The standard MSI methodThe standard MSI method estimates the synchronization between the EEG signal and the reference signal as a potential indicator to identify the stimulus frequency. Since the SSVEP signal spectrum not only shows maximum amplitude at the fundamental frequency of the stimulus, but also local peaks at the harmonics, the harmonic components of the stimulus signal are incorporated in the reference signal.Let \(X\in {R}^{{N}_{c}\times M}\) denotes the multichannel EEG signal. \({N}_{c}\) represents the number of channels and \(M\) is the number of time samples. The reference signal \(Y\in {R}^{2{N}_{h}\times M}\) is defined as:$$Y={Y}_{n}=\left[\begin{array}{c}sin\left(2\pi {f}_{n}t\right)\\ cos\left(2\pi {f}_{n}t\right)\\ \vdots \\ sin\left(2\pi {N}_{h}{f}_{n}t\right)\\ cos\left(2\pi {N}_{h}{f}_{n}t\right)\end{array}\right],t=\frac{1}{{F}_{S}},\frac{2}{{F}_{S}},\cdots ,\frac{M}{{F}_{S}}$$
(1)
where \({N}_{h}\) is the number of harmonics, \({f}_{n}\) represents the \(n\) th stimulus frequency, and \({F}_{s}\) represents the sampling frequency. In this study, there are 40 stimulus frequencies, \({f}_{n}\in \left\{{f}_{1},{f}_{2},\ldots ,{f}_{40}\right\}\). Without loss of generality, the EEG signal \(X\) and the reference signal \(Y\) need to be normalized to have zero mean and unit variance. The joint covariance matrix of \(X\) and \(Y\) can be calculated as:$$C=\left[\begin{array}{c}{C}_{11}=\frac{1}{M}X{X}^{T}{C}_{12}=\frac{1}{M}X{Y}^{T}\\ {C}_{21}=\frac{1}{M}Y{X}^{T}{C}_{22}=\frac{1}{M}Y{Y}^{T}\end{array}\right]$$
(2)
To reduce the impact of the autocorrelation of \(X\) and \(Y\) on subsequent synchronization calculations, the following linear transformation is applied:$$Q=\left[\begin{array}{cc}{C}_{11}^{-\frac{1}{2}}& 0\\ 0& {C}_{22}^{-\frac{1}{2}}\end{array}\right]$$
(3)
Then, the new joint correlation matrix can be described as:$$R=QC{Q}^{T}=\left(\begin{array}{cc}{\text{I}}_{{N}_{c}}& {C}_{11}^{-\frac{1}{2}}{C}_{12}{C}_{22}^{-\frac{1}{2}}\\ {C}_{22}^{-\frac{1}{2}}{C}_{21}{C}_{11}^{-\frac{1}{2}}& {\text{I}}_{2{N}_{h}}\end{array}\right)$$
(4)
where \({\text{I}}_{{N}_{c}}\) and \({\text{I}}_{2{N}_{h}}\) are identity matrices.\({\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{p}\) are the eigenvalues of the matrix \(R\). The normalized eigenvalue \({\lambda }_{i}^{{{\prime}}}\) is calculated as follows:$${\lambda }_{i}^{{{\prime}}}=\frac{{\lambda }_{i}}{\sum_{i=1}^{P}{\lambda }_{i} }=\frac{{\lambda }_{i}}{tr\left(R\right)}$$
(5)
Let \(P={N}_{c}+2{N}_{h}\). The synchronization index, i.e., the normalized entropy between two multi-channel signals can be obtained as:$$S=1+\frac{\sum_{i=1}^{P}{\lambda }_{i}^{{{\prime}}}\text{log}\left({\lambda }_{i}^{{{\prime}}}\right)}{\text{log}\left(P\right)}$$
(6)
According to (6), we can calculate the multivariate synchronization index \({S}_{n}\) between the EEG signal \(X\) and the different reference frequency signals \({Y}_{n}\). Finally, the target frequency \({f}_{t}\) can be determined as the reference frequency corresponding to the maximum value of \({S}_{n}\).$${f}_{t}=\underset{n}{arg\text{max}} {S}_{n}, n=\text{1,2},\cdots ,40$$
(7)
Temporally local MSIIn the standard MSI method, the mutual synchronization of multichannel EEG signals is evaluated by calculating the eigenvalues and eigenvectors of the covariance matrix, as well as the normalized entropy, which has been shown to be an effective method for frequency identification in SSVEP-BCI systems. However, the standard MSI method ignores the temporal local information of EEG signals. To address this problem, the TMSI algorithm was proposed which has shown superior frequency recognition accuracy compared with standard MSI [20].The TMSI algorithm defines the adjacency matrix \(W\in {R}^{M\times M}\) and multiple signals \(Z=\left[{z}_{1},{z}_{2},\ldots ,{z}_{M}\right]\in {R}^{{N}_{c}\times M}\), respectively. \({N}_{c}\) denotes the number of variables or channels while \(M\) denotes the number of time samples. Then, the temporal local covariance matrix is expressed as follows:$${C}^{{{\prime}}}=\frac{1}{2M}\sum_{i=1}^{M}\sum_{j=1}^{M}{W}_{i,j}\left({z}_{i}-{z}_{j}\right){\left({z}_{i}-{z}_{j}\right)}^{T}$$
