Mohammad Shahin Mahanta

Separable Common Spatio-Spectral Patterns for Motor Imagery BCI Systems

Objective: Feature extraction is one of the most important steps in any brain-computer interface (BCI) system. In particular, spatio-spectral feature extraction for motor-imagery BCIs (MI-BCI) has been the focus of several works in the past decade. This paper proposes a novel method, called separable common spatio-spectral patterns (SCSSP), for extraction of discriminant spatio-spectral EEG features in MI-BCIs. Methods: Assuming a binary classification problem, SCSSP uses a heteroscedastic matrix-variate Gaussian model for the multiband EEG rhythms, and seeks the spatio-spectral features whose variance is maximized for one brain task and minimized for the other task. Therefore, SCSSP can be considered as a spatio-spectral generalization of the conventional common spatial patterns (CSP) algorithm. Results: The experimental results on two-class and multiclass motor-imagery data from publicly available BCI Competition datasets demonstrate that the proposed computationally efficient method competes closely with filter-bank CSP (FBCSP), and can even outperform the FBCSP if enough training data are available. Furthermore, SCSSP provides us with a simple measure for ranking the discriminant power of extracted spatio-spectral features, which is not possible in FBCSP. Conclusion: The matrix-variate Gaussian assumption allows the SCSSP method to jointly process the EEG data in both spatial and spectral domains. As a result, compared to the similar solutions in the literature such as FBCSP, the proposed SCSSP method requires significantly lower computations. Significance: The proposed computationally efficient spatio-spectral feature extractor is particularly suitable for applications in which the computational power is limited, such as emerging wearable mobile BCI systems.

A Bayes optimal matrix-variate LDA for extraction of spatio-spectral features from EEG signals

Classification of mental states from electroencephalogram (EEG) signals is used for many applications in areas such as brain-computer interfacing (BCI). When represented in the frequency domain, the multichannel EEG signal can be considered as a two-directional spatio-spectral data of high dimensionality. Extraction of salient features using feature extractors such as the commonly used linear discriminant analysis (LDA) is an essential step for the classification of these signals. However, multichannel EEG is naturally in matrix-variate format, while LDA and other traditional feature extractors are designed for vector-variate input. Consequently, these methods require a prior vectorization of the EEG signals, which ignores the inherent matrix-variate structure in the data and leads to high computational complexity. A matrix-variate formulation of LDA have previously been proposed. However, this heuristic formulation does not provide the Bayes optimality benefits of LDA. The current paper proposes a Bayes optimal matrix-variate formulation of LDA based on a matrix-variate model for the spatio-spectral EEG patterns. The proposed formulation also provides a strategy to select the most significant features among the different rows and columns.

Heteroscedastic linear feature extraction based on sufficiency conditions

Classification of high-dimensional data typically requires extraction of discriminant features. This paper proposes a linear feature extractor, called whitened linear sufficient statistic (WLSS), which is based on the sufficiency conditions for heteroscedastic Gaussian distributions. WLSS approximates, in the least squares sense, an operator providing a sufficient statistic. The proposed method retains covariance discriminance in heteroscedastic data, while it reduces to the commonly used linear discriminant analysis (LDA) in the homoscedastic case. Compared to similar heteroscedastic methods, WLSS imposes a low computational complexity, and is highly generalizable as confirmed by its consistent competence over various data sets.

Regularized LDA based on separable scatter matrices for classification of spatio-spectral EEG patterns

Linear discriminant analysis (LDA) is a commonly-used feature extraction technique. For matrix-variate data such as spatio-spectral electroencephalogram (EEG), matrix-variate LDA formulations have been proposed. Compared to the standard vector-variate LDA, these formulations assume a separable structure for the within-class and between-class scatter matrices; these structured parameters can be estimated more accurately with a limited number of training samples. However, separable scatters do not fit some data, resulting in aggravated performance for matrix-variate methods. This paper first proposes a common framework for the vector-variate LDA with non-separable scatters and our previously proposed solution with separable scatters. Then, a regularization of the non-separable scatter estimates toward the separable estimates is introduced. This novel regularized framework integrates vector-variate and matrix-variate approaches, and allows the estimated scatter matrices to adapt to the data characteristics. Experiments on data set V from BCI competition III demonstrate that the proposed framework achieves a considerable classification performance gain.

Linear feature extraction using sufficient statistic

The objective in feature extraction is to compress the data while maintaining the same Bayes classification error as on the original data. This objective is achieved by a sufficient statistic with the minimum dimension. This paper derives a non-iterative linear feature extractor that approximates the minimal-dimension linear sufficient statistic operator for the classification of Gaussian distributions. This new framework alleviates the bias of an existing similar formulation towards the parameters of a reference class. Moreover, it is a heteroscedastic extension of linear discriminant analysis and captures the discriminative information in the first and second central moments of the data. The proposed method can improve the performance of the similar feature extractors while imposing equal, or even lower, computational complexity.

