# Motor-imagery classification model for brain-computer interface: a sparse
group filter bank representation model

Cancheng Li1,2 Chuanbo Qin3 Jing Fang4 1School of Biological Science and
Medical Engineering, Beihang University, Beijing, China  2School of
Information Science and Engineering, Lanzhou University, Lanzhou, China
3Department of Intelligent Manufacturing, Wuyi University, Jiangmen, China
4Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences,
Shenzhen, China

###### Abstract

Background: Common spatial pattern (CSP) has been widely used for feature
extraction in the case of motor imagery (MI) electroencephalogram (EEG)
recordings and in MI classification of brain-computer interface (BCI)
applications. BCI usually requires relatively long EEG data for reliable
classifier training. More specifically, before using general spatial patterns
for feature extraction, a training dictionary from two different classes is
used to construct a compound dictionary matrix, and the representation of the
test samples in the filter band is estimated as a linear combination of the
columns in the dictionary matrix.

New method: To alleviate the problem of sparse small sample (SS) between
frequency bands. We propose a novel sparse group filter bank model (SGFB) for
motor imagery in BCI system.

Results: We perform a task by representing residuals based on the categories
corresponding to the non-zero correlation coefficients. Besides, we also
perform joint sparse optimization with constrained filter bands in three
different time windows to extract robust CSP features in a multi-task learning
framework. To verify the effectiveness of our model, we conduct an experiment
on the public EEG dataset of BCI competition to compare it with other
competitive methods.

Comparison with existing methods: Decent classification performance for
different subbands confirms that our algorithm is a promising candidate for
improving MI-based BCI performance.

###### keywords:

Brain-computer interface (BCI) , sparse grouped filter bank (SGFB) , common
spatial pattern (CSP) , sparse representation classification (SRC).

††journal: Journal of Neuroscience Methods

##  1 Introduction

Brain-computer interface (BCI) is a new method for the communication and
control between the human brain and external devices [1, 2]. Compared with the
evoked potential-based BCI, motor imagery (MI) is easily operated and does not
rely on external stimuli [3, 4]. It has been suggested to be suitable for the
mechanical control and exercise rehabilitation training. MI system shows
various applications, such as controlling the movement of a wheelchair, the
mouse cursor on the computer screen, and the movement of the left and right
direction by imagining the left and right hands [5, 6]. Electroencephalogram
(EEG) signal of the brain activities can effectively control the execution of
MI tasks. In the past decade, its low cost, non-invasiveness and wide
availability have attracted the interest of many researchers [7].

At present, the widely used EEG signals for BCI system control include
sensorimotor rhythms (SMRs), event-related potentials (ERP), and visual evoked
potentials (VEP) [8, 9, 10, 11, 12]. Particularly, event-related
desynchronization / synchronization (ERD / ERS) utilizes the $mu$ rhythm power
of sensory motor rhythm (SMR). Recently, a series of BCI has been established
based on rhythmic activity recorded on the sensorimotor cortex. SMR draws the
attention from BCI using non-invasive neural recording like EEG. SMR is a
feature as a band power change within a particular EEG frequency band [3]. At
the same time, the EEG band appears in the brain region of sensory organ
motion imaging. Therefore, the EEG power conversion can be correlated as the
control of MI task.

Until now, many algorithms have been applied to the EEG classification in BCI
system [10, 14, 15, 16, 17, 18, 19]. For example, a common spatial pattern
(CSP) as a classic feature extraction method was introduced in this field. It
uses the spatial features of ERD/ERS, which consists of a spatial filtering
technique and simultaneously detects filters maximizing the variance for one
class and minimizing the variance for another class [20]. The CSP is greatly
effective to classify motor imagery EEG, since the variance of the bandpass
filtered signals is equal to the bandpower. This is why CSP plays a key role
in discriminating the information of SMR related EEG data [21, 22]. Besides,
algorithms combining filter-bank structure with regularization common spatial
pattern (RCSP) have also been studied by many researchers. In [10], Park et
al. presented RCSP feature extraction on each frequency band, and used mutual
information as the individual feature algorithm in a small sample (SS).
Moreover, a sub-band regularized common spatial pattern (SBRCSP) used
principal component analysis (PCA) to extract RCSP features from all frequency
sub-bands [9]. In [10], the author used the filter-bank regularized common
spatial pattern (FBRCSP) selected optimum frequency bands for extracting
mutual information of RCSP feature. Apart from the methods of feature
extraction, another research interest in this field is on the complex
classification algorithm to improve the accuracy of EEG classification and
provide good robustness [23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In earlier
studies, with the assumption that sample covariance matrix of different
classes are similar, linear discriminant analysis (LDA) was used as the main
algorithm to distinguish the two types of motor imagination [33, 34]. In order
to improve the generalization ability of the model, numerous regularization
classifications have been applied for SMR classification. For instance,
Mahanta et al. improved the reliability of the classifier by regularized LDA
to correct inaccuracy of estimated covariance matrices [35]. Moreover, the
sparse representation classification (SRC) which has been successfully applied
in the image field [36, 37] is also employed in many studies. The SRC aims to
estimate the sparse representation of the test samples as a linear combination
of the columns (ie, training samples) in the dictionary matrix. Then, we
identify the category by minimizing the reconstruction error, the labels of
the test samples are determined by detecting which class the training samples
providing the smallest residual norm belong to [43]. Many studies show that by
introducing the SRC scheme, the classification accuracy on SMR can be
distinctly improved [38, 39, 40, 41, 42].

Many previous researchers focused on SMR feature extraction and pattern
recognition. Then, when the dimensions of the training data are insufficient,
the number of training samples cannot truly reflect the distribution of
features, which can lead to unsatisfactory results. In order to obtain
relatively large data, a long calibration time is required during BCI
experiment acquisition, which will affect the practicality of the system. On
the other hand, some research try to reduce the sample size as much as
possible without sacrificing classification accuracy.

