Learning transfer operators by kernel density estimation.

Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The conventional description, known as the Ulam-Galerkin method, involves projecting onto basis functions represented as characteristic functions supported over a fine grid of rectangles. From this perspective, the Ulam-Galerkin approach can be interpreted as density estimation using the histogram method. In this study, we recast the problem within the framework of statistical density estimation. This alternative perspective allows for an explicit and rigorous analysis of bias and variance, thereby facilitating a discussion on the mean square error. Through comprehensive examples utilizing the logistic map and a Markov map, we demonstrate the validity and effectiveness of this approach in estimating the eigenvectors of the Frobenius-Perron operator. We compare the performance of histogram density estimation (HDE) and kernel density estimation (KDE) methods and find that KDE generally outperforms HDE in terms of accuracy. However, it is important to note that KDE exhibits limitations around boundary points and jumps. Based on our research findings, we suggest the possibility of incorporating other density estimation methods into this field and propose future investigations into the application of KDE-based estimation for high-dimensional maps. These findings provide valuable insights for researchers and practitioners working on estimating the Frobenius-Perron operator and highlight the potential of density estimation techniques in this area of study.

Fractal basins as a mechanism for the nimble brain.

An interesting feature of the brain is its ability to respond to disparate sensory signals from the environment in unique ways depending on the environmental context or current brain state. In dynamical systems, this is an example of multi-stability, the ability to switch between multiple stable states corresponding to specific patterns of brain activity/connectivity. In this article, we describe chimera states, which are patterns consisting of mixed synchrony and incoherence, in a brain-inspired dynamical systems model composed of a network with weak individual interactions and chaotic/periodic local dynamics. We illustrate the mechanism using synthetic time series interacting on a realistic anatomical brain network derived from human diffusion tensor imaging. We introduce the so-called vector pattern state (VPS) as an efficient way of identifying chimera states and mapping basin structures. Clustering similar VPSs for different initial conditions, we show that coexisting attractors of such states reveal intricately "mingled" fractal basin boundaries that are immediately reachable. This could explain the nimble brain's ability to rapidly switch patterns between coexisting attractors.

Beach-level 24-hour forecasts of Florida red tide-induced respiratory irritation.

An accurate forecast of the red tide respiratory irritation level would improve the lives of many people living in areas affected by algal blooms. Using a decades-long database of daily beach conditions, two conceptually different models to forecast the respiratory irritation risk level one day ahead of time are trained. One model is wind-based, using the current days' respiratory level and the predicted wind direction of the following day. The other model is a probabilistic self-exciting Hawkes process model. Both models are trained on beaches in Florida during 2011--2017 and applied to the red tide bloom during 2018-2019. For beaches where there is enough historical data to develop a model, the model which performs best depends on the beach. The wind-based model is the most accurate at half the beaches, correctly predicting the respiratory risk level on average about 84% of the time. The Hawkes model is the most accurate (81% accuracy) at nearly all of the remaining beaches.

Entropic regression with neurologically motivated applications.

The ultimate goal of cognitive neuroscience is to understand the mechanistic neural processes underlying the functional organization of the brain. The key to this study is understanding the structure of both the structural and functional connectivity between anatomical regions. In this paper, we use an information theoretic approach, which defines direct information flow in terms of causation entropy, to improve upon the accuracy of the recovery of the true network structure over popularly used methods for this task such as correlation and least absolute shrinkage and selection operator regression. The method outlined above is tested on synthetic data, which is produced by following previous work in which a simple dynamical model of the brain is used, simulated on top of a real network of anatomical brain regions reconstructed from diffusion tensor imaging. We demonstrate the effectiveness of the method of AlMomani et al. [Chaos 30, 013107 (2020)] when applied to data simulated on the realistic diffusion tensor imaging network, as well as on randomly generated small-world and Erdös-Rényi networks.

The essential synchronization backbone problem.

Network optimization strategies for the process of synchronization have generally focused on the re-wiring or re-weighting of links in order to (1) expand the range of coupling strengths that achieve synchronization, (2) expand the basin of attraction for the synchronization manifold, or (3) lower the average time to synchronization. A new optimization goal is proposed in seeking the minimum subset of the edge set of the original network that enables the same essential ability to synchronize in that the synchronization manifolds have conjugate stability. We call this type of minimal spanning subgraph an essential synchronization backbone of the original system, and we present two algorithms: one is a strategy for an exhaustive search for a true solution, while the other is a method of approximation for this combinatorial problem. The solution spaces that result from different choices of dynamical systems and coupling schemes vary with the level of a hierarchical structure present and also the number of interwoven central cycles. Applications can include the important problem in civil engineering of power grid hardening, where new link creation may be costly, and the defense of certain key links to the functional process may be prioritized.

Fast weighted centroid algorithm for single particle localization near the information limit.

A simple weighting scheme that enhances the localization precision of center of mass calculations for radially symmetric intensity distributions is presented. The algorithm effectively removes the biasing that is common in such center of mass calculations. Localization precision compares favorably with other localization algorithms used in super-resolution microscopy and particle tracking, while significantly reducing the processing time and memory usage. We expect that the algorithm presented will be of significant utility when fast computationally lightweight particle localization or tracking is desired.