Ning Lipeng
Department of Psychiatry, Brigham and Women's Hospital, Harvard Medical School, Boston, MA 02115 USA.
IEEE Trans Automat Contr. 2019 Aug;64(8):3184-3193. doi: 10.1109/tac.2018.2879597. Epub 2018 Nov 5.
We propose an approach to use the state covariance of autonomous linear systems to track time-varying covariance matrices of nonstationary time series. Following concepts from the Riemannian geometry, we investigate three types of covariance paths obtained by using different quadratic regularizations of system matrices. The first quadratic form induces the geodesics based on the Hellinger-Bures metric related to optimal mass transport (OMT) theory and quantum mechanics. The second type of quadratic form leads to the geodesics based on the Fisher-Rao metric from information geometry. In the process, we introduce a weighted-OMT interpretation of the Fisher-Rao metric for multivariate Gaussian distributions. A main contribution of this work is the introduction of the third type of covariance paths, which are steered by system matrices with rotating eigenspaces. The three types of covariance paths are compared using two examples with synthetic data and real data from resting-state functional magnetic resonance imaging, respectively.
我们提出一种利用自主线性系统的状态协方差来跟踪非平稳时间序列时变协方差矩阵的方法。遵循黎曼几何的概念，我们研究了通过对系统矩阵使用不同二次正则化得到的三种协方差路径。第一种二次形式基于与最优质量传输（OMT）理论和量子力学相关的Hellinger-Bures度量诱导出测地线。第二种二次形式基于信息几何中的Fisher-Rao度量产生测地线。在此过程中，我们引入了多元高斯分布的Fisher-Rao度量的加权OMT解释。这项工作的一个主要贡献是引入了第三种协方差路径，其由具有旋转特征空间的系统矩阵引导。分别使用两个包含合成数据和来自静息态功能磁共振成像的真实数据的例子对这三种协方差路径进行了比较。