In order to address the SMR classification in the SS situation and decrease
the calibration time of the BCI system, we propose a novel sparse group filter
bank representation model (SGFB). The most compact representation of the test
sample is estimated as a linear combination of columns in the dictionary
matrix. Moreover, unlike the SRC scheme using only $L_{1}$-norm
regularization, the SGFB introduces two penalty factors ($L_{1,2}$-norm) to
control the sparsity by frequency bands, which effectively and automatically
select frequency bands from a sparse group representation of the test samples
and exclude those which provide no contribution. To be specific, the EEG
signals in the range of 4-40Hz are divided into subbands with a range of 4Hz,
i.e., 4-8Hz, 8-12Hz, 12-16Hz, 16-20Hz,$\cdots$,36-40Hz by a filter bank that
consists of a five-order butterworth bandpass filter. Furthermore, CSP is
applied to the divided signals by the filter bank. Finally, we apply the SGFB
method for the BCIs and the output of classification. The performance of the
SGFB algorithm is evaluated by the classification accuracy for five subjects
in the BCI competition III dataset IVa. The good results we obtained suggest
our method provides a new direction for the classification of the small sample
sets for MI. Furthermore, for the filter band optimization, the EEG
segmentation time window is an important issue and has a certain
interpretation of the correlation among features [44]. In an MI system,
subjects is usually required to complete certain tasks. However, the brain’s
response time to psychological tasks is an unknown parameter. In the MI tasks,
0-1 s is considered to be the preparatory stage before the mission, while the
period between 3.5 and 4 seconds is a later stage.

The remainder of this paper is organized as follows. The previous work is
introduced in Section 2.3. In Section 2, mathematical methods which include
the CSP feature extraction method, the sparse filter bank, and the filter bank
group sparse representation are introduced. In Section 3, a brief description
on the experimental procedures is provided. Results and discussion are given
in Section 4. This paper ends with a conclusion in Section 5.

##  2 Problem formulations and the proposal of SGFB model

###  2.1 Mathematical Symbols

To state the rest of our study more clearly, we first list the mathematical
symbols and their meanings in this paper as below. We denote vectors and
matrices as ltalian italics. Furthermore, we define the main notations in the
tab 1.

Table 1: Definitions of main notations Parameter | Description | Definition
---|---|---
$\it{X}\in{\mathbb{R}^{m\times n}}$ | The EEG signal with K class | $\it{{{X}=[X_{1},X_{2},\cdots,X_{K}]}}$
$\it{{X_{K}}}\in{R^{m*{n_{k}}}}$ | The training samples labeled | $\it{{{X_{K}}=[X_{1}^{k},X_{2}^{k},\cdots,X_{nk}^{k}]}}$
| with the $k$th class |
$\it{{W}}\in\mathbb{R}^{N\times 2M}$ | Spatial filter | $\it{{W=\left[{{w_{1}},\cdots,{w_{2M}}}\right]}}$
$D\in\mathbb{R}^{2M\times(N1+N2)}$ | Composite matrix of single | $D=[D_{Left},D_{Right}]$
| frequency band |
$\it{\widetilde{D}_{f}}\in{\mathbb{R}^{2M\times F}}$ | The composite matrix with 8 | $\it{\widetilde{D}_{f}=[D_{f1},D_{f2},\cdots,D_{f9}]}$
| frequency bands |
$\it{y}\in{\mathbb{R}^{m\times 1}}$ | Test sample | ${y}=[y_{1},y_{2},\cdots,y_{n}]$
$\it{{\widetilde{D}}}\in\mathbb{R}^{2M\times i}$ | Component dictionary matrix | $\it{{\widetilde{D}=\left[{{D_{l,1}},{D_{l,2}},\cdots,{D_{l,{N_{l}}}}}\right]}}$
$\it{u^{*}}$ | Band sparse weight | $\it{{u^{*}}=\left[{u_{1}^{*},u_{2}^{*},\cdots,u_{n}^{*}}\right]}$
$\it{{X_{L}},{X_{R}}}$ | Left and right motor- | $\it{{X_{L}},{X_{R}}}\in{\mathbb{R}^{2n\times T}}$
| imagery signal |
$\it{X_{L}^{CSP},X_{R}^{CSP}}$ | Left and right motor-imagery | $\it{X_{L}^{CSP},X_{R}^{CSP}}\in{\mathbb{R}^{2n\times T}}$
| signal extracted by CSP |
$\it{{\delta_{l}}}:{\mathbb{R}^{N}}\to{\mathbb{R}^{N}}$ | Characteristic function | -

###  2.2 Common spatial pattern (CSP)

We assume an EEG epoch with $N_{t}$ time samples from $N_{n}$ channels. This
EEG signal first passes through a set of $N_{f}$ bandpass filters. Denote the
data set of training samples labeled with the $k$th class as

${\it{{X_{k}}=[X_{1}^{k},X_{2}^{k},\cdots,X_{nk}^{k}]}}\in{\mathbb{R}^{m\times{n_{k}}}}$
(1)

where each element is a training sample, $m$ is the dimension of the feature
space and $n_{k}$ is the total number of $X_{k}$. It supposes that the
dictionary define the samples to be classified as
${\it{{X}=[X_{1},X_{2},\cdots,X_{K}]}}\in{\mathbb{R}^{m*n}}$, $K$ denotes the
number of class and $n=\sum\nolimits_{i=1}^{K}{{n_{i}}}$. Given a query sample
$\it{y}\in{\mathbb{R}^{m}}$, the task of pattern recognition is to determine
which class $y$ belongs to.

