with weights W, using St i = W (W T W) -1 W T S ivec, where St i denotes the geodesic filtered SPD matrices from S i . These filtered matrices St i are then projected back onto the manifold using specific narrow frequency band. Since the alpha rhythm (8-12 Hz) is known to vary according to both workload [4] and emotions [28], we applied CSP after bandpass filtering in 8-12 Hz. FBCSPs The FBCSP is an algorithm that optimizes both spatial and spectral filters. To do so, it first filters EEG signals into multiple frequency bands using a filter bank. Here, we used nine bandpass filters in 4-Hz-wide bands (in 4-8 Hz, 8-12 Hz, ..., 36-40 Hz) as in [13]. Then, for each bandpassed signal, CSP optimizes two spatial filter pairs. From the resulting 36 features (nine bands multiplied by four CSP filters per band), the four most relevant were selected using minimal redundancy maximal relevance (mRMR) [36] and used as input to an LDA. The FBCSP algorithm proved its efficiency when winning the Fifth International BCI competition [13]. t i are the filtered SPD matrices projected onto where C the manifold and exp m ^M h denotes the exponential of matrix M. Finally, this approach uses a minimum distance to the mean classifier to classify testing geot i . To do so, during the traindesic filtered matrices C ing step, the class centroids G k of each class k are computed by averaging the geodesic filtered covarit ik from each class k. During testing, ance matrices C the Riemannian distances between the testing geodet j and each class centroid G k are sic filtered matrix C t j is assigned first calculated by using (2). The matrix C k class label for which the centroid G k is the nearest. In our study, FgMDM was applied on EEG bandpass filtered in 8-12 Hz, as for the CSP. ◆◆ TSC: TSC first projects all training SPD matrices C i onto the tangent space at point G (the mean of all training matrices). Then, it uses any classifier such as LDA, SVM, or LR on the vectorized upper-triangular elements of the projected matrices [39]. We used LR with L2 regularization (with the default C = 1.0 in scikit-learn [40]). For FgMDM, TSC used data filtered in 8-12 Hz. Riemannian Geometry Riemannian approaches represent EEG trials as covariance matrices, which are symmetric positive definite (SPD) matrices, and manipulate them with an appropriate geometry, the Riemannian geometry [32], [37]. Classifiers based on such geometry are called RGCs. First, in a Riemannian manifold we can estimate intrinsic non-Euclidean distances between two SPD matrices, i.e., two points (here, C 1, and C 2 ), using the Riemannian distance d 2 ^C 1, C 2 h = | log 2 m n ^C -1 1 C 2 h, (2) n where m n (M) is the nth eigenvalue of matrix M. The set of tangent vectors to point G on the manifold defines the manifold tangent space at G. More generally, any SPD matrix C i can be projected onto the tangent space at point G using S i = Log G ^C i h = G 1/2 log m ^G -1/2 C i G -1/2 h G 1/2, (3) where S i is the projection of C i on the tangent plane, and log m ^ · h denotes the logarithm of a matrix. In this article, we study two existing RGCs, the mean classifier (FgMDM) and the TSC, and introduce two new ones, the FBTSC and the FBFgMDM. Existing Methods ◆◆ FGMDM: FgMDM [38] projects training matrices C i onto the tangent space at point G (the mean of all training data) using (3) to obtain matrices S i . Then, a Fisher geodesic filter is obtained by optimizing an LDA classifier on S ivec, the vectorized upper-triangular elements of S i, to discriminate classes using such vectors. This results in a matrix of weights W = LDA ^S ivec h . The projected SPD matrices S i from both the training and the testing sets are then filtered t i = Exp C ^St i h = G 1/2 exp m ^ G -1/2 St i G -1/2 h G 1/2, (4) C New Methods ◆◆ FBFgMDM: Contrary to FgMDM, which exploits EEG signals in a single frequency band, FBFgMDM applies FgMDM in multiple bands separately and combines the resulting distances to exploit additional spectral information. This approach should possibly improve classification performances, as FBCSP did to improve CSP. To do so, FBFgMDM first filters EEG signals in multiple bands using a filter bank, as for FBCSP. Here, we used the same bands as the FBCSP, i.e., 4-8 Hz, 8-12 Hz, ..., 36-40 Hz. Then, for the EEG signals in each frequency band j, this method first uses a regular FgMDM, i.e., it computes the Riemannian distances t ij i between a geodesic filtered SPD matrix d 2 _ G kj, C t ij and each class centroid G kj . Then, from all nine C bands j, the four that are most useful for classification are selected with mRMR feature selection [36] on the t ij i used as features, on Riemannian distances d 2 _ G kj, C the training set. For testing, we compute the squared Riemannian distances for the four bands selected using mRMR only and sum them as t i i = |d 2 _ G kj, C t ij i, (5) c 2 _ G k, C j!X where Ω is the set of frequency bands selected with t i i to mRMR. We thus obtain k new distances c 2 _ G k, C Ju ly 2020 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 33

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