Ranking 2DLDA features based on fisher discriminance

In classification of matrix-variate data, two-directional linear discriminant analysis (2DLDA) methods extract discriminant features while preserving and utilizing the matrix structure. These methods provide computational efficiency and improved performance in small sample size problems. Existing 2DLDA solutions produce a feature matrix which is commonly vectorized for processing by conventional vector-based classifiers. However, the vectorization step requires a one-dimensional ranking of features according to their discriminance power. We first demonstrate that independent column-wise and row-wise ranking provided by 2DLDA is not sufficient for uniquely sorting the resulting features, and does not guarantee the selection of the most discriminant features. Then, we theoretically derive the desired global ranking score based on Fishers criterion. The current results focus on non-iterative solutions, but future extensions to iterative 2DLDA variants are possible. Face recognition experiments using images from the PIE data set are used to demonstrate the theoretically proved improvements over the existing solutions.

A heteroscedastic extension of LDA based on multi-class matusita affinity

Linear discriminant analysis (LDA), a conventional feature extraction technique, is a homoscedastic solution and ignores the second order information of the data. A heteroscedastic extension of LDA has been previously proposed which relies on the average pairwise Chernoff distances of the classes. However, in a multi-class scenario with number of classesC >; 2, the average of pairwise distances is not directly related to the classification error rate. Furthermore, the corresponding method imposes a high computational complexity of order O(C(C - 1)). This paper proposes an inherently multi-class heteroscedastic extension of LDA based on Matusitas separability measure, a multi-class generalization of the Chernoff distance which is related to multi-class error bounds. The proposed feature extractor can be trained non-iteratively with computational complexity of O(C). Experimental comparisons with the Chernoffmethod demonstrate both a performance improvement when estimated parameters are used, and a reduction of factor C - 1 in the computational load as predicted.

Separable common spatio-spectral pattern algorithm for classification of EEG signals

This paper proposes a novel method for extraction of discriminant spatio-spectral EEG features in motor imagery brain-computer interfaces. Considering a heteroscedastic binary classification setup, this method extracts the spatio-spectral features whose variance is maximized for one brain task and minimized for the other task. Therefore, our method can be considered as a spatio-spectral generalization of the conventional common spatial patterns (CSP) algorithm. In comparison to the similar solutions in the literature, such as filter-bank CSP (FBCSP) method, the proposed method benefits from joint processing of both spatial and spectral features, which improves the overall performance of the BCI while reducing its computational cost. Furthermore, our algorithm provides a simple measure that allows for ranking the discriminant power of extracted spatio-spectral features, which is not possible in FBCSP method. The experimental results demonstrate that the proposed method outperforms FBCSP for both raw EEG and preprocessed EEG data.

Statistical characterization of complex-valued EEG spectrum during mental imagery tasks

Electroencephalogram (EEG) recordings of brain activities can be processed in order to augment the brains cognitive, sensory, or motor functionality. A representative, yet analytically tractable, model is essential to EEG processing. Several studies have examined different statistical models for EEG power spectrum. But recent studies have shown that not only the power, but also the phase of the spectrum, carries relevant information on brain activities. As a result, this paper focuses on the complex-valued spectrum of EEG, and proposes a general non-circularly-symmetric multivariate Gaussian model for this spectrum. This simple model can encapsulate the information in both power and phase of the spectrum, and its validity for EEG data has been verified using standard statistical tests.

2DLDA as matrix-variate formulation of a separable 1DLDA

Necessary and sufficient conditions for separability of 1DLDA are derived.The commonly used separable scatter model is proved as a special case.Separability of the MVLDA operator is proved.1DLDA and 2DLDA solutions are theoretically related and compared.Zigzag sorting procedure is proposed for row & column-sorted 2DLDA features. Two-directional (2D) variants of the linear discriminant analysis (LDA) algorithm have been widely used to extract features of matrix-variate signals. This paper derives the theoretical relationship between 2DLDA and one-directional (1D) LDA based on the separable transformation framework. Separable transforms such as separable 2DDCT are widely used for image compression in the JPEG standard; therefore, a similar framework for 2DLDA provides the corresponding parallel foundation for separable image feature extraction. There are existing 2DLDA methods providing a separable transformation, however they are not directly related to the 1DLDA solution. We will derive a 2DLDA method as a matrix-variate representation of a separable 1DLDA operator. Furthermore, we derive the necessary and sufficient conditions for separability of 1DLDA. These conditions will be helpful to clarify both limitations and advantages of 2DLDA. Also, a 2DLDA framework in parallel to 2DDCT allows us to exploit related techniques developed for 2DDCT, such as the feature selection procedure.