Matrix $\it{X_{j}}$ refers to the EEG signal filtered in the $j^{th}$ filter
bank for a single trial, the sample covariance matrix $\it{C}$ for
$\it{X_{j}}$ is calculated as

$C=\it{\frac{{{X_{j}}X_{j}^{T}}}{{tr\left({{X_{j}}X_{j}^{T}}\right)}}}$ (2)

Suppose the EEG data obtained from a temporal interval in the $ith$ trial of
class $l$ ($l=-1~{}or~{}+1$) and assume that the EEG samples have removed the
mean within a given frequency band, then the spatial covariance matrix of the
category $l$ can be calculated as

$\it{\sum\nolimits_{l}=\frac{1}{{{N_{l}}}}\sum\nolimits_{i=1}^{N{}_{l}}{{X_{i,l}}}X_{i,l}^{T}}$
(3)

where $N_{l}$ is the number of trials belonging to the class $l$, and $T$
denotes the transpose operator. CSP aims at seeking spatial filters (i.e.,
linear transforms) to maximize the ratio of transformed data variance between
two classes as

$\it{\mathop{\max}\limits_{w}J\left(w\right)=\frac{{{w^{\rm
T}}\sum\nolimits_{1}w}}{{{w^{\rm
T}}\sum\nolimits_{2}w}}~{}~{}~{}~{}~{}~{}~{}{\rm{s}}{\rm{.t}}{\rm{.}}~{}~{}~{}~{}{\left\|w\right\|_{2}}}=1$
(4)

where ${\it{w}\in\mathbb{R}^{C}}$ is a spatial filter and $\|{\cdot}\|$ is the
$l_{2}$-norm. The CSP filter matrix $W$ consists of the column vector
$\it{{w_{i}}}$. This can be achieved by equivalently solving a generalized
eigenvalue problem
$\sum\nolimits_{1}{\it{w}}=\lambda\sum\nolimits_{2}{\it{w}}$. By collecting
eigenvectors corresponding to the largest and smallest generalized eigenvalues
of the $M^{*}$, a set of spatial filters are obtained from $\it{W}$. For a
given EEG sample ${\it{X}}$, the feature vector is formed as ${\it{Z}}$ with
entries as follow,

${{\rm Z}_{m}}=\log\left({{\mathop{\rm
var}}\left({w_{m}^{T}X}\right)}\right)~{}~{}~{}~{}~{}~{}~{}{\rm{m=1,}}\cdots{\rm{,2M^{*}}}$
(5)

where var($\cdot$) denotes the variance.

Given two types of EEG training signals, ${\it{X_{L}}}$ and ${\it{X_{R}}}$, we
define the CSP filtered signal as

$\begin{array}[]{l}{\it{X_{R}^{CSP}}}=W_{CSP}^{T}{X_{R}}\\\
{\it{X_{L}^{CSP}}}=W_{CSP}^{T}{X_{L}}\end{array}$ (6)

The EEG samples ${\it{{X_{i,l}}}}$ is the bandpass filtered on different bands
instead of just a single frequency band, the CSP features can be extracted on
each frequency band. Hence, the dimensionality of the feature vector is
expanded to be 2M$\times$ N by concatenating all of the features. Note that,
the successful application of CSP highly depends on the selection of filter
bands. The optimal filter band is generally subject-specific. An increasing
number of studies have suggested that the accuracy of MI classification can be
significantly improved by the optimization of the filter band.

Figure 1: The block diagram of the SVM and sparse representation method.

###  2.3 Sparse representation classification (SRC)

We have mentioned that $X$ is the training sample. In addition, $\it{X_{ij}}$
and ${n_{i}}$ represent the $j$th training sample and the number of training
samples from the $i$th class respectively. Furthermore, the test signal
$\it{y}\in\mathbb{R}^{m}$ can be sparsely represented as a linear combination
of some columns of $\it{X}$, where ${\vartheta}\in{\mathbb{R}^{n}}$ is a
coefficient vector corresponding to $\it{X}$. For a test sample
${\vartheta}\in{\mathbb{R}^{n}}$, its representation under all the training
samples is

$\begin{array}[]{l}y={X_{1}}{\vartheta_{1}}+{X_{2}}{\vartheta_{2}}+\cdots+{X_{c}}{\vartheta_{c}}\\\
{\rm{}}={X_{11}}{\vartheta_{11}}+{X_{12}}{\vartheta_{12}}+\cdots+{X_{c{n_{c}}}}{\vartheta_{c{n_{c}}}}\\\
{\rm{}}=X\vartheta\end{array}$ (7)

Therefore, the sparse solution of Eq. 7 can be represented by Eq. 8.

$\mathop{\arg\min}\limits_{\vartheta}\left\|{\varphi-Z\vartheta}\right\|_{2}^{2}+\lambda
f\left({{{\left\|\vartheta\right\|}_{p}}}\right)$ (8)

where $\varphi$ and $Z$ are the test sample and the training samples,
respectively. $f\left({{{\left\|\vartheta\right\|}_{p}}}\right)$ serves as an
adjustable penalty function of $l_{p}$-norm constrained, and $\lambda>0$ is a
limiting parameter, a larger $\lambda$ can obtain a greater degree of sparse
solution. According to the changes in $\it{Z}$, $\it{\varphi}$ and $p$, when
Equation 9 is minimized, we get the sparse ground coefficient vector.
$\it{{\vartheta^{*}}}$. Let $\it{{\delta_{i}}\left({{\vartheta^{*}}}\right)}$
be a vector whose only nonzero entries are associated with class $i$, the
class label of $\it{y}$ can be decided as $\it{y}$ that gives the minimum
reconstruction error, i.e.,

$y=\mathop{\arg\min}\limits_{\vartheta}{\left\|{\varphi-Z{\delta_{i}}\left({{\vartheta^{*}}}\right)}\right\|_{2}}$
(9)

Besides, we show the commonly used SVM and sparse representation frameworks
using the BCI in the Fig. 1.

Figure 2: The block diagram of the filter bank group sparse representation
method. We define a component dictionary matrix
$\widetilde{D}=\left[{{D_{l,1}},{D_{l,2}},\cdots,{D_{l,{N_{l}}}}}\right]$, and
the $\widetilde{D}$ is a matirx of size $2M\times N$, $u^{*}$ is of size
$N\times 1$ and is divided into $K+1$ non-overlapping group
${u_{0}},{u_{1}},{u_{2}},\cdots,{u_{K}}$.

###  2.4 Sparse group filter bank (SGFB)

Our proposed method is based on the SGFB framework, i.e., we use the different
frequency bands (4-8Hz, 8-12Hz, … , 36-40Hz). The block diagram of the filter
bank group sparse representation method is shown in Fig. 2. SRC is used in
many applications [48], especially in face recognition. SGFB is an extended
approach of the SRC, which can provide the new idea in MI classification
tasks. The CSP feature of the test samples are in the form of a linear
combination and are superior to the LDA method in the MI classification of
BCI. In our algorithm, we design the dictionary matrix to represent the test
samples.

