I Introduction
A braincomputer interface (BCI), which has been extensively used in neuroscience, neural engineering and clinical rehabilitation, is a communication pathway that allows people to interact with computers using brain signals directly[1, 2, 3, 4]. There are many approaches to collecting signals from the brain. Electroencephalogram (EEG), usually measured from the scalp, may be the most popular one due to its simplicity and low cost[5].
An EEGbased BCI speller allows a user to input text to a computer by thought[6, 7]. It can help severely disabled individuals, e.g., amyotrophic lateral sclerosis patients, to communicate with computers or other people. There are mainly two categories of EEGbased BCI spellers, P300 spellers[6] and steadystate visual evoked potential (SSVEP) spellers[7], which elicit different EEG patterns, as illustrated in Fig. 1a.
A P300 speller, which uses P300 evoked potentials as its input signal[8], was first invented by Farwell and Donchin in 1988[6] and further developed by many others[9, 10, 11, 12]. P300 is a positive deflection in voltage, typically appearing around 250 to 500 ms after a rare target stimulus occurs[13]. It is an endogenous potential linked to people’s cognitive processes, such as information processing and decision making[14, 15]
. The standard oddball paradigm is usually used to elicit P300, in which rare target stimuli are mixed with highprobability nontarget ones. The P300 speller considered in this article uses a
character matrix, which consists of 26 letters and 10 other symbols, as shown in Fig. 1b. The user stares at the character he/she wants to input, while a row or column is rapidly intensified sequentially. The corresponding EEG signals are recorded and classified as a target (containing P300) or nontarget (not containing P300) for each intensification. Then, the computer identifies the character at the intersection of the target row and the target column, which elicit the largest P300s, as the output. For reliable performance, each row and column may have to be intensified multiple times, which reduces the speed of the P300 speller.
Compared with the P300 speller, an SSVEP speller has the advantages of high information transfer rate (ITR), little user training, and some immunity to artifacts[16, 17, 18]. When the user stares at a visual target flickering at a specific frequency, usually between 3.5 Hz and 75 Hz, electrical signals of the same frequency, as well as its corresponding harmonics, can be observed from the EEG signals[16]. In an SSVEP speller, the pictures of different characters are flickering at different frequencies, so that a classifier can directly identify the output character from a large number of candidates by matching their flickering frequencies with the user’s EEG oscillation frequency. Since all characters in an SSVEP speller are flickering simultaneously (in contrast to sequential intensification in a P300 speller), they can have much higher ITRs. The SSVEP speller considered in this article has 40 characters (Fig. 1c), whose stimulation frequencies are from 8 Hz to 15.8 Hz with 0.2 Hz increment[19].
Machine learning is used in BCI spellers to construct the classifiers. Most studies so far focused on making the BCI classifiers faster and more reliable; however, few have considered their security. It has been found in other application domains that adversarial examples[20]
, which are normal examples contaminated by deliberately designed tiny perturbations, can easily fool machine learning models. These perturbations are usually so small that they are indistinguishable to human eyes. Existing studies on adversarial examples focused largely on deep learning models for computer vision. For example, it was found that a picture of a panda, after adding a weak adversarial perturbation, can be misclassified as a gibbon by a deep learning classifier
[21]. Kurakin et al.[22]found that printed photos of adversarial examples can degrade the performance of an ImageNet Inception classifier. Athalye
et al.[23] 3D printed a turtle with an adversarial texture, which was classified as a riffle from almost every viewpoint. Recently, adversarial examples were also found in traditional machine learning models[24] and in many other application domains, e.g., speech recognition[25], text classification[26], malware identification[27], etc.This article aims to expose a critical security concern in EEGbased BCI spellers, and more broadly, EEGbased BCIs, which has received little attention before. It shows that one can generate tiny adversarial EEG perturbation templates for target attacks for both P300 and SSVEP spellers, i.e., mislead the classification to any character the attacker wants, regardless of what the user intended character is. The consequence could range from merely user frustration to severe misdiagnosis in clinical applications[28]. We believe a new and more detailed understanding of how adversarial EEG perturbations affect BCI classification can inform the design of BCIs to defend such attacks.
