provided for the competition. The electrode placement for recording EEG data and the timing scheme of each trial are illustrated in Fig. 1 (a) and Fig. 2 (a), respectively.

Figure 1.

(a) The placement of the 118 electrodes in the EEG data set according to extended international 10/20 system; (b) The placement of the 8 × 8 electrode grid used for recording ECoG data of the second patient in the ECoG data set. It was placed on the right hemisphere.

Figure 2.

(a) The timing scheme of each trial for the EEG dataset. In each trial, the duration of motor imagery was 3.5 s and the next 1.75-2.25 s was the time for a subject to relax; (b) The timing scheme of each trial for the ECoG dataset. In each trial, the duration of motor imagery was 4 s and the next 2 s was the time for a subject to relax.

The ECoG data set was recorded from three epileptic patients (AM, JS, SS) with intracranial electrodes. All patients suffered from focal epilepsy and had to undergo surgical operation to have their foci resected. Prior to the surgery, the localization of epileptic foci required placing electrodes onto the surface of the cortex and into deeper regions of the brain. After several days of recovery and follow-up examinations due to implantation surgery, the BCI experiments were carried out in the hospital.

For the experiment, each subject was asked to repeatedly imagine two different limb movements according to the visual cue. Each trial started with a fixation cross displayed in the center of screen and lasted for 7 s. At second 1, a visual cue appeared on the screen indicating the MI task to be performed. The cue for patient SS was an arrow pointing to the left or right hand, whereas that for patients AM and JS was a picture showing either a tongue or a little finger. The imagination phase lasted 4 s. In the final 2 s of each trial, the patient could relax.

All three patients had grid electrodes implanted, but patients JS and SS had additional strip electrodes. The electrode grids were placed on the cortex under the dura master, covering the primary motor and premotor areas as well as the frontotemporal region of either the right or left hemisphere. The electrodes were connected to an EEG amplifier by cables. The ECoG signals were recorded at a sampling rate of 1000 Hz and re-sampled to 100 Hz for offline analysis. The number and positions of implanted electrodes, the tasks performed, and the number of trials recorded from each patient are listed in Table 1. The electrode placement for recording ECoG data of the second patient and the timing scheme of each trial are illustrated in Fig. 1 (b) and Fig. 2 (b), respectively.

The particle swarm optimization (PSO) developed by Kennedy and Eberhart (1995) is a population-based evolutionary search method. The main idea of the algorithm comes from the social behavior of animals, such as bird flocking, fish schooling, and animal herding. The original PSO designed for continuous search space was modified to be applicable to discrete binary search space, thus termed binary PSO (BPSO) (Kennedy and Eberhart, 1997). From the perspective of quantum mechanics, Shin et al. (2004) adapted the PSO algorithm to develop a novel quantum-behaved particle swarm optimization (QPSO), using the quantum uncertainty principle to describe the motion state of particles. Subsequently, they further generalized the QPSO algorithm to discrete binary search spaces, developing the binary QPSO (BQPSO) (Xi et al., 2010).

3.1.1 Binary particle swarm optimization

In PSO, a particle swarm consists of M particles that denote potential problems, X = {X₁,X₂,··· ,X_M}. A potential solution to a problem is expressed as a particle flying in a D-dimensional space having the position vector X_i ⁼ {X_i1,X_i2,··· ,X_iD} and the velocity vector V_i ⁼ {V_i1,V_i2,··· ,V_iD}. Each particle maintains a record of the position of its previous best performance (i.e. the position with the best fitness value) in a vector, pbest_i={pbest_i1,pbest_i2,··· ,pbest_iD}. At each iteration, each particle competes with others in the population for the best position, denoted as gbest_i= {gbest_i1,gbest_i2,··· ,gbest_iD}. Thereby, particles move in the search space according to the following:
(1) ${{V}_{id}}=w{{V}_{id}}+{{\varphi }_{1}}(pbes{{t}_{id}}-{{X}_{id}})+{{\varphi }_{2}}(gbes{{t}_{d}}-{{X}_{id}})$
(2) ${{X}_{id}}={{X}_{id}}+{{V}_{id}}$
where i = 1,2,··· ,M and d = 1,2,··· ,D. The w is the inertia weight introduced for accelerating the convergence speed of PSO. φ₁and φ₂ are two random positive numbers generated for each i and d. At each iteration, the value of V_id is confined to [-V_max, V_max] .