Although many algorithms have achieved significant results in MI
classification, it is difficult to provide satisfactory results when the size
of available train samples is small. Furthermore, to achieve better
classification performance, relatively large system calibration time is
required for larger EEG data. Therefore, to avoid sacrificing classification
accuracy and obtain significant results with small data, we propose a new
classification framework based on SGFB.

Suppose that the number of training samples for each class $l$ is $N_{l}$ ($l$
= 1 for the left hand, and the $l=2$ for the right hand). Denote
$\widetilde{D}=\left[{{D_{l,1}},{D_{l,2}},\cdots,{D_{l,{N_{l}}}}}\right]$ as
the combined dictionary matrix of class $l$, where the column vectors
$\it{{D_{l,i}}}\in{\mathbb{R}^{2M\times 1}}$ $(i=1,2,\cdots,N_{l})$ are the
CSP features obtained by 4. Then, with the constructed dictionary matrix, we
will use SGFB to find the best sparse representation vector $u^{*}$.

The coherence measures the correlation between the two component dictionaries
defined in the following way:

$M\left({{D_{L}},{D_{R}}}\right)\buildrel\Delta\over{=}\max\left\\{{\left|{\left\langle{{D_{L,j}},{D_{R,k}}}\right\rangle}\right|:j,k=1,2,\cdots,{N_{l}}}\right\\}$
(10)

The vector ${D_{L,j}}$ is the $j^{th}$ column of ${D_{L}}$; similarly,
${D_{R,k}}$ is the $k^{th}$ column of ${D_{R}}$. The notation
$\left\langle{{D_{L,j}},{D_{R,k}}}\right\rangle$ denotes the inner product of
two vectors. We call $M$ the measure of mutual coherence of the two component
dictionaries; when $M$ is small, we say that the complete dictionary is
incoherent.

By this method, we construct a complex dictionary matrix from different
frequency bands, which includes not only the training samples collected on the
frequency band (8-13Hz) that many researchers have already investigated, but
also on the frequency bands that are rarely considered before. As a result,
good classification effect can be obtained even for a relatively limited
amount of the training samples. Applying sparsity to the entire training set,
the SGFB estimates the optimal representation vector, and the $\ell_{1}$ norm
regularization is employed to further exclude those insignificant frequency
bands. A detailed description of the SGFB method is given below.

We collect EEG training data from 9 different frequency bands. We then use the
CSP method for feature extraction and derive K+1 dictionaries
$D_{f1},D_{f2},...D_{f9}$ in different frequency bands, which are concatenated
later into a conforming matrix $\widetilde{D}$. The compounding different
frequency bands to form a composite matrix and our SGFB method is designed to
estimate the best representation vector $u^{*}$ as:

${\it{{u^{*}}}}=\mathop{\arg\min}\limits_{u}\frac{1}{2}\left\|{y-\widetilde{D}{u}}\right\|_{F}^{2}+\lambda{\left\|u\right\|_{1}}$
(11)

where $\lambda$ denotes the Hyperparameter. In order to generate a sparse
vector for the samples of $u^{*}$, the quadratic constraint is

$\mathop{\arg\min}\limits_{\it{u}}\sum\limits_{i=1}^{n}{\left\|{{\it{{u_{i}}}}-\frac{1}{n}\sum\limits_{j=1}^{n}{{\it{{u_{j}}}}}}\right\|}_{2}^{2}$
(12)

To make the sparse vector as close as possible to its concentrated vector, we
update Eq. 11 as shown in Eq. 13.

$\mathop{\arg\min}\limits_{u}\frac{1}{2}\left\|{y-\widetilde{D}{u}}\right\|_{F}^{2}+\lambda{\left\|u\right\|_{1}}+\frac{{{\lambda_{1}}}}{2}\sum\limits_{i=1}^{n}{\left\|{{u_{i}}-\frac{1}{n}\sum\limits_{j=1}^{n}{{u_{j}}}}\right\|_{2}^{2}}$
(13)

The hyperparameter $\lambda_{1}$ is the constraint parameter, which improves
the generalization ability of the model by controlling the sparsity of $u^{*}$
and the sparsity between groups of these subbands. The optimal sparse matrix
${u^{*}}$ can generate not only similar sparse decompositions, but also
similar weights. Therefore, if $u_{ik}^{*}$ and $u_{jk}^{*}$ are not zero,
they should be with the same sign, namely, $u_{ik}^{*}\times u_{jk}^{*}>0$.

Given data $y$, $\widetilde{D}_{f}$, and the parameters $\lambda$ and
$\lambda_{1}$, if $u_{ik}^{*}\times u_{jk}^{*}>0$, then $u_{ik}^{*}$ and
$u_{jk}^{*}$ have ${\mathop{\rm sgn}}\left({u_{ik}^{*}}\right)={\mathop{\rm
sgn}}\left({u_{jk}^{*}}\right)$, where ${\mathop{\rm
sgn}}\left(\bullet\right)$ denotes the sign function, Eq. 11 is transformed
into

$L\left({{c_{i}},\lambda,{\lambda_{1}}}\right)=\frac{{\rm{1}}}{{\rm{2}}}\left\|{y-\widetilde{D}u}\right\|_{F}^{2}+\lambda{\left\|{{u_{i}}}\right\|_{1}}+\frac{{{\lambda_{1}}}}{{\rm{2}}}\sum\limits_{i=1}^{n}{\left\|{{u_{i}}-\frac{1}{n}\sum\limits_{j=1}^{n}{{u_{j}}}}\right\|}_{2}^{2}$
(14)

in this case, $u_{i}^{*}$ should satisfy $(if\left({\it{{u_{{i_{k}}}}}^{*}\neq
0}\right))$

$\frac{{\partial
L\left({{u_{i}},\lambda,{\lambda_{1}}}\right)}}{{\partial{u_{ik}}}}\left|\begin{array}[]{l}\\\
{u_{i}}=u_{i}^{*}\end{array}\right.=0$ (15)

It obtains

$\begin{array}[]{l}-\alpha_{k}^{T}\left({{y_{i}}-\widetilde{D}u_{i}^{*}}\right)+\lambda{\mathop{\rm
sgn}}\left({u_{ik}^{*}}\right)+\\\
{\lambda_{1}}\left({1-\frac{1}{n}}\right)\left({u_{ik}^{*}-\frac{1}{n}\sum\nolimits_{t=1}^{n}{u_{tk}^{*}}}\right)=0\end{array}$
(16)

