Delay Differential Analysis of Time Series

Abstract

Nonlinear dynamical system analysis based on embedding theory has been used for modeling and prediction, but it also has applications to signal detection and classification of time series. An embedding creates a multidimensional geometrical object from a single time series. Traditionally either delay or derivative embeddings have been used. The delay embedding is composed of delayed versions of the signal, and the derivative embedding is composed of successive derivatives of the signal. The delay embedding has been extended to nonuniform embeddings to take multiple timescales into account. Both embeddings provide information on the underlying dynamical system without having direct access to all the system variables. Delay differential analysis is based on functional embeddings, a combination of the derivative embedding with nonuniform delay embeddings. Small delay differential equation (DDE) models that best represent relevant dynamic features of time series data are selected from a pool of candidate models for detection or classification. We show that the properties of DDEs support spectral analysis in the time domain where nonlinear correlation functions are used to detect frequencies, frequency and phase couplings, and bispectra. These can be efficiently computed with short time windows and are robust to noise. For frequency analysis, this framework is a multivariate extension of discrete Fourier transform (DFT), and for higher-order spectra, it is a linear and multivariate alternative to multidimensional fast Fourier transform of multidimensional correlations. This method can be applied to short or sparse time series and can be extended to cross-trial and cross-channel spectra if multiple short data segments of the same experiment are available. Together, this time-domain toolbox provides higher temporal resolution, increased frequency and phase coupling information, and it allows an easy and straightforward implementation of higher-order spectra across time compared with frequency-based methods such as the DFT and cross-spectral analysis.

Multivariate cross-frequency coupling via generalized eigendecomposition

“…In practice, more than one delay embedding is useful. Time-delay embedding was developed in dynamical systems analysis, and has been successfully applied to psychometric and neural data (Brunton et al, 2016; de Cheveigné and Simon, 2007; Lainscsek and Sejnowski, 2015; Tome et al, 2004; von Oertzen and Boker, 2010). After creating this time-delay embedded data matrix, Method 4 proceeds similarly as Method 1: two covariance matrices are computed, one from data surrounding troughs and one from the entire time series, and GED is applied to those two matrices.…”

Derivative Delay Embedding

“…In addition, the reconstruction is performed in an incremental fashion, facilitating online processing, because only recent observations are considered when reconstructing the dynamical system from a streaming time series. The reconstructed dynamical system is usually represented in a higher dimensional space, in which the dynamics presents recursive patterns [8,18,9] regardless of the length of the original time series. Therefore, an infinite streaming time series can potentially be stored in a finite memory through the delay embedding because of the recursiveness of the reconstructed dynamical system.…”

“…Then, the label of a testing time series could be decided by comparing with those learned trajectories. Many existing works related to delay embedding would model the trajectories by a group of differential functions, parametric models [9], or topological features [18], e.g., barcodes from persistent homology. However, they all perform in an offline manner, and it is difficult to find a parametric model that is suitable for all applications.…”

Delay Differential Analysis of Electroencephalographic Data

“…In a companion paper (Lainscsek & Sejnowski, 2015), we make the connection between nonlinear dynamics and spectral analysis using functional embeddings or delay differential equations (DDE) of time-series data. This idea was introduced by Lainscsek and Sejnowski (2013).…”

“…Considering a signal $$xfalse(tfalse)=𝒮+𝒫;𝒫=\text{cos}false(\mathrm{normal\Omega }t+\varphi false).$$ where 𝒮 is the signal under investigation and 𝒫 is a probing signal, we demonstrate that the function L (Ω) for $𝒮={\sum }_{i=1}^{\infty }{A}_i\text{cos}false({\omega }_{j}t+{\phi }_{i}false)$, where $$Lfalse(\mathrm{normal\Omega }false)=\underset{\varphi }{\text{max}}true(false\langle x^{2}false\rangle -false\langle {𝒮}^{2}false\rangle -\frac{1}{2}true),$$ can be used as a frequency detector, similar to the Goertzel algorithm (Goertzel, 1958; Jacobsen & Lyons, 2003). In the companion paper (Lainscsek & Sejnowski, 2015), we also give the theoretical basis for a novel time-domain bispectrum (TDB) B (Ω) for $𝒮={\sum }_{i=1}^{\infty }{A}_i\text{cos}false({\omega }_{j}t+{\phi }_{i}false)+A_{j,k}\text{cos}false(false({\omega }_j+{\omega }_{k}false)t+{\phi }_j+{\phi }_{k}false)$ where $$Bfalse(\mathrm{normal\Omega }false)=Lfalse(\mathrm{normal\Omega }false)·\underset{\varphi }{\text{max}}false(false\langle x^{3}false\rangle -false\langle {𝒮}^{3}false\rangle false).$$…”