In a discrete binary space, the velocity of a particle can be described by the number of bits changed per iteration, or the Hamming distance of a particle, between time t and t + 1. A particle with zero bits flipped does not move, while it moves the “farthest” with all bitsflipped. Accordingly, the velocity of a particle can be defined in terms of the probabilities that a bit will be in one state or the other. That is to say, a particle moves in a state space with each dimension confined to 0 and 1, where each V_id denotes the probability of bit X_id taking the value 1.

In summary, the particle swarm Eqn. (1) remains unchanged in BPSO, but now pbest_id, gbest_id, X_idand X_idare taken as integers in {0, 1}. Since V_id is a probability, it must be confined to [0.0, 1.0]. This can be implemented by a sigmoid limiting transformation function S(v) = 1/(1+exp(-v)). Thus, the main difference between PSO and BPSO is that formula (2) is replaced by the following Eqn. (3):
(3) if rand() < S(V_id) then X_id = 1 else X_id= 0
where rand() is a random number selected from a uniform distribution in [0, 1]. In BPSO, V_max is retained, i.e. |V_id| < V_max, which limits the ultimate probability that bit X_id will take on a binary value.

3.1.2 Binary quantum-behaved particle swarm optimization

Quantum-behaved particle swarm optimization (QPSO) is a novel variant of PSO and outperforms PSO in search abilities. In QPSO, there are no the concepts of velocity and trajectory, but those of position and distance. A particle moves in the continuous search space according to the following equations:
(4) $mbest=\frac{1}{M}\sum\limits_{i=1}^{M}{pbes{{t}_{i}}}=\frac{1}{M}\left( \sum\limits_{i=1}^{M}{pbes{{t}_{i1}}},\sum\limits_{i=1}^{M}{pbes{{t}_{i2}}},\cdot \cdot \cdot ,\sum\limits_{i=1}^{M}{pbes{{t}_{id}}} \right) $ (5) ${{p}_{id}}=\varphi pbes{{t}_{id}}+(1-\varphi )gbes{{t}_{d}},\varphi =rand()$ (6) ${{X}_{id}}={{p}_{id}}\pm \alpha |mbes{{t}_{d}}-{{X}_{id}}|\ln (1/\mu ),\mu =Rand()$
where mbest is the mean best position among the particles. p_id is a stochastic point between pbest_id and gbest_d, i.e. the dth coordinate of the local attractor of the ith particle pi. φ and µare two random numbers distributed in [0, 1], and α is a parameter of QPSO called contraction-expansion coefficient.

Since the iteration equations of QPSO are far different from those of PSO, the methodology of BPSO does not apply to QPSO. Because the position of a particle in a discrete space is expressed as a binary string, the key problem of designing BQPSO is to define the distance between two positions and the corresponding transformation. In BQPSO, the distance is defined as the Hamming distance between two binary strings X and Y, i.e. |X-Y| = d_H(X,Y), where d_H() is the function for computing the Hamming distance, i.e. the sum of different bits between the two strings. In BQPSO, the variable X_id stands for the dth substring (i.e. dth decision variable) of the ith particle, rather than dth bit of a binary string. Let the length of X_id be l_d, then the length of X_i can be calculated as
(7) $l=\sum\limits_{d=1}^{D}{{{l}_{d}}},d=1,2,\cdot \cdot \cdot ,D$
The remaining problem for BQPSO design is in adapting the continuous evolution Eqns. (4)-(6) to discrete binary spaces. In QPSO, the mean best position of all particles (mbest) is derived from Eqn. (4), whereas in BQPSO, the jth bit of mbestis determined by the states of jth bits of all particles’ pbests. The jth bit of m best is 1 if mbest_j > 0.5, 0 if mbest_j < 0.5, and randomly taken as 1 or 0 if mbest_j = 0.5. In BQPSO, P_i can be generated through crossover operation of pbesti and gbest, which can be divided into one-point operations and multi-point operations.