$\begin{array}[]{l}-\alpha_{k}^{T}\left({{y_{j}}-\widetilde{D}u_{j}^{*}}\right)+\lambda{\mathop{\rm
sgn}}\left({u_{jk}^{*}}\right)+\\\
{\lambda_{1}}\left({1-\frac{1}{n}}\right)\left({u_{jk}^{*}-\frac{1}{n}\sum\nolimits_{t=1}^{n}{u_{tk}^{*}}}\right)=0\end{array}$
(17)

Subtract equation 16 and 17 as follow,

$\alpha_{k}^{T}\left({{r_{j}}-{r_{i}}}\right)+{\lambda_{1}}\left({1-\frac{1}{n}}\right)\left({u_{ik}^{*}-u_{jk}^{*}}\right)=0$
(18)

which gives an equivalent form as

$u_{ik}^{*}-u_{jk}^{*}=\frac{n}{{\left({n-1}\right){\lambda_{1}}}}\alpha_{k}^{T}\left({{r_{i}}-{r_{j}}}\right)$
(19)

where ${y_{i}}-\widetilde{D}u_{i}^{*}$ is the residual vector. We can update
the Eq. 19 and implies that

$\begin{array}[]{l}\left|{u_{{i_{k}}}^{*}-u_{{j_{k}}}^{*}}\right|=\frac{n}{{\left({n-1}\right){\lambda_{1}}}}\left|{\alpha_{k}^{T}\left({{r_{i}}-{r_{j}}}\right)}\right|\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq\frac{n}{{\left({n-1}\right){\lambda_{1}}}}{\left\|{{\alpha_{k}}}\right\|_{2}}{\left\|{{r_{i}}-{r_{j}}}\right\|_{2}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq\frac{1}{{\left({n-1}\right){\lambda_{1}}}}{\left\|{{r_{i}}-{r_{j}}}\right\|_{2}}\end{array}$
(20)

For calculating the optimal solution, we have rewritten Eq. 12 as Eq. 21

$\begin{array}[]{l}{\rm{}}\sum\limits_{i=1}^{n}{\left\|{{u_{i}}-\frac{1}{n}\sum\limits_{j=1}^{n}{{u_{j}}}}\right\|_{2}^{2}}\\\
=\left\|{u-u\frac{1}{n}{{11}^{\text{T}}}}\right\|_{F}^{2}\\\
=\left\|{u\left({I-\frac{1}{n}{{11}^{T}}}\right)}\right\|_{F}^{2}\\\
=Tr\left({u{{\left({I-\frac{1}{n}{{11}^{T}}}\right)}^{2}}{u^{T}}}\right)\\\
=Tr\left({uM{u^{T}}}\right)\end{array}$ (21)

where $Tr\left(\cdot\right)$ denotes the trace and ${\left(\cdot\right)^{T}}$
denotes the transpose of a matrix, the
$M={\left({I-\frac{1}{T}{{11}^{T}}}\right)^{2}}\in{\mathbb{R}^{T\times T}}$
$(I\in\mathbb{R}^{T\times T}\text{is the unit matrix
and}~{}1=[1,\cdots,1]^{T}\in\mathbb{R}^{T})$. Hence, we can update the Eq. 13
as

$\mathop{\arg\min}\limits_{i}\frac{1}{2}\left\|{y-\widetilde{D}{u^{*}}}\right\|_{F}^{2}+\lambda{\left\|u\right\|_{1}}+\frac{{{\lambda_{1}}}}{2}Tr\left({uM{u^{T}}}\right)$
(22)

The $c_{i}$ is updated by fixing the vector
${\left\\{{{u_{j}}}\right\\}_{j\neq i}}$, and the optimization process is as

$\mathop{\arg\min}\limits_{i}\frac{1}{2}\left\|{y-\widetilde{D}{u_{i}}^{*}}\right\|_{2}^{2}+\frac{{{\lambda_{1}}}}{2}{M_{ii}}u_{i}^{T}{u_{i}}+u_{i}^{T}{h_{i}}+\lambda{\left\|{{u_{i}}}\right\|_{1}}$
(23)

where ${h_{i}}={\lambda_{1}}\left({\sum\limits_{j\neq
i}{{M_{ij}}{c_{j}}}}\right)$. Here, we need to look for the signs of the
coefficients $c_{i}^{j}$ of $c_{i}$. Once all the signs of $c_{i}^{j}$ are
determined, the Eq. 23 can be transformed into an unconstrained optimization
problem. In this paper, we define
$h\left({{u_{i}}}\right)=\frac{1}{2}\left\|{y-\widetilde{D}{u_{i}}}\right\|_{2}^{2}+\frac{{{\lambda_{1}}}}{2}{M_{ii}}u_{i}^{T}{u_{i}}+u_{i}^{T}{h_{i}}$
and $\nabla_{i}^{h}\left|{h\left({{u_{i}}}\right)}\right|=\frac{{\partial
h\left({{u_{i}}}\right)}}{{\partial u_{i}^{j}}}$

If $\widetilde{D}$ is large, the values in the calculated system vector will
be small. In order to solve this problem, we normalize each column of
$\widetilde{D}$ so that the $\ell 2$ norm value of each column is less than or
equal to 1. The normalized matrix set $\widetilde{D}$ is obtained by the Eq.
24.

$\widetilde{D}=\left\\{{\widetilde{D}\in{R^{m\times k}}~{}~{}s.t.~{}\forall
j=1,\cdots,k,d_{j}^{T}{d_{j}}\leq 1}\right\\}$ (24)

In principle, we calculate the minimum reconstruction error to determine the
classification label that the test sample belongs to. Specifically,
${\delta_{l}}:{\mathbb{R}^{N}}\to{\mathbb{R}^{N}}$ as a characteristic
function to select the coefficient related to class $l$. $u^{*}$ with all non-
zero coefficients are associated with the columns of a single category $l$ in
the dictionary matrix $\widetilde{D}$. Then, the calculation of the sparse
coefficient vector is realized by the Lars algorithm[49].

