Intrinsic modeling of stochastic dynamical systems using empirical geometry

In a broad range of natural and real-world dynamical systems, measured signals are controlled by underlying processes or drivers. As a result, these signals exhibit highly redundant representations, while their temporal evolution can often be compactly described by dynamical processes on a low-dimensional manifold. In this paper, we propose a graph-based method for revealing the low-dimensional manifold and inferring the processes. This method provides intrinsic models for measured signals, which are noise resilient and invariant under different random measurements and instrumental modalities. Such intrinsic models may enable mathematical calibration of complex measurements and build an empirical geometry driven by the observations, which is especially suitable for applications without a priori knowledge of models and solutions. We exploit the temporal dynamics and natural small perturbations of the signals to explore the local tangent spaces of the low-dimensional manifold of empirical probability densities. This information is used to define an intrinsic Riemannian metric, which in turn gives rise to the construction of a graph that represents the desired low-dimensional manifold. Such a construction is equivalent to an inverse problem, which is formulated as a nonlinear differential equation and is solved empirically through eigenvectors of an appropriate Laplace operator. We examine our method on two nonlinear filtering applications: a nonlinear and non-Gaussian tracking problem as well as a non-stationary hidden Markov chain scheme. The experimental results demonstrate the power of our theory by extracting the underlying processes, which were measured through different nonlinear instrumental conditions, in an entirely data-driven nonparametric way.