The update Eqn. (6) for QPSO can be rewritten as |X_id- pid| = α |mbestd- X_id| ln(1/µ), µ = Rand(). It can be further adapted for use in BQPSO as follows
(8) d_H(X_id,P_id) = ⌈b⌉
where b = αd_H(X_id,P_id)ln(1/µ). Function ⌈ ⌉is a round sign used for rounding towards nearest decimal or integer. According to the above equations, a new substring X_id can be calculated with time complexity O(bl_d). To reduce the computation cost, X_id is generated by mutating each bit of P_id with the mutation probability,
(9) ${P}_{r}=\begin{cases} b/{{l}_{d}} & \\1,\quad if \quad b/{{l}_{d}}>1 \end{cases}$

3.1.3 Fitness function

When BPSO/BQPSO is used for channel selection, individuals in a population are represented in terms of n-bit binary strings, corresponding to n channels used for data recording. The BPSO/BQPSO operates on a population of binary strings and chooses channels by optimizing a fitness (or objective) function. There are two goals in channel selection: improving classification accuracy, and reducing the number of channels. Accordingly, the fitness function, f(z), can be defined as the weighted sum of two decision variables, the error rate of 10-fold cross-validation, f₁(z), and the relative number of channels, f₂(z), for a minimization problem (Hasan et al., 2010; Reyes-Sierra and Coello, 2006)
(10) $f(z)={{w}_{1}}{{f}_{1}}(z)+{{w}_{2}}{{f}_{2}}(z) $
where the weights w_i are normalized, i.e.,$\sum\limits_{i=1}^{2}{{{\text{w}}_{i}}}\text{=}1$. ${{f}_{1}}(z)$ is obtained from the given channel subset denoted by an individual, z, and f₂(z) is derived by dividing the number of channels chosen in the individual z by the total number of channels in raw data set, i.e. the length of the binary string, n. Since the numerical value of f₁(z) and f₂(z) ranges from 0 to 1, so does that of f(z).

Several important steps for BPSO/BQPSO based channel selection are explained below:

1) Coding. Each particle in a population is coded as a binary string, whose length is equal to the total number of channels in a raw data set. When any bit of the binary string is 1, the corresponding channel is retained; otherwise the corresponding channel is removed. Thus, each particle denotes a different subset of channels, which is a candidate solution to the problem of channel selection.

2) Initialization. An initial population with i particles (i = 20 in this study) is randomly generated, and each bit of binary string for every particle is randomly set to 1 or 0.

3) Selection. Channel selection is equivalent to finding a global minimum of the fitness function. At each generation, the particle with the smallest fitness value is found. After each iteration, the positions of all particles are updated, and the current best position of each particle is compared with that of the best particle of the previous generation to find the global optimal position, i.e. the best particle at the current generation. The best particle at the final generation includes the channels selected by BPSO/BQPSO.

3.2.1 Data preprocessing

Prior to channel selection, both the EEG and ECoG data sets were preprocessed with respect to time windowing, temporal filtering, and electrode referencing. In the EEG dataset, the raw data in a time window of 1-2 s after the visual cue were segmented from each channel for classification (Shin et al., 2012). The windowed EEG data were band-pass filtered between 8 to 15 Hz to extract µ rhythm signals associated with MI (Shin et al., 2012). In the ECOG dataset, the raw data in a time window of 0.5-2.5 s following the visual cue were segmented from each channel for classification. The data segments used for classifying the two data sets were not optimized, but were determined experimentally and heuristically. Common average reference (CAR) was used to re-reference the windowed data to reduce sensitivity to artifacts (Ludwig et al., 2009). Re-referenced ECoG data were band-pass filtered between 8 and 30 Hz to extract both µ and β rhythm signals associated with MI (Wei et al., 2007).

3.2.2 Common spatial pattern

Common spatial pattern (CSP) is a powerful algorithm for spatial filtering, which has been successfully employed in MI-based BCIs for discriminating between two classes of EEG data. By spatially filtering multi-channel EEG signals, CSP maximizes the variance of one class while minimizing the variance of the other class, making subsequent classification more effective (Blankertz et al., 2008; Lotte and Guan, 2011; Muller-Gerking et al., 1999).