$\begin{array}[]{l}{r_{i}}\left({\overline{y}}\right)={\left\|{\overline{y}-{{\overline{y}}_{l}}}\right\|_{2}}\\\
~{}~{}~{}~{}~{}~{}~{}{\rm{=}}~{}{\left\|{\overline{y}-\widetilde{D}{\delta_{l}}\left({{u^{*}}}\right)}\right\|_{2}}\end{array}$
(25)

we approximate the test sample as
${{\it{{{\overline{y}}_{l}}}}}={\widetilde{D}{\delta_{l}}\left({{u^{*}}}\right)}$
and calculate the residuals between ${\overline{y}}$ and
${{{\overline{y}}_{l}}}$ to determine the category which the minimum
reconstruction error belongs to.

${\rm{class}}\left(\bar{y}\right)=\mathop{\arg\min}\limits_{i}{r_{i}}\left({\it{\bar{y}}}\right)$
(26)

The column vectors in $D$ corresponding to those zero entries in $u^{*}$ are
excluded to form a optimized feature set $\tilde{D}$ that is of lower
dimensionality. The given $\lambda$ and $\lambda_{1}$ determine the sparsity
degree of $u^{*}$, correspondingly the selection of CSP features.

For the new test sample, the corresponding subband feature vector
${D_{l,{N_{l}}}}\in{R^{2M\times N}}$ is estimated by CSP. According to the
sparse vector $u^{*}$, the optimal feature is selected, and then MI
classification is determined by the minimum redundancy error.

Besides, we summarize the process of the algorithm of the SGFB optimization
and the classification process in Algorithm $1$ and Algorithm $3$
respectively.

##  3 Experimental datasets and evaluation

###  3.1 EEG data description

In this paper, we use the BCI competition III dataset IVa that is publicly
available to evaluate the performance of the proposed algorithm. More details
can be found at http://www.bbci.de/competition. These datasets are effective
to evaluate the performance of the algorithms, which has been used in many
studies.

The BCI competition III dataset IVa is available in 1,000Hz version and 100Hz
version. In this experiment, we use the 100Hz version and 1-2s interval of
3.5s motor imagery EEG. The EEG signal is obtained from five healthy subjects
(aa, al, av, aw and ay). The subjects sit in a comfortable chair and perform
motor imagery experiments. The EEG signal is recorded by using 118 channels
and 140 trials for each class. The classes consist of the right hand and foot,
i.e., the EEG signal is provided with a total of 280 trials for each subject.

Instead of performing band-pass filtering on the original EEG segment, we
perform a set of sub-band filtering, selecting 9 sub-bands from the frequency
range of 4-40Hz, and the bandpass is 4Hz, i.e., 4-8Hz, 6-10Hz, … , 36-40Hz.
Subsequently, the CSP features can be extracted from the EEG segments of each
subband.

###  3.2 Experimental evaluation

Our experiment consists of five steps. Firstly, the EEG signal is divided by
the frequency range of 4-40Hz into 9 bands. This filter bank consists of a
bandpass filter. The range of each band is 4-8Hz, 8-12Hz, $\cdots$, 36-40Hz
respectively. Shin et al. explored the effectiveness of different CSP numbers
for MI classification tasks [42]. Secondly, we apply CSP to the signal derived
from the filter bank and the 32 spatial filters. As a next, the band power is
calculated in different filter banks. Thirdly, the dictionary learning with
$\ell 1$ minimization is employed in different filter bank. Finally, the label
for each data is assigned by using the ensemble.

The Classification performance is evaluated based on classification accuracy
(ACC), sensitivity (SEN), and specificity (SPE). These statistical measures
are defined as follow:

$ACC=\frac{TP+TN}{TP+FP+TN+FN}$ (27)

$SEN=\frac{TP}{TP+FN}$ (28)

$SPE=\frac{TN}{TN+FP}$ (29)

where TP, TN, FP and FN denote the true positive, true negative, false
positive and false negative, respectively. Thus, ACC measures the proportion
of subjects correctly classified among all subject, SEN and SPE correspond to
the proportions of left hand and right hand classified, respectively.

##  4 Results and discussion

In the past few years, researchers have developed a series of algorithms in
MI. In this paper, We proposed a complex model of SGFB to improve MI
classification accuracy. The relevant results are described in Table 3.
Besides, we performed a statistical comparison of the performance of (C3-C4),
CSP, FBCSP, and SGFB. Moreover, we used the 10-fold cross-validation. We
marked the average accuracy in the SGFB method in bold for each sample.
Besides, We will discuss the impact of different parameters on classification
accuracy in Section 4.1.

Table 2: Procedures required by each of the compared methods. Method | Procedures
---|---
(C3-C4)+SVM | Bandpass filtering + Variance calculation + SVM
CSP+SVM | Bandpass filtering + CSP + SVM with parameter selection
FBCSP+SVM | Multi-subband filtering + CSP + SVM with parameter selection
FBCSP+SRC | Multi-subband filtering + SRC
SGFB | Multi-subband filtering + CSP + Sparse learning with parameter selection

To comprehensively investigate the effectiveness of our proposed method on
training data reduction, we further tested the classification accuracy by
using different numbers of training samples from the target subject. In
particular, we randomly select 30%, 50%, 70% and 100% samples (TS) from each
target subject and evaluate the classification accuracy based on repeating the
above steps 10 times. The tab. 4, 5, 6 and 7 describe the average MI
classification accuracy obtained by SGFB for 30%, 50% 70%, and 100% samples,
respectively. The results show that even if only 30% of the samples can be
used for training to obtain 81.16% accuracy, this confirms the potential
prospect of our model in MI classification with only a few training samples.

