Geodesic Paths for Time-Dependent Covariance Matrices in a Riemannian Manifold

Time-dependent covariance matrices are important in remote sensing and hyperspectral detection theory. The difficulty is that C(t) is usually available only at two endpoints C(t(0)) = A and C(t(1)) = B where C(t(0) < t < t(1)) is needed. We present the Riemannian manifold of positive definite symmetric matrices as a framework for predicting a geodesic time-dependent covariance matrix. The geodesic path A -> B is the shortest and most efficient path (minimum energy). Although there is no guarantee that data will necessarily follow a geodesic path, the predicted geodesic C(t) is of value as a concept. The path for the inverse covariance is also geodesic and is easily computed. We present an interpretation of C(t) with coloring and whitening operators to be a sum of scaled, stretched, contracted, and rotated ellipses.