Delay Differential Analysis

Delay Differential Analysis (DDA) is a nonlinear time-series analysis framework that models the dynamics of a system using delay differential models. Instead of relying on statistical or spectral features, DDA assumes that the current rate of change of a measured signal depends on its present value and on its past values with typically two time delays. By fitting a compact delay differential model directly to the data, DDA extracts dynamical signatures that characterize the underlying system.

A major advantage of DDA is that it performs well with relatively short, noisy datasets while capturing both linear and nonlinear interactions. The resulting model parameters provide interpretable features that can be used for classification, detection, and causal inference. Originally developed from concepts in nonlinear dynamics and chaos theory, DDA has been successfully applied to (i)EEG, ECG, and movement data for applications such as seizure charactrization and prediction, Parkinson’s disease characterization, functional brain connectivity, and the analysis of information flow in complex biological networks.

Single time series analysis (ST-DDA)

By using a simple approximated delay differential model

it was shown that, although the model contains too few monomials and cannot be integrated to reproduce the dynamics underlying the data, its coefficients, obtained using a least squares method, reflect these dynamics sufficiently well to allow for classification. This has been applied to numerous cases [1] [2]
Let us consider the case where the simple delay differential equation

is used for classifying different time windows extracted from a single time series \(\{ x(t) \}_{t = 0}^T\). Since the derivative is computed at time \(t\), the corresponding embedding is spanned by \((x, x_{\tau_1}, x_{\tau_2})\). This is, for instance, particularly useful for detecting epileptic seizures from EEG [3]. For each time window, the derivatives \(\{ \dot{x}_{\rm d}(t) \}\) are computed from the measured data ; then coefficients \(a_i\)s are determined by solving