Figure 3: Performance of different classification tasks. (a) Discriminant vectors $u^{*}$ and feature distributions of subject ay, derived by SGFB. The vertical axis of the upper and lower subgraphs represents the weight of the discrimination vector. Large weights indicate that the corresponding features are more important for MI classification. (b)The scatter plot of two category features. (c) Comparison of the recognition accuracy for different TS situations.  Table 3: Recognition accuracy (%) obtained by (C3-C4)+SVM, CSP+SVM (8-12Hz), CSP+SVM (4-40), FBCSP+SVM (multi-subband) and SGFB (multi-subband), respectively, at different time window (TW), for the BCI Competition IV dataset IVb. TW |  Method | Subject | Average
---|---|---|---
aa | al | av | aw | ay
1-2 (1s) |  | (C3-C4)+SVM (8-12 Hz)
---
54.62 | 57.36 | 57.82 | 59.31 | 54.47 | 56.71
| CSP+SVM (8-12 Hz)
---
75.52 | 79.1 | 73.67 | 76.33 | 71.05 | 75.13
| CSP+SVM (4-40 Hz)
---
79.9 | 84.4 | 89.2 | 80.61 | 80.7 | 84.96
| FBCSP+SVM (multi-subband)
---
55.91 | 79.58 | 63.25 | 73.53 | 78.56 | 70.17
| SGFB (multi-subband)
---
83.57 | 95.36 | 88.21 | 81.07 | 93.21 | 88.28
2-3 (1s) |  | (C3-C4)+SVM (8-12 Hz)
---
57.65 | 73.5 | 68.66 | 70.1 | 72.67 | 68.52
| CSP+SVM (8-12 Hz)
---
66.9 | 87.65 | 71.85 | 82.5 | 82 | 77.98
| CSP+SVM (4-40 Hz)
---
77.7 | 78.2 | 74.44 | 77.12 | 78.57 | 77.2
| FBCSP+SVM (multi-subband)
---
55.85 | 61.19 | 60.05 | 65.24 | 71.46 | 62.78
| SGFB (multi-subband)
---
74.64 | 91.07 | 77.86 | 84.64 | 92.86 | 84.21
Table 4: Classification results achieved by the SGFB for left and right EEG signals (30% TS). | evluation | Subject | Average |
---|---|---|---|---
aa | av | aw | ay | al |
| Acc | 73.57 | 81.43 | 84.02 | 78.21 | 88.57 | 81.16 |
Spec | 82.86 | 94.29 | 77.86 | 97.86 |  85 | 87.58 |
Sen | 64.29 | 68.57 | 90.71 | 58.57 | 92.14 | 74.86 |
Table 5: Classification results achieved by the SGFB for left and right EEG signals (50% TS). | evluation | Subject | Average |
---|---|---|---|---
aa | av | aw | ay | al |
| Acc | 75.36 |  79.29 | 71.43 | 79.29 | 89.29 |  79.83 |
Spec | 86.59 |  97.86 | 73.57 | 94.29 | 85.71 |  87.6 |
Sen |  70 |  60.71 | 69.29 | 64.29 | 92.86 |  71.43 |
Table 6: Classification results achieved by the SGFB for left and right EEG signals (70% TS). | evluation | Subject | Average |
---|---|---|---|---
aa | av | aw | ay | al |
|  Acc |  79.64 |  80 |  77.14 | 87.86 | 95.36 |  84 |
Spec |  69.14 |  90 |  57.86 | 99.29 | 97.86 |  82.83 |
Sen |  92.14 |  65 |  96.43 | 76.43 | 92.86 |  84.57 |
Table 7: Classification results achieved by the SGFB for left and right EEG signals (Full TS). | evluation | Subject | Average |
---|---|---|---|---
aa | av | aw | ay | al |
|  Acc |  83.57 |  88.21 |  81.07 | 93.21 | 95.36 |  88.28 |
Spec |  80.07 |  82.66 |  98.57 | 92.14 | 91.43 |  88.97 |
Sen |  86.43 |  83.57 |  63.57 | 94.29 | 99.29 |  85.43 |

Figure 4: (a). Classification accuracy with different values of parameters
$\lambda$ and $\lambda_{1}$ respectively. The results indicate that the
performance of our method slightly fluctuates when the values of
hyperparameters vary in a relatively large range. In most cases, the
classification accuracies are generally stable with respect to the changes of
the two hyperparameters, indicating the robustness of our proposed SGFB
algorithm to the parameter values. (b). Average computational time of
different methods including SGFB, FBCSP+SVM (multi-subband), (C3-C4)+SVM,
CSP+SVM (8-13Hz) and (C3-C4)+SVM respectively.

###  4.1 Parameter sensitivity

In order to evaluate the effectiveness of the proposed EEG classification
method and superior to the existing methods, five other different algorithms
are compared to the proposed algorithm using the same dataset.

The 10 Fold cross validation (10FCV) scheme is adopted for evaluation of
algorithm performance. In each fold of 10FCV procedure, an additional inner
10FCV is also carried out on the training data to determine the optimal
hyperparameters (i.e., $\lambda$ for SFBG and the soft-margin parameter C for
SVM). The selection ranges of $\lambda$ and $\lambda_{1}$ are
$[0.1,0.2,\cdots,0.9]$ and $\text{[}0.1,\cdots,0.4]$ respectively, while C is
selected from $[0.05,0.1,\cdots,1]$. The effectiveness of our proposed method
is affected by the selection of hyperparameters, i.e., $\lambda$ for weighted
group sparsity and $\lambda_{1}$ for the pattern similarity. In our
experiment, we implement a grid search to select the optimal parameter values
on the training data using inner 10FCV. To investigate the parameter
sensitivity of our proposed algorithm, we evaluate effects of varying values
of these two hyperparameters on classification accuracy using 10FCV with all
subjects. The best accuracy of 94.64% is achieved by using $\lambda$ = 0.3 for
strength weighted sparsity and $\lambda_{1}=0.1$ for similarity constraint.
Our future study will further validate performance of the proposed algorithm
on a completely independent dataset.

###  4.2 Computational efficiency

We also compare the computational efficiency of above-mentioned algorithms.
Fig. 4-b shows the calculation time evaluated in a MATLAB R2019a environment
on a desktop computer with a 3.8 Ghz CPU (i7-7700k, 16G RAM). It suggests that
these algorithms can be run on internal loop cross-validation, which is needed
for training models. We note that the FBCSP spent the longest calculation time
on MI classification. The proposed SGFB algorithm shows comparable
computational efficiency with the best accuracy and comparable computational
efficiency. At present, in order to explore more complex relationships in the
feature space, many subspace regularization algorithms have appeared for
various applications including imaging processing and encephalopathy diagnosis
[50, 51, 52, 53, 54, 55, 56, 57]. A popular subspace learning method provides
an effective method for characterizing feature distributions and has recently
been used to improve the correlation of a single event latent classification
[58, 59]. This kind of relationship constraint based on manifold learning can
further improve the accuracy of classification, which is worthy of our future
research [60, 61, 62].

