Manifold learning techniques for dynamical systems and time series have shown their utility for a broad spectrum of applications in recent years. While these methods are effective at learning a low-dimensional representation, they are often insufficient for visualizing the global and local structure of the data. In this paper, we present DIG (Dynamical Information Geometry), a visualization method for multivariate time series data that extracts an information geometry from a diffusion framework. Specifically, we implement a novel group of distances in the context of diffusion operators, which may be useful to reveal structure in the data that may not be accessible by the commonly used diffusion distances. Finally, we present a case study applying our visualization tool to EEG data to visualize sleep stages.
Dimensionality reduction in multivariate time series analysis has broad applications, ranging from financial data analysis to biomedical research. However, high levels of ambient noise and various interferences result in nonstationary signals, which may lead to inefficient performance of conventional methods. In this paper, we propose a nonlinear dimensionality reduction framework using diffusion maps on a learned statistical manifold, which gives rise to the construction of a low-dimensional representation of the high-dimensional nonstationary time series. We show that diffusion maps, with affinity kernels based on the Kullback-Leibler divergence between the local statistics of samples, allow for efficient approximation of pairwise geodesic distances. To construct the statistical manifold, we estimate time-evolving parametric distributions by designing a family of Bayesian generative models. The proposed framework can be applied to problems in which the time-evolving distributions (of temporally localized data), rather than the samples themselves, are driven by a low-dimensional underlying process. We provide efficient parameter estimation and dimensionality reduction methodologies, and apply them to two applications: music analysis and epileptic-seizure prediction. Author-HighlightsWe build a class of Bayesian models to learn the evolving statistics of time series.We construct diffusion maps based on the time-evolving distributional information.The proposed method recovers the underlying process controlling the time series.The proposed framework is applied to the analysis of music and icEEG recordings. In this paper, we present a method for time series analysis based on empirical intrinsic geometry (EIG). EIG enables one to reveal the low-dimensional parametric manifold as well as to infer the underlying dynamics of high-dimensional time series. By incorporating concepts of information geometry, this method extends existing geometric analysis tools to support stochastic settings and parametrizes the geometry of empirical distributions. However, the statistical models are not required as priors; hence, EIG may be applied to a wide range of real signals without existing definitive models. We show that the inferred model is noise-resilient and invariant under different observation and instrumental modalities. In addition, we show that it can be extended efficiently to newly acquired measurements in a sequential manner. These two advantages enable us to revisit the Bayesian approach and incorporate empirical dynamics and intrinsic geometry into a nonlinear filtering framework. We show applications to nonlinear and non-Gaussian tracking problems as well as to acoustic signal localization. This paper presents a data-driven approach for analyzing multivariate time series. It relies on the hypothesis that highdimensional data often lie on a low-dimensional manifold whose geometry may be revealed using manifold learning techniques. We define a notion of distance between multivariate time series and use it to determine a low-dimensional embedding capable of describing the statistics of the signals at hand using just a few parameters. We illustrate our method on two simulated examples and two real datasets containing electroencephalographic recordings (EEG). Abstract 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.