The purpose of CSP is to extract task-related signal components and suppress task-unrelated components and noise. Assume that there are two-class EEG signals evoked by two mental tasks, e.g. MI of left hand and right hand. Let X₁ and X₂ respectively denote a single-trial EEG signal of the two classes 1 and 2, with the dimension of N(channels)T(sampling points). Two normalized spatial covariance matrices, R₁ and R₂, are calculated with X₁ and X₂, respectively, as
(11) ${{R}_{i}}=\frac{{{X}_{i}}X_{i}^{T}}{trace({{X}_{i}}X_{i}^{T})},i=1,2$
where superscript T denotes the transpose operation, and trace(A) stands for the trace operation, i.e. the sum of diagonal elements of matrix A. The averaged spatial covariance matrix, ${{\bar{R}}_{i}}$ across all training trials can be obtained for each class. Subsequently, the composite spatial covariance matrix R_c can be calculated as
(12) ${{R}_{c}}={{\bar{R}}_{1}}+{{\bar{R}}_{2}}$
Since R_c is a real and symmetric matrix, it can be factored as ${{R}_{c}}={{U}_{c}}{{\Sigma }_{c}}U_{c}^{T}$, where U_c is the eigenvector matrix and Σ_cis the diagonal eigenvalue matrix. U_c and Σ_c can be used for calculatingthe whitening transform matrix as $P=\sqrt{\Sigma _{c}^{-1}}U_{c}^{T}$, which transforms ${{\bar{R}}_{i}}$ as follows
(13) ${{S}_{i}}=P{{\bar{R}}_{i}}{{P}^{T}},i=1,2$
Consequently, S₁ and S₂ will share the same eigenvector. If S₁ is factored as S₁ = BΣ₁B^T ,S₂ will be factored as S₂ = BΣ₂B^T , and Σ₁ +Σ₂ = I, where I is the identity matrix. Given that the sum of two eigenvalues corresponding to the two-class EEG signals is always equal to one, eigenvectors with the largest eigenvalues for S₁ will correspond to those with the smallest eigenvalues for S₂, and vice versa. This property is extremely important for classification of EEG signals, because it means that when the signal variance for one class is maximized, that for the other class will be minimized.

The CSP algorithm leads to a spatial filter matrix as follows:
(14) $W={{({{B}^{T}}P)}^{T}}$
where $W\in {{R}^{N\times N}}$. In general, the first and the last m rows are used as two spatial filters W₁ and W₂ for the two mental tasks respectively. The two spatial filters are optimal in the sense that they extract task-related components and eliminate common components.

3.2.3 Feature definition

The last step of feature extraction is to define feature signals for classification. Suppose that task 1 causes a relatively increased EEG variance over a specific area of the brain, and the variance of the EEG component filtered by W₁ is greatly enhanced compared with that filtered by W₂, and vice versa. Given a single-trial spatiotemporal signal matrix, X, with unknown label, two runs of spatial filtering by W₁ and W₂ are applied. Then, features f₁ and f₂ are defined as follows:
(15) ${{f}_{i}}=\log \left( \frac{\operatorname{var}({{W}_{i}}X)}{\operatorname{var}({{W}_{1}}X)+\operatorname{var}({{W}_{2}}X)} \right),i=1,2$
where f_i takes value between 0 and 1 before logarithmic operation. In theory, f₁ takes value 0 for trials from task 2 and takes 1 for trials from task 1. Contrary results will be yielded for f₂. The logarithmic operation is adopted to make the distribution of elements in f_i more normal. Ultimately, the feature vector used for classification can be structured as $f=[f_{1}^{1},f_{1}^{2},\cdot \cdot \cdot f_{1}^{m},f_{2}^{1},f_{2}^{2},\cdot \cdot \cdot f_{2}^{m}]\in {{R}^{1\times 2m}}$.

A linear support vector machine (SVM) was used as the classifier in this study. Proposed by (Cortes and Vapnik (1995), SVM is a superior classification algorithm in the field of pattern recognition and machine learning. In the field of BCI studies, SVM has exhibited robust classification performance (Blankertz et al., 2003; Kaper et al., 2004; Schlgl et al., 2005). The purpose of SVM is to design a hyperplane g(V) = w^TV+w₀ = 0, which maximizes the margin between two classes of training data, where w is a weight vector and w₀ is an offset. Due to this characteristic, the generalization performance of the classifier is guaranteed.