There have been some studies on adversarial attacks of timeseries signals[25, 28]; however, they need to know the full timeseries before computing the adversarial perturbations, which means these approaches are not causal and hence cannot be implemented in realworld applications. For example, to attack a voice command, previous approaches need to record the entire voice command first, and then design the perturbation. However, once the perturbation is obtained, the voice command has already been sent out (e.g., to a smart phone or Amazon Echo), so there is no chance to add the perturbation to the voice command to actually perform the attack.
What distinguishes the attack approaches in this article most from previous ones is that this article explicitly considers the causality in designing the perturbations. The adversarial perturbation template is constructed directly from the training set and then fixed. So, there is no need to know the test EEG trial and compute the perturbation specifically for it. The perturbation can be directly added to a test EEG trial as soon as it starts, and hence satisfies causality and can be implemented in practice. Thus, it calls for an urgent need to be aware of such attacks and defend them.
A closely related concept is universal adversarial perturbations[29]
, which can also be viewed as adversarial perturbation templates and have been used to attack deep learning models in image classification. In this article, we focus on the security of a traditional and most frequently used BCI pipeline, which consists of separate feature extraction and classification steps, whereas universal adversarial perturbations are usually designed for nontarget attacks of endtoend deep learning models.
Ii Results
Iia Performance evaluation
We used two measures to evaluate the performance of a BCI speller. The first is classification accuracy, and the other is the ITR[30], which measures the speed of the speller:
(1) 
where is the average time (minutes) spent to input a user character, the number of different characters (which was 36 in our P300 speller and 40 in the SSVEP speller), and the classification accuracy. The unit of ITR is bits/min. When the classification accuracy is lower than random guess, i.e., , the ITR is directly set to 0.
To distinguish between the character the user wants to spell, and the one the attacker wants to mislead to, we denote the former user character, and the latter attacker character. Accordingly, user score and user ITR are used to describe the classification accuracy of user characters and the corresponding ITR, respectively. An attacker score is defined as the ratio that the perturbation template leads the speller to output an attacker character, and the corresponding attacker ITR is calculated by replacing in equation (1) with the attacker score. A higher attacker score or attacker ITR represents a better target attack performance.
IiB Security of the P300 speller
Dataset: We used a public P300 dataset (dataset II) introduced by by Rrusienski and Schalk[31]. It recorded 64channel EEG signals from two subjects (A and B). The EEG data were sampled at 240 Hz, bandpass filtered to 0.140 Hz, then normalized for each channel. There were 85 training character trials and 100 test ones for each subject. For each trial, a set of 12 random intensifications (six rows and six columns) were repeated 15 times (i.e., each row was intensified 15 times, and each column was also intensified 15 times). Each intensification lasted for 100 ms, after which the character matrix was blanked for 75 ms. So, it took ms, or 31.5 s, to input a character. The spelling speed can be improved by using fewer repeats, e.g., 10 or 5; however, the spelling accuracy generally decreases with a smaller number of repeats.
The victim model: The victim model was a Riemannian geometry based approach, which won the Kaggle BCI challenge^{1}^{1}1https://www.kaggle.com/c/inriabcichallenge in 2015. First, 16 xDAWN spatial filters[32]
, eight for the target trials and another eight for the nontarget trials, were designed to filter all the trials. The templatesignal covariance matrices of the EEG epochs were projected onto the tangent space of a Riemannian manifold
[33, 34, 35], using Affine Invariant Riemannian Metric as its distance metric. Finally, we classified the feature vectors with a Logistic Regression model in the tangent space. The details can be found in the Supplementary Information. The model was trained with classspecific weights to accommodate class imbalance. All operations in these blocks are differentiable, so we reimplemented them using Tensorflow
[36] to facilitate the gradient calculation.To get the label (target or nontarget) of an intensification, an epoch between 0600 ms from the beginning of the intensification was extracted and fed into the victim model to calculate the target probability. Because each row and column was intensified multiple times, voting was performed for each trial to get the target row and target column, and hence the target character.
