Browsing by Author "Spendlove, Kelly T."

Now showing 1 - 3 of 3

On the Efficacy of State Space Reconstruction Methods in Determining Causality
(2015-03) Cummins, Bree; Gedeon, Tomas; Spendlove, Kelly T.
We present a theoretical framework for inferring dynamical interactions between weakly or moderately coupled variables in systems where deterministic dynamics plays a dominating role. The variables in such a system can be arranged into an interaction graph, which is a set of nodes connected by directed edges wherever one variable directly drives another. In a system of ordinary differential equations, a variable $x$ directly drives $y$ if it appears nontrivially on the right-hand side of the equation for the derivative of $y$. Ideally, given time series measurements of the variables in a system, we would like to recover the interaction graph. We introduce a comprehensive theory showing that the transitive closure of the interaction graph is the best outcome that can be obtained from state space reconstructions in a purely deterministic system. Our work depends on extensions of Takens\' theorem and the results of Sauer et al. [J. Stat. Phys., 65 (1991), pp. 579--616] that characterize the properties of time-delay reconstructions of invariant manifolds and attractors. Along with the theory, we discuss practical implementations of our results. One method for empirical recovery of the interaction graph is presented by Sugihara et al. [Science, 338 (2012), pp. 496--500], called convergent cross-mapping. We show that the continuity detection algorithm of Pecora et al. [Phys. Rev. E, 52 (1995), pp. 3420--3439] is a viable alternative to convergent cross-mapping that is more consistent with the underlying theory. We examine two examples of dynamical systems for which we can recover the transitive closure of the interaction graph using the continuity detection technique. The strongly connected components of the recovered graph represent distinct dynamical subsystems coupled through one-way driving relationships that may correspond to causal relationships in the underlying physical scenario.
Predicting Critical Transitions in Complex Dynamical Systems
(2013-03) Spendlove, Kelly T.; Gedeon, Tomas
Complex dynamical systems, ranging from physiological diseases to financial markets and Earth's climate, often exhibit radical changes in their behavior following small changes in their parameters. In physiology, there are spontaneous system failures such as asthma attacks and epileptic seizures; in ecology, sudden collapses of wildlife populations. Data indicate Earth's climate has swung between a 'snowball' and 'tropical' Earth, occurring rapidly on a geologic scale. The common theme in these systems is a drastic change in the behavior due to potentially imperceptible changes in the conditions or parameters. Even with robust mathematical models, predicting such critical transitions prior to their occurrence is notoriously difficult. Recently, a topological approach has been developed which coarsely characterizes the dynamics of these systems. Our research builds upon this framework, using machine learning algorithms in combination with rigorous theorems regarding the underlying dynamics to construct a database which detects and catalogs critical transitions. Exploiting state-of-the-art paradigms in parallel computing, we are making this database efficiently computable for increasingly complex systems.
Predicting High-Codimension Critical Transitions In Dynamical Systems Using Active Learning
(2013-05) Spendlove, Kelly T.; Berwald, Jesse; Gedeon, Tomas
Complex dynamical systems, from those appearing in physiology and ecology to Earth system modelling, often experience critical transitions in their behaviour due to potentially minute changes in their parameters. While the focus of much recent work, predicting such bifurcations is still notoriously difficult. We propose an active learning approach to the classification of parameter space of dynamical systems for which the codimension of bifurcations is high. Using elementary notions regarding the dynamics, in combination with the nearest-neighbour algorithm and Conley index theory to classify the dynamics at a predefined scale, we are able to predict with high accuracy the boundaries between regions in parameter space that produce critical transitions.