Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimize the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, Significant Times on Persistent Strands (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic, and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, [Formula: see text]-step predictors trained on embeddings constructed with SToPS were found to outperform other embedding methods when predicting fast-slow time series.

We study swarms as dynamical systems for reservoir computing (RC). By example of a modified Reynolds boids model, the specific symmetries and dynamical properties of a swarm are explored with respect to a nonlinear time-series prediction task. Specifically, we seek to extract meaningful information about a predator-like driving signal from the swarm’s response to that signal. We find that the naïve implementation of a swarm for computation is very inefficient, as permutation symmetry of the individual agents reduces the computational capacity. To circumvent this, we distinguish between the computational substrate of the swarm and a separate observation layer, in which the swarm’s response is measured for use in the task. We demonstrate the implementation of a radial basis-localized observation layer for this task. The behavior of the swarm is characterized by order parameters and measures of consistency and related to the performance of the swarm as a reservoir. The relationship between RC performance and swarm behavior demonstrates that optimal computational properties are obtained near a phase transition regime.

Positioning and tracking a moving target from limited positional information is a frequently-encountered problem. For given noisy observations of the target’s position, one wants to estimate the true trajectory and reconstruct the full phase space including velocity and acceleration. The shadowing filter offers a robust methodology to achieve such an estimation and reconstruction. Here, we highlight and validate important merits of this methodology for real-life applications. In particular, we explore the filter’s performance when dealing with correlated or uncorrelated noise, irregular sampling in time and how it can be optimised even when the true dynamics of the system are not known.

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To understand the collective motion of many individuals, we often rely on agent-based models with rules that may be computationally complex and involved. For biologically inspired systems in particular, this raises questions about whether the imposed rules are necessarily an accurate reflection of what is being followed. The basic premise of updating one’s state according to some underlying motivation is well suited to the realm of reservoir computing; however, entire swarms of individuals are yet to be tasked with learning movement in this framework. This work focuses on the specific case of many selfish individuals simultaneously optimizing their domains in a manner conducive to reducing their personal risk of predation. Using an echo state network and data generated from the agent-based model, we show that, with an appropriate representation of input and output states, this selfish movement can be learned. This suggests that a more sophisticated neural network, such as a brain, could also learn this behavior and provides an avenue to further the search for realistic movement rules in systems of autonomous individuals.

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We propose a robust algorithm for constructing first return maps of dynamical systems from time series without the need for embedding. A first return map is typically constructed using a convenient heuristic (maxima or zero-crossings of the time series, for example) or a computationally nuanced geometric approach (explicitly constructing a Poincaré section from a hyper-surface normal to the flow and then interpolating to determine intersections with trajectories). Our method is based on ordinal partitions of the time series, and the first return map is constructed from successive intersections with specific ordinal sequences. We can obtain distinct first return maps for each ordinal sequence in general. We define entropy-based measures to guide our selection of the ordinal sequence for a “good” first return map and show that this method can robustly be applied to time series from classical chaotic systems to extract the underlying first return map dynamics. The results are shown for several well-known dynamical systems (Lorenz, Rössler, and Mackey–Glass in chaotic regimes).

This work outlines a pipeline for time series analysis that incorporates a measure of similarity not previously applied between homological summaries. Specifically, the well-established, but disparate, methods of persistent homology and TrAnsformation Cost Time Series (TACTS) are combined to provide a metric for tracking dynamics via changing homological features. TACTS allows subtle changes in dynamics to be accounted for, gives a quantitative output that can be directly interpreted, and is tunable to provide several complementary perspectives simultaneously. Our method is demonstrated first with known dynamical systems and then with a real-world electrocardiogram dataset. This paper highlights inadequacies in existing persistent homology metrics and describes circumstances where TACTS can be more sensitive and better suited to detecting a variety of regime changes.