Baseline performance: The first part of Table I shows the baseline performance of the clean EEG data (without adding any perturbations). As the number of intensification repeats increased, the user score increased, indicating that the classification accuracy of the user characters increased. Meanwhile, the user ITR decreased, because the time needed to input each character significantly increased.
Subject  Number  Before attack  After attack  
of  Clean  Gaussian noise  User  Attacker  Period  Trial  
repeats  Score  ITR  Score  ITR  SPR  Score  ITR  Score  ITR  SPR  SPR  
5  0.64  13.07  0.65  13.40  20.8  0.072  0.248  0.825  19.8  20.8  25.8  
A  10  0.85  10.62  0.84  10.40  21.0  0.049  0.052  0.900  11.7  21.0  25.9 
15  0.91  8.03  0.92  8.19  21.0  0.040  0.021  0.950  8.7  21.0  25.9  
5  0.79  18.41  0.79  18.41  25.2  0.107  0.578  0.713  15.6  25.2  30.2  
B  10  0.91  11.96  0.89  11.50  25.5  0.061  0.093  0.860  10.9  25.5  30.4 
15  0.93  8.35  0.91  8.03  25.6  0.049  0.034  0.907  8.0  25.6  30.5 
The second part of Table I shows the baseline performance when we added Gaussian noise to the raw EEG data, averaged over 10 runs. The Gaussian noise perturbations were preprocessed in the same way as the adversarial perturbations, by replacing in equation (6) with Gaussian noise, so that they had the same energy. We use signaltoperturbation ratio (SPR) to quantify the magnitude of the perturbation, which is also presented in the second part of Table I. Gaussian noise perturbations had almost no impact on the user score and the user ITR at all, not to mention forcing the P300 speller to output a specific attacker character. These results suggest that more sophisticated adversarial perturbations are needed to attack the P300 speller.
Performance under adversarial attacks: We added the adversarial perturbation template to the test EEG trials to validate whether it was effective in misleading the P300 speller. Fig. 2 shows the attacker scores of the 36 characters. The attacker can manipulate the P300 speller to spell whatever character he/she wants, regardless of what the user intended character is, with higher than 90% average success rate.
The third part of Table I shows the average user scores and ITRs with different number of intensification repeats. The user scores and ITRs were close to zero, suggesting that the user almost cannot correctly input the character he/she wanted.
The fourth part of Table I shows the average attacker scores and ITRs with different numbers of intensification repeats. The attacker score increased with the number of intensification repeats, because more repeats increased the number of times that the attacker can inject the perturbation into the benign EEG trial.
To better quantify the magnitude of the perturbations, we also calculated two SPRs. The adversarial perturbation template was only added at some specific periods of the EEG trial, as shown in Fig. 3a, therefore we defined a period SPR to measure the SPR of the perturbed period, and also a trial SPR to measure the SPR of the entire trial. The last part of Table I shows these SPRs. They were higher than 20 dB, suggesting that the adversarial perturbation template may be undetectable when added to benign EEG trials.
Visualization of the adversarial perturbations: In addition to high attack performance, another requirement in adversarial attacks is that the perturbations should not be detected easily.
Fig. 3a shows a typical EEG trial before and after the adversarial perturbation on Subject A. For clarity, we only show channels F3, F4, Cz, P3 and P4, which evenly distribute on the scalp. One can barely distinguish the adversarial EEG trial from the original EEG trial.
A traditional way to visualize the P300 signal is to take the average of multiple P300 trials. We also took this approach to check if there was a noticeable difference between the average target (or nontarget) trials, before and after perturbation. Fig. 3b shows the results from the Cz channel. One can hardly observe any differences. Fig. 3b also shows the spectrograms and topoplots of the difference between the average target EEG trial and the average nontarget EEG trial. The original and adversarial spectrograms (or topoplots) show very similar energy distributions, and are hardly distinguishable by human eyes.