A linear SVM can be summarized as the following optimization problem:
(16) ${P}_{r}=\begin{cases} minimize \quad J(w)=\frac{1}{2}{{\left\| w \right\|}^{2}}+C\sum\limits_{i=1}^{N}{\xi _{i}^{2}}& \\ subjectto \quad {{y}_{i}}({{w}^{T}}{{V}^{(i)}}+{{w}_{0}})\ge 1-{{\xi }_{\iota }}, & i=1,2,\cdot \cdot \cdot ,N \end{cases}$
where i is the index of training trials, ζ_i is a slack variable and C is a regularization parameter. The role of ζ_i is to slack the requirement of linear separability, whereas that of C is to make a compromise between the bias and variance of classification results.

A linear SVM classifier (Mller et al., 2003) is trained with the function fitcsvm in Statistics and Machine Learning Toolbox^TM. Usually, a model selection procedure is required for determining the regularization parameter C, in order to improve classification accuracy. Since the purpose of this research is to evaluate the search algorithm for channel selection, we adopted the default parameter in fitcsvm, i.e. C = 1.

In this study, both the error rate and the number of chosen channels yielded by BQPSO-CSP and BPSO-CSP were the results of 10-fold cross-validatiobss 5 independent executions (Xi et al., 2016). The setting of w₁ and w₂ in the fitness function (10) is a dynamically changing process, i.e. one weight coefficient changed with the other. We tested 9 combinations of w₁ and w₂ in which w₁ increased from 0.1 to 0.9 in increments of 0.1, while simultaneously, w₂ decreased from 0.9 to 0.1 in sequential reductions of 0.1. Thus, for each subject or patient, both the BQPSO-CSP and the BPSO-CSP had 9 sets of classification results.

Fig. 4 and Fig. 5 depict the classification error rate and the number of channels yielded by BQPSO-CSP and BPSO-CSP at the nine pairs of weight coefficients on the two data sets. Each mark (cycle or asterisk) in each subplot represents the error rate and the number of channels achieved at one pair of weight coefficients. When the weight of error rate (w₁) was assigned the maximum value (0.9), the two methods for channel selection produced the least (or near least) classification error rates, by excluding the most redundant channels. On the contrary, when the weight of channel number (w₂) was assigned the maximum value (0.9), the two methods retained minimal (or near minimal) number of channels, without increasing the error rates as compared to the CSP method using all channels.

Figure 4.

The evolution of the classification error rates yielded by BQPSO-CSP and BPSO-CSP with the number of channels at the nine pairs of weight coefficients (marked by red cycles and blue asterisks) on the EEG data set.

Figure 5.

The evolution of classification error rates yielded by BQPSO-CSP and BPSO-CSP with the number of channels at the nine pairs of weight coefficients (marked by red cycles and blue asterisks) on the ECoG data set.

From each subplot of Fig. 4 and Fig. 5, it can be observed that the changing curve of error rate with the number of channels yielded by BQPSO-CSP is always located on the left of that yielded by BPSO-CSP. This means that to obtain a roughly equal error rate, the latter needs to select many more channels than the former. Examining data from subject al as an example, a 2.17% error rate required an average of 9.6 channels for the former, whereas a 2.42% error rate required an average of 27.6 channels for the latter. Therefore, the proposed BQPSO-CSP method outperformed the BPSO-CSP method, particularly when the number of channels is small.

Fig. 6 and Fig. 7 display the topological analyses of spatial patterns yielded by the three methods for channel selection in representative subjects in the two data sets. The spatial patterns are derived from the CSP filters, i.e., the inverse of the CSP filter matrix (Eqn. 14). The first and the last columns of the inverse matrix constitute the most important spatial patterns. In the two figures, the first row plots the topological maps obtained from all channels, whereas the second and third rows show the topological maps obtained from channels selected by BPSO-CSP and BQPSO-CSP, respectively. The dots in each topological map represent the positions of total channels or chosen channels.

Figure 6.

Visualization of the most important spatial patterns of the two MI tasks derived from three CSP methods for subjects aa and al in the EEG data set (118 raw electrodes in total). The dots in each topological map represent the whole channels in raw data or the channels selected by BPSO/BQPSO. The color at each electrode denotes the magnitude of spatial patterns.

Figure 7.

Visualization of the most important spatial patterns of the two MI tasks derived from three CSP methods for the patient AM in the ECoG data set (64 raw electrodes in total). The dots in each topological map represent the whole channels in raw data or the channels selected by BPSO/BQPSO.

