Smooth Interpolation of Covariance Matrices and Brain Network Estimation.

Area of Science:

• Statistics
• Machine Learning
• Signal Processing

Background:

• Nonstationary time series exhibit changing statistical properties over time.
• Tracking time-varying covariance matrices is crucial for analyzing such data.
• Existing methods may not fully capture complex covariance dynamics.

Purpose of the Study:

• To develop an approach for tracking time-varying covariance matrices of nonstationary time series.
• To investigate covariance paths derived from Riemannian geometry and quadratic regularizations.
• To introduce a novel covariance path type steered by system matrices with rotating eigenspaces.

Main Methods:

• Utilizing state covariance of autonomous linear systems.
• Applying concepts from Riemannian geometry to define covariance paths.
• Investigating geodesics based on Hellinger-Bures and Fisher-Rao metrics.
• Introducing a new covariance path type using rotating eigenspaces.

Main Results:

• Three distinct types of covariance paths were explored and compared.
• A weighted-OMT interpretation of the Fisher-Rao metric was established.
• The proposed methods were validated using synthetic and functional magnetic resonance imaging (fMRI) data.

Conclusions:

• The study presents a robust framework for analyzing time-varying covariance in nonstationary data.
• The novel covariance path type offers new possibilities for dynamic system analysis.
• The findings have implications for fields utilizing time series analysis, including neuroimaging.