Objective: Electroencephalography signals are recorded as a multidimensional dataset. We propose a new framework based on the augmented covariance extracted from an autoregressive model to improve motor imagery classification. Methods: From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to extend the standard approach using these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. Results: The augmented covariance matrix performed noticeably better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. Conclusion: The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information, incorporating nonlinear components of the signal through an embedding procedure, which allows the leveraging of dynamical systems algorithms. Significance: These results extend the concepts and the results of the Riemannian distance based classification algorithm.

翻译：目的：脑电图信号被记录为多维数据集。我们提出一种基于自回归模型提取的增广协方差的新框架，以改进运动想象分类。方法：由自回归模型可推导出尤尔-沃克方程，该方程揭示了一个对称正定矩阵的出现：即增广协方差矩阵。当前对协方差矩阵进行分类的最先进技术基于黎曼几何。因此，一个相当自然的想法是扩展使用增广协方差矩阵的标准方法。构建增广协方差矩阵的方法与Takens针对动力系统提出的延迟嵌入定理展现了天然联系。这种嵌入方法基于两个参数的确定：延迟时间和嵌入维数，分别对应于自回归模型的滞后期和阶数。该途径除标准网格搜索外，还提供了计算超参数的新方法。结果：增广协方差矩阵的性能显著优于任何现有最先进方法。我们将使用MOABB框架，在多个数据集和多名受试者上，分别通过会话内和跨会话评估来测试我们的方法。结论：结果的改进归因于增广协方差矩阵不仅包含空间信息，还通过嵌入过程整合了时间信息及信号的非线性成分，这有助于利用动力系统算法。意义：这些结果扩展了基于黎曼距离分类算法的概念与成果。