(8)
Equation (8) can be transformed into$$\begin{aligned}{C}^{{{\prime}}}&=\frac{1}{M}\sum_{i=1}^{M} {z}_{i}{z}_{i}^{T}\sum_{j=1}^{M} {W}_{i,j}-\sum_{i=1}^{M} \sum_{j=1}^{M} {W}_{i,j}{z}_{i}{z}_{j}^{T}\\ &=\frac{1}{M}\left(Z(D-W){Z}^{T}\right)\\ &=\frac{1}{M}ZL{Z}^{T}\end{aligned}$$
(9)
where \(D\) is the diagonal matrix, for \(i=1,2,\ldots ,M\),\({D}_{i,i}=\sum_{j=1}^{M} {W}_{i,j}\). \(L\) is the Laplacian matrix and\(L=D-W\). The adjacency matrix \(W\) can be generated by in a variety of ways. In this paper, it is determined by Tukey’s tricube weighting function [26]:$${W}_{i,j}=\left\{\begin{array}{c}{\left(1-{\left|\frac{i-j}{\tau }\right|}^{3}\right)}^{3},\left|\frac{i-j}{\tau }\right|<1\\ 0, else\end{array}\right.$$
(10)
Based on (2) and (9), a new temporal local covariance matrix can be calculated as$${C}^{{{\prime}}}=\left[\begin{array}{cc}\frac{1}{M}XL{X}^{T}& \frac{1}{M}XL{Y}^{T}\\ \frac{1}{M}YL{X}^{T}& \frac{1}{M}YL{Y}^{T}\end{array}\right]$$
(11)
When \({C}{\prime}\) is determined, we can use (2)–(6) to calculate the synchronization index, and use (7) to implement frequency identification.FBTMSITMSI benefits from exploiting the temporal local structure of EEG signals. However, it ignores the harmonic components of SSVEP, which have been shown to be informative for frequency identification [16, 27]. Filter bank approach has been successfully applied in previous studies to extract the information contained in harmonics [16, 19, 23]. Based on these ideas, we propose a novel FBTMSI method that takes into account both the temporal local structure and the harmonic components in SSVEP signal to improve frequency detection performance. The flowchart of FBTMSI algorithm is shown in Fig. 1.Fig. 1Flow chart of FBTMSI methodFirst, filter banks are applied on the original EEG signal to decompose the signal into multiple frequency subbands. According to the filter bank design method in [16], a zero-phase Chebyshev type I IIR filter is employed to extract each subband component (\(F{B}_{l},l=1,2,\cdots ,{N}_{sb}\), where \(l\) denotes the subband index) from the original EEG signal \(X\). After the filter bank analysis, the temporally local multivariate synchronization index between each subband component and the reference signal corresponding to each stimulus frequency is then calculated respectively.In this study, the bandwidth of the stimulation frequency (8–15.8 Hz) was 8 Hz. According to [16], the fundamental and harmonic components exhibit high signal-to-noise ratio (SNR) in the upper-frequency band from the stimulation frequency to about 90 Hz. Therefore, we chose the frequency range within [8 Hz, 88 Hz] (10 times the bandwidth of the stimulus frequency) as the filter bank. The frequency range is divided into \({N}_{sb}\) subbands, each with a frequency range of\(\left[\text{8,88}\right]\text{Hz}, \left[\text{16,88}\right]\text{Hz}, \left[l,88\right]\text{Hz }\ldots [{N}_{sb}\times 8,88\text{Hz}]\), as shown in Fig. 2. Then, TMSI is performed on each subband signal and the reference signal\({Y}_{n}\), to calculate the corresponding TMSI index\({S}_{n}^{l}\). Since the SNR of the harmonic component of the SSVEP signal decreases with increasing frequency [23], we performed a weighted sum of \({S}_{n}^{l}\) corresponding to each subband, and the weighting factor \({\omega }_{l}\) corresponding to each \({S}_{n}^{l}\) is shown in (12):$${\omega }_{l}={l}^{-a}+b$$
(12)
where \(a\) and \(b\) are constants. Then, the weighted sum \({\text{S}}_{n}\) of each subband’s TMSI is calculated as an indicator of FBTMSI, as shown in (13). Finally, the same (7) is used to implement frequency identification.Fig. 2Frequency range corresponding to each subband of the FBTMSI algorithm$${S}_{n}=\left[{S}_{n}^{1},{S}_{n}^{2},\cdots ,{S}_{n}^{Nsb}\right]\left[\begin{array}{c}{\omega }_{1}\\ {\omega }_{2}\\ \vdots \\ {\omega }_{Nsb}\end{array}\right]$$
(13)
Filter bank temporally local multivariate synchronization index for SSVEP-based BCI | BMC Bioinformatics