##  5 Conclusion

In this paper, we introduce a novel SGFB algorithm to construct an efficient
classification model in motor imagery-based BCI applications. We construct a
compound matrix dictionary for the frequency bands of different rhythms. By
combining $L_{2}$-norm and $L_{1}$-norm, a sparse group representation model
is designed to control the sparseness between frequency bands to automatically
determine the best model. With the public motor imagery-related EEG datasets,
extensive experimental comparisons are carried out between the proposed
algorithm and two other state-of-the-art methods. The results illustrate the
effectiveness of our method in motor imagery BCI applications.

##  6 Conflict of interest

We declare that we do not have any commercial or associative interest that
represents a conflict of interest in connection with the work submitted.

##  7 CRediT author statement

Cancheng Li: Conceptualization, Methodology, Software, Formal analysis,
Investigation, Writing, Original Draft, Visualization. Chuanbo Qin:
Investigation, Visualization, Writing Review. Cancheng Li: Formal analysis,
Investigation, Visualization, Writing Review. Jing Fang: Writing Review,
Resources, Data Curation, Supervision.

##  8 Acknowledgments

The authors would like to thank the editors and anonymous reviewers for their
valuable suggestions and constructive comments, which have really helped the
authors improve very much the presentation and quality of this paper. The
author would like to acknowledge and thank the organizers of the BCI
Competitions III.

##  9 appendix

Algorithm 1 Pseudo-code Implementation of the Sparse Group Filter Bank
Representation (SGFB) Model for Motor Imagery EEG Classification.

0:
$\mathbf{\tilde{D}=\left[{{\tilde{D}_{f1}},{\tilde{D}_{f2}},\cdots,{\tilde{D}_{f9}}}\right]}$,
hyperparameter $\lambda$ and $\lambda_{1}$.

0: The optimal sparse matrix
${u^{*}}={u^{(L+1)}}=\left[{u_{1}^{(L+1)},u_{2}^{(L+1)},\cdots,u_{n}^{(L+1)}}\right]$

1: Initialize the sparse coefficients and its corresponding sign set;

2: for $i\leftarrow 1,\cdots,L$ do

3: for $i\leftarrow 1,\cdots,T$ do

4: if $the~{}coefficient~{}c_{i}^{(n)}=0$ then

5: choose
$j=\mathop{\arg\max}\limits_{i}\left|{\frac{{\partial{{\left\|{y-\widetilde{D}{u^{*}}}\right\|}^{2}}}}{{\partial
u_{j}^{*}}}}\right|$;

6: if
$\frac{{\partial{{\left\|{y-\widetilde{D}{u^{*}}}\right\|}^{2}}}}{{\partial
u_{j}^{*}}}>\lambda$ then

7: set $c_{i}^{j(n)}=-1$;

8: else if
$\frac{{\partial{{\left\|{y-\widetilde{D}{u^{*}}}\right\|}^{2}}}}{{\partial
u_{j}^{*}}}<\lambda$ then

9: set $c_{i}^{j(n)}=1$;

10: end if

11: end if

12: end for

13: Set feature sign search step according to Algorithm 2

14: end for

15: return coefficient $s$.

Algorithm 2 Set feature sign search step.

1: (a) $\widehat{{u^{*}}}$ is a sub-column vector corresponding to the active
set in the complete atomic dictionary ${u^{*}}$

2: (b) Calculate the solution of Eq. 11
$\mathop{\arg\min}\limits_{{{\widehat{u}}_{i}}}\left\|{y-\widetilde{D}{u^{*}}}\right\|+\lambda\widehat{\theta}{\widehat{u}_{i}}$;
$\widehat{u}_{i}^{\left(n\right)}={\left({{{\widehat{u}}^{{*^{T}}}}T{{\widehat{u}}^{*}}}\right)^{-1}}\left({{{\widehat{u}}^{{*^{T}}}}y-\frac{{\lambda\widehat{\theta}}}{2}}\right)$;

3: (c) Perform a discrete linear search on the close line segment from
$\widehat{u}$ to ${\widehat{u}_{new}}$

4: (1) Check the target function value of ${\widehat{u}_{new}}$ and the point
in which all coefficient;

5: (2) Update the value of $\widehat{u}$ to the point with the smallest
objective function value.

6: (d) remove the coefficient of zero and update $\theta={\mathop{\rm
sgn}}\left({\widehat{u}_{i}^{*\left(n\right)}}\right)$ in the active set;

7: if $\forall~{}u_{j}^{*}\neq 0$ (for non-zero coefficients) then

8: $\frac{{\partial{{\left\|{y-\widetilde{D}{u^{*}}}\right\|}^{2}}}}{{\partial
u_{j}^{*}}}+\lambda sign\left({u_{j}^{*}}\right)=0$;

9: end if

10: if $\forall~{}u_{j}^{*}=0$ (for zero coefficients ) then

11: $\left|{\frac{{\partial\left\|{y-\widetilde{D}{u^{*}}}\right\|}}{{\partial
u_{j}^{*}}}}\right|<\lambda$;

12: else

13: continue feature-sign search, go to step (b);

14: end if

Algorithm 3 SGFB for the EEG Classification.

0: coefficient s and the test sample signal $\mathbf{y}\in{\mathbb{R}^{m\times
1}}$.

0:
$class(\overline{y})=\mathop{\arg\min}\limits_{i}{r_{i}}\left({\overline{y}}\right)$.

1: Normalize $\tilde{D}$ and $\overline{y}$ according to Eq.(24)

2: The optimal representation vector $\mathbf{u^{*}}$ according to Algorithm 1

3: Compute
${\rm{class}}\left(\bar{y}\right)=\mathop{\arg\min}\limits_{i}{r_{i}}\left(\bar{y}\right)$

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