IiC Security of the SSVEP Speller
Dataset: The dataset was first introduced by Wang et al.[19] as a benchmark dataset for SSVEPbased BCIs. The 64channel signals were recorded from 35 subjects using an extended 1020 system. During the experiments, the subjects were facing a monitor, in which a character matrix was flickering. Different flickering frequencies were assigned to the 40 characters respectively, ranging from 8 Hz to 15.8 Hz with 0.2 Hz increment, as shown in Fig. 1c. Six blocks of EEG signals were recorded from each subject, each with 40 trials, corresponding to the 40 target characters. Each trial was downsampled to 250 Hz and lasted 6 seconds, including 0.5 s before stimulus onset, 5 s for stimulation, and 0.5 s after stimulus offset.
Chen et al.[37] showed that an SSVEP at the stimulation frequency and its harmonics usually starts to be evoked with a delay around 130140 ms; hence, we extracted EEG signals between [0.13, 1.38] s after the stimulus onset as the input to the victim model. Nine channels over the occipital and parietal areas (Pz, POz, PO3, PO4, PO5, PO6, Oz, O1 and O2) were chosen. The signals were bandpass filtered to 790 Hz with a fourthorder butterworth filter.
The victim model
: Extracting the frequency information of SSVEPs is an essential step in recognizing the stimulation frequency, and hence the user character. A natural solution is to utilize fast Fourier transform to estimate the spectrum, so that the energy peaks can be matched to the stimulation frequency; however, canonical correlation analysis (CCA) was recently shown to be more promising in identifying the stimulation frequency
[38, 37]. Thus, CCAbased frequency recognition was used in the victim model.CCA is a statistic approach which can be used to extract the underlying correlation between two multichannel time series[39]. Its main idea is to find a linear combination of channels for each time series, so that their correlation is maximized. When applied to SSVEP spellers, CCA is utilized to calculate the maximum correlation between the input EEG signals and a standard reference signal, which consists of the sinusoidal signal of a stimulation frequency and its harmonics ( in our case).
Mathematically, let denote an EEG trial with channels and samples, and a standard reference signal of stimulation frequency . The th entry of is:
(2) 
where is the sampling rate, , and . To calculate the maximum correlation coefficient , and are first normalized, and then
is computed as the square root of the largest eigenvalue of matrix
(3) 
i.e.,
(4) 
More detailed derivations can be found in the Supplementary Information.
Let be the set of candidate stimulation frequencies ( in our case). Then, the SSVEP speller outputs the character corresponding to the following stimulation frequency:
(5) 
Baseline performance: Among the 35 subjects, eight with the best baseline performances (shown in the first part of Table II) were used in our experiments.
Subject  Before attack  After attack  
Clean  Gaussian Noise  Periodic Noise  SPR  User  Attacker  SPR  
Score  ITR  Score  ITR  Score  ITR  Score  ITR  Score  ITR  
3  0.88  182.5  0.88  181.6  0.73  135.6  25.0  0.44  61.1  0.58  93.3  25.3 
4  0.90  186.9  0.90  187.1  0.71  127.8  25.0  0.07  2.3  0.95  210.1  25.7 
12  0.90  188.8  0.90  188.0  0.82  160.7  25.0  0.26  26.6  0.75  139.5  25.5 
22  0.82  160.0  0.79  150.2  0.77  146.5  25.0  0.11  6.1  0.91  191.0  25.1 
25  0.90  189.1  0.89  184.1  0.87  178.1  25.0  0.78  148.2  0.17  13.8  26.7 
26  0.90  187.8  0.88  180.3  0.60  98.8  25.0  0.03  0.1  1.00  229.9  24.8 
32  0.87  176.9  0.87  179.6  0.64  110.3  25.0  0.03  0.0  1.00  231.4  24.9 
34  0.80  154.7  0.79  151.8  0.47  71.3  25.0  0.03  0.0  1.00  231.2  25.9 
Because SSVEPs are highly susceptible to periodic noise, we evaluated the robustness of the victim model to both Gaussian noise and sinusoidal noise of a random frequency chosen from 40 stimulation frequencies, and a random phase chosen from to . The SPRs were all set to 25 dB, so that the energy of the Gaussian noise and periodic noise was comparable to that of the adversarial perturbation templates. The second and third parts of Table II show the results on these noisy data, averaged over 10 runs, respectively. The victim model was almost completely immune to the Gaussian noise. The periodic noise degraded the model performance more than the Gaussian noise.
