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.
 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). Recently, emotion classification from EEG data has attracted much attention with the rapid development of dry electrode techniques, machine learning algorithms, and various real-world applications of brain-computer interface for normal people. Until now, however, researchers had little understanding of the details of relationship between different emotional states and various EEG features. To improve the accuracy of EEG-based emotion classification and visualize the changes of emotional states with time, this paper systematically compares three kinds of existing EEG features for emotion classification, introduces an efficient feature smoothing method for removing the noise unrelated to emotion task, and proposes a simple approach to tracking the trajectory of emotion changes with manifold learning. To examine the effectiveness of these methods introduced in this paper, we design a movie induction experiment that spontaneously leads subjects to real emotional states and collect an EEG data set of six subjects. From experimental results on our EEG data set, we found that (a) power spectrum feature is superior to other two kinds of features; (b) a linear dynamic system based feature smoothing method can significantly improve emotion classification accuracy; and (c) the trajectory of emotion changes can be visualized by reducing subject-independent features with manifold learning. In this paper, two modern adaptive signal processing techniques, empirical intrinsic geometry and synchrosqueezing transform, are applied to quantify different dynamical features of the respiratory and electroencephalographic signals. We show that the proposed features are theoretically rigorously supported, as well as capture the sleep information hidden inside the signals. The features are used as input to multiclass support vector machines with the radial basis function to automatically classify sleep stages. The effectiveness of the classification based on the proposed features is shown to be comparable to human expert classification—the proposed classification of awake, REM, N1, N2, and N3 sleeping stages based on the respiratory signal (resp. respiratory and EEG signals) has the overall accuracy @math (resp. @math ) in the relatively normal subject group. In addition, by examining the combination of the respiratory signal with the electroencephalographic signal, we conclude that the respiratory signal consists of ample sleep information, which supplements to the information stored in the electroencephalographic signal. We study the inference of latent intrinsic variables of dynamical systems from output signal measurements. The primary focus is the construction of an intrinsic distance between signal measurements, which is independent of the measurement device. This distance enables us to infer the latent intrinsic variables through the solution of an eigenvector problem with a Laplace operator based on a kernel. The signal geometry and its dynamics are represented with nonlinear observers. An analysis of the properties of the observers that allow for accurate recovery of the latent variables is given, and a way to test whether these properties are satisfied from the measurements is proposed. Scattering and window Fourier transform observers are compared. Applications are shown on simulated data, and on real intracranial Electroencephalography (EEG) signals of epileptic patients recorded prior to seizures.