Time Series Analysis

Time series analysis based on ordinal pattern statistics

Previous work by Rosso et al. [1] showed that time series from low dimensional chaotic systems can be distinguished from stochastic processes (“noise”) by their location in the complexity-entropy plane spanned by the normalized entropy \( H_S [P] = S[P] / S_{max} \) given by the Shannon entropy \( S[P] = -\sum_{j=1}^{m!} p_j \log p_j \) divided by its maximal value \( S_{max} = 1 / \log(m!) \) and the statistical complexity \( C_{JS} [P] = H_S \cdot Q_{JS} [P,P_e] \), where \( P= \{ p_j \} \) denotes the distribution of ordinal patterns of length \( m \) and

\( Q_{JS}[P,P_e] = D_{JS}[P,P_e] / D_{JS,max} \)

stands for the normalized Jensen-Shannon divergence

\( D_{JS}[P,P_e] = S[\frac{P+P_e}{2}] - \frac{S[P]}{2} - \frac{S[P_e]}{2} \)

quantifying the difference between $P$ and the uniform distribution $P_e$.

To address the question whether this distinction is also possible for high-dimensional chaos we applied this approach to time series generated by high-dimensional systems with Kaplan-Yorke dimensions $\Delta^{(KY)}$ in the range from 1 to 50 and found that dense sampling of continuous signals and too few data points may lead to non-uniform ordinal pattern distributions and spuriously low entropies and high complexities [2]. In particular, phase randomized surrogate data show the same path in the CE-plane as the original data upon lag variation (variation of sampling time), as illustrated for the Mackey-Glass delay differential equation

$\dot x(t) = \beta \frac{x(t-\tau)}{1 + x(t-\tau)^\nu} - \gamma x(t)$

and the one-dimensional Kuramoto–Sivashinsky equation [5]

$\partial_t u(x,t) = -\frac{1}{2} \nabla\left[ u^2(x,t)\right] - \nabla^2 u(x,t) - \nabla^4 u(x,t)$

with periodic boundary conditions.

Predicting chaotic time evolution using reservoir computing

Reservoir computing exploits the response of a dynamical system driven by a given (observed) input signal to predict a desired target signal or to classify the input. To achieve reproducible results the driven system has to fulfill the echo state property [3]. If the input signal is generated by another dynamical system the pair of unidirectional systems has to exhibit generalized synchronization [4] with a global basin of attraction of the response dynamics [5].

Using the Lorenz-63 system [5], the Hindmarsh-Rose model, and the Lorenz-96 equations for generating input signals and an echo state network as reservoir system we demonstrated that delayed values of input and reservoir state improve prediction performance if only partial knowledge about the state of the driving system is available or if non-optimal hyperparameters are used [6].

Furthermore, we showed [7] that efficient iterated prediction of spatio-temporal chaos can be achieved by combining parallel reservoir computing with dimension reduction, as shown for the one-dimensional Kuramoto–Sivashinsky equation with periodic boundary conditions.

Interestingly, for small reservoir systems (< 1000 nodes), dimension reduction even improves the performance of the prediction.

Analysing complex spiral wave dynamics using wave tracking
To analyse the creation, annihilation and break-up of (spiral) waves in excitable media we have developed a wave tracker approach which provides an in-depth analysis of complex spatio-temporal dynamics which can also be applied to experimental imaging data.