Performance under adversarial attacks: We generated 40 adversarial perturbation templates, each forcing the SSVEP speller to output a specific character. Fig. 4 shows their attacker scores. For six of the eight subjects, their output character can be manipulated to any character the attacker wanted, at 70%100% success rate. Interestingly, due to individual differences, Subjects 3 and 25 showed some resistance to adversarial perturbation templates.
The fifth and sixth parts of Table II show the averaged user and attacker performances, respectively. The adversarial perturbation templates were very effective on most subjects (except Subjects 3 and 25), reducing both the user scores and the user ITRs to almost zero, i.e., the user almost cannot correctly input any character he/she wanted. The attacker scores for five subjects were close to one, i.e., the attacker was able to force the SSVEP speller to output any character he/she wanted. The SPRs were all around 25 dB, comparable to the SPRs for random noise.
Visualization of the adversarial perturbations: This subsection shows the characteristics of the adversarial perturbation templates, and verifies their imperceptibility to some widelyused approaches for evaluating the quality of SSVEPs.
Fig. 5a shows the EEG signals before and after adversarial perturbations, along with the magnified difference. The SSVEP speller misclassified the user character, which was supposed to be Y (8.6 Hz), into N (13.2 Hz). Human eyes can barely recognize the difference between the benign and the adversarial EEG trials. After being magnified by 10 times, the perturbation looks periodical, which can modify the user frequency to the attacker frequency.
We compared the clean and adversarial EEG signals with standard sinusoidal signals in Fig. 5b, using Subject 26 as an example. We took the average of the clean temporal waveforms of 8 Hz SSVEPs from Channel POz, and did the same for their adversarial signals with added (which forced the SSVEP speller to output the character of 13 Hz stimulation frequency). We chose Channel POz because the adversarial perturbation on this channel had one of the largest amplitudes, as shown in Fig. 5a. Fig. 5b shows that both clean and adversarial EEG signals were synchronized with the standard 8 Hz sinusoidal signal, indicated by the green dotdashed lines. Comparing the 13 Hz sinusoidal signal with the magnified difference, the synchronization can also be observed, suggesting that the adversarial perturbation template introduced a frequency component matching the attacker character, which was imperceptible to human eyes but powerful enough to mislead the SSVEP speller.
Fig. 5c shows the spectrum analysis of SSVEPs for 40 stimulation frequencies. We averaged the spectra of the benign EEG signals of the same stimulation frequency from all 40 subjects and all chosen channels, so that background activities can be suppressed. The left column of Fig. 5c, for benign trials, clearly shows that the visual stimulus, flickering at a stimulation frequency, can evoke SSVEP of the same frequency and its harmonics. The right column of Fig. 5c shows the same property of adversarial trials, whose attacker character was randomly chosen and fixed for each stimulation frequency. We cannot observe noticeable differences between the two columns in Fig. 5c, demonstrating the challenge in detecting the adversarial perturbation templates.
Iii Discussion
This article aims to expose a critical security concern in EEGbased BCI spellers, and more broadly, EEGbased BCIs, which has received little attention before. It shows that one can generate adversarial EEG perturbation templates for target attacks for both P300 and SSVEP spellers, i.e., deliberatelydesigned tiny perturbations can manipulate an EEGbased BCI speller to output anything the attacker wants with high success rate, demonstrating the vulnerability of BCI spellers. We should emphasize that these attacks are not specific to the victim models used in this article. They may also be used to attack many other classifiers in BCIs with little modification.
It should be noted that the current approaches have two limitations: (a) they require some subjectspecific EEG trials to construct the adversarial perturbation template; and, (b) they need to know the exact timing of the stimulus to achieve the best attack performance. They could be more dangerous if these limitations are resolved.