It can be observed from Fig. 6 that the spatial patterns obtained from all channels (1st row) have large weights scattered in several locations irrelevant to the MI tasks. Especially for subject aa, spatial patterns yielded from MI of foot movement appear messy, displaying no clear focus. After BPSO- or BQPSO-based channel selection, the focus of spatial patterns (2nd and 3rd rows) is clearer than that using all channels. Moreover, the foci of these patterns are moved to (nearby) locations related to the MI tasks.

With respect to these two methods for channel selection, while BPSO could reduce the number of channels employed, the positions of the chosen channels were relatively scattered. In addition, some channels outside the focus area were also selected, raising the potential to introduce noise into data used for classification. By contrast, BQPSO selected fewer channels and these channels were concentrated mainly on the focus area. The positions and number of chosen channels explained the decrease in error rate compared to that of BPSO and total channel method. Moreover, channels selected by BQPSO were almost identical to those selected in the focus area by BPSO, especially for subject AM in Fig. 7. Together, these results indicate that the proposed BQPSOCSP method robustly selects channels which are more relevant to mental tasks and interpretable from a neurophysiological point of view.

As shown in Fig. 4 and Fig. 5, the nine combinations of weight coefficients resulted in nine pairs of error rate and number of channels. Thus, channels in a BCI system can be configured according to these results and the requirement of the error rate for a specific application. As an example, the error rate and the number of channels yielded by BQPSO-CSP and BPSO-CSP at the weight coefficients of w₁ = w₂ = 0.5 on the EEG and ECoG data sets are listed in Table 3 and Table 4, respectively. As a comparison, the error rate and number of channels yielded by Basic-CSP-1 and Basic-CSP-2 are listed in Table 3, and those yielded by Basic-CSP-1 only are listed in Table 4. (As Basic-CSP-2 contains results from the 18 benchmarked channels around electrodes C3 and C4 for the EEG data set, there are no Basic-CSP-2 values for the ECoG data set in Table 4). To compare the proposed method with previously presented methods for channel selection, the error rate and number of channels yielded by sparse CSP for channel selection (SCSP1) (Arvaneh et al., 2011) and recursive channel elimination (RCE cross-val.) (Lal et al., 2005), are also listed in Table 3 and Table 4, respectively.

It is observed from Table 3 that BQPSO-CSP yielded the lowest error rate for each subject among the four CSP methods. In particular, subject av demonstrated a substantial drop in error rate from 32.79% (yielded by the full complement of 118 channels) to 18% (by an average of 14.5 channels selected by BQPSO). On average, BQPSO-CSP achieved a reduction of 2.12%, 5.74%, and 7.71% in error rate compared to BPSO-CSP, Basic-CSP-1, and Basic-CSP-2, respectively. These decreases are remarkable in terms of MI-based BCIs. Paired Wilcoxon signed rank tests at 95% confidence level establish a significant difference in error rate between BQPSO-CSP and the other three CSP methods, and between BPSO-CSP and the two Basic-CSP methods, with p values all equaling 0.043. In addition, the average number of channels used by BQPSO-CSP was considerably decreased to an average of 14.9, as compared to 30.8 (in BPSO) and 118 (in Basic-CSP-1). Paired Wilcoxon signed rank tests at 95% confidence level reveal a significant difference in the number of channels between any two of the former three CSP methods, with p values all equaling 0.043. Finally, compared with SCSP-1, BQPSO-CSP remarkably reduced the average error rate by 9.66% and the average number of channels by 7.7.

In the ECoG data set, Table 4 reveals that BQPSO-CSP yielded an average reduction of 8.63% in the error rate of Basic-CSP-1, by decreasing the average number of channels from 74 to 7.9. The decrease in error rate is especially large for subject AM (14.98%). Likewise, BPSO-CSP reduced the average error rate by 3.63% with a remarkable drop in the average number of channels from 74 to 16.9. Hence, both BQPSO-CSP and BPSO-CSP are capable of reducing the error rate by removing a large number of redundant channels. However, BQPSO-CSP was considerably more effective than BPSO-CSP, evidenced by its considerably lower error rate with fewer channels selecte for each of the three patients. Compared with REC cross-val., BQPSO-CSP reduced the average error rate by 8.75% and the average number of channels by 2.9.