The first limitation may be alleviated by utilizing the transferability of adversarial examples, which was one of the most dangerous properties of adversarial examples. It was first discovered by Szegedy et al.[20] in 2014 and further investigated by many others[40, 24, 41, 42]. The transferability means that adversarial examples generated from one model can also be used to attack another model, which may have a completely different architecture and/or be trained from a different dataset. Thus, it may be possible to construct the adversarial perturbation template from some existing subjects and then apply it to a new subject.
The second limitation is that the attacker needs to know the precise time synchronization between adversarial perturbation templates and EEG signals. To study how the synchronization time delay affects the attack performance, we show the relationship between the user/attacker scores and the time delay in adding the perturbation template in Fig. 6. For the P300 speller, when the time delay increased, the user scores increased rapidly and the attacker score decreased rapidly, suggesting that the adversarial template was sensitive to the synchronization. The SSVEP speller was more robust to the time delay. These results may suggest that hiding the time synchronization information can help defend adversarial attacks in EEGbased BCI spellers. However, attacks insensitive to the synchronization may also be possible. For example, the idea of adversarial patch[43], which is a tiny picture patch that can mislead the classifier when added anywhere to a large picture to be classified, may be used to increase the robustness to the synchronization time delay. Thus, defending the attackers may not be an easy task.
Finally, we need to emphasize again that the goal of this study is not to damage EEGbased BCIs. We aim to demonstrate that serious adversarial attacks to EEGbased BCIs are possible, and hence expose a critical security concern, which received little attention before. Our future research will develop strategies to defend such attacks. Meanwhile, we hope our study can attract more researchers’ attention to the security of EEGbased BCIs.
Iv Methods
Iva Attack the P300 Speller
The main idea to construct the adversarial perturbation template was to find a universal perturbation which leads the P300 classifier to classify nontarget epochs into target ones. The approach was to get the directions pointing from nontarget epochs to the decision boundary of the victim model, and then sum up these directions as the universal perturbation. These directions can be identified by simply calculating the gradients of the loss with respect to the input nontarget EEG epochs, assuming the decision boundary is linear. Though the victim model includes nonlinear operations, the attack approach still worked surprisingly well.
Let be an EEG trial, its label (0 for nontarget, and 1 for target), the victim model which gives the label probability for each input ,
the loss function (crossentropy loss in our case), and
the dataset containing all nontarget epochs in the training set. Then, the overall direction can be computed as:(6) 
After obtaining , we filtered it by a fourthorder Butterworth bandpass filter of Hz, extracted the first 350ms signal, and then normalized it in each channel so that the L2 norm is 1. Denote the result as . Then, the adversarial perturbation was computed as:
(7) 
where is a constant controlling the energy of the perturbation ( in our experiments).
To mislead the P300 speller, one only needs to tamper with some specific signal periods according to the onset of the target stimuli. Because in a practical P300 speller the same row or column is never intensified successively, the perturbation template can last more than one intensification period. In our experiments, the template lasted ms, i.e., two intensification periods.
Fig. 7 illustrates the attack procedure. The benign EEG trial would output character 7, since the last row and the third column of the character matrix have the highest P300 probability, and their intersection is 7. However, after applying the perturbation template, the trial outputs the character Z, because the fifth row and the second column have the highest P300 probability. Interestingly, the adversarial template acts like random noise when it is not synchronized with an intensification onset. As shown in Fig. 7, the last 175 ms of the template does not influence the classification of the corresponding intensification.
IvB Attack the SSVEP Speller
There are two difficulties in attacking the victim model of the SSVEP speller. First, the victim model is not fixed, as the parameters of CCA vary in different EEG trials. Second, unlike the P300 speller whose base victim model only needs to classify the input into two classes, there are many more classes in the SSVEP speller. These make adversarial attacks of the SSVEP speller much more challenging.
The purpose was to generate the adversarial perturbation template , which can lead the SSVEP speller to output the attacker character of stimulation frequency . For each user, we used the first block to craft , and the remaining five blocks to evaluate its attack performance.
According to the victim model, should be able to maximize in equation (4), such that
(8) 
In other words, can be crafted by solving
(9) 
where is defined in equation (3).
Since is not symmetric, it is difficult to calculate the derivatives of its largest eigenvalue, resulting in challenges in optimization. Because of the fact that the largest eigenvalue is always no smaller than the average of all eigenvalues:
(10) 
instead of solving equation (9) directly, we can maximize its lower bound to reduce the optimization difficulty:
(11) 
Because the effective frequency band of SSVEP signals is 790 Hz, we introduced a new variable so that
(12) 
where means retaining only the 790 Hz effective signal frequency components. As a result, we can ensure the integrity of the adversarial template during signal filtering. In addition, we added to penalize the energy of the perturbation, where is the penalty coefficient.
Finally, the problem becomes:
(13) 
Gradient descent was used to update , and the iteration stopped when the SPR was lower than a threshold, which was set to 25dB in our experiments.
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Appendix A The victim model of the P300 speller
The details of the victim model of the P300 speller are introduced.
Aa xDAWN spatial filters
The original xDAWN filter[32] was designed for P300 evoked potentials by enhancing the target response with respect to the nontarget response. We used a generalized version in our experiments, which was implemented in pyRiemann^{2}^{2}2https://pyriemann.readthedocs.io/en/latest/index.html.
More specifically, let be the training set, where is the th meancentered EEG epoch ( is the number of channels, and the number of time domain samples), and its corresponding label ( for nontarget, and for target). The average epoch , , is first calculated. Spatial filters were then designed to maximize the signal to signalplusnoise ratio for each class:
(14) 
where is the number of filters ( was used in our experiments), is obtained by concatenating all EEG epochs in along the channels, and is the trace of a matrix. Generalized eigenvalue decomposition can be used to solve equation (14).
After obtaining the filters for both classes, the concatenated spatial filters can be used to filter each EEG epoch:
(15) 
AB Tangent space projection
Covariance matrices of the EEG trials are widelyused in BCIs. However, they lie on a Riemannian manifold of Symmetric Positive Definite (SPD) matrices, and hence cannot be directly used by Euclidean space classifiers, such as Logistic Regression and Support Vector Machines. To solve this problem, the covariance matrices are projected onto the Euclidean tangent space of a reference SPD matrix, and then the vectorized features are used by Euclidean space classifiers.
More specifically, we first calculate the augmented covariance matrix for each :
(16) 
where . Then, is projected onto the tangent space of the reference SPD matrix
, which is the geometric mean of
, i.e.,(17) 
where is the Affine Invariant Riemannian Metric distance:
(18) 
The vectorized features are:
(19) 
where vectorizes the upper triangular part of a symmetric matrix. A weight of is applied to the offdiagonal elements, and a weight of to the rest, during the vectorization. can then be fed into any Euclidean space classifier.
Appendix B Canonical correlation analysis (CCA)
This section introduces CCA, which can be used to extract the underlying correlation between two time series.
Ba Problem setup
Let and be two multichannel time series, where and represent the number of channels, and the number of time domain samples. and are normalized in each channel.
The main idea of CCA is to find a pair of canonical variables, denoted as and , for and respectively, so that the correlation coefficient between and can be maximized. The problem can be mathematically formulated as:
(20) 
which can be reexpressed as:
(21)  
s.t. 
BB Solution of CCA
There are several approaches to solve equation (21). Here we introduce the Lagrange multiplier method[44].
Denote by . Then, equation (21) can be rewritten as:
(22)  
s.t. 
According to the Lagrange multiplier method, equation (22) is equivalent to , where:
(23) 
By setting the first partial derivatives to zero, i.e.,
(24)  
(25)  
(26)  
(27) 
we have
(28) 
It should be noted that equation (28) is also the definition of the correlation coefficient .
According to equations (24) and (25), we have:
(29) 
which implies that equals the largest eigenvalue of , and
is the corresponding eigenvector.
can be obtained in a similar way.
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