A review of symbolic dynamics and symbolic reconstruction of dynamical systems

Identifying low-dimensional trajectories of mechanically-ventilated patient systems: Empirical phenotypes of joint patient+care processes to enhance temporal analysis in ARDS research

Refined management of mechanically ventilation is an obvious target for improving patient outcomes, but is impeded by the nature of data for study and hypothesis generation. The connections between clinical outcomes and temporal development of iatrogenic injuries current lung-protective ventilator settings remain poorly understood. Analysis of lung-ventilator system (LVS) evolution at relevant timescales is frustrated by data volume and multiple sources of heterogeneity. This work motivates, presents, and validates a computational pipeline for resolving LVS systems into the joint evolution of data-conditioned model parameters and ventilator information. Applied to individuals, the workflow yields a concise low-dimensional representation of LVS behavior expressed in phenotypic breath waveforms suitable for analysis. The effectiveness of this approach is demonstrated through application to multi-day observational series of 35 patients. Individual patient analyses reveal multiple types of patient-oriented dynamics and breath behavior to expose the complexity of LVS evolution; less than 10% of phenotype changes related to ventilator settings changes. Dynamics are shown to including both stable and unstable phenotype transitions as well as both discrete and continuous changes unrelated to ventilator settings. At a cohort scale, 721 phenotypes constructed from individual data are condensed into a set of 16 groups that empirically organize around certain settings (positive end-expository pressure and ventilator mode) and structurally similar pressure-volume loop characterizations. Individual and cohort scale phenotypes, which may be refined by hypothesis-specific constructions, provide a common framework for ongoing temporal analysis and investigation of LVS dynamics.

Machine-Learning Optimized Measurements of Chaotic Dynamical Systems via the Information Bottleneck

Deterministic chaos permits a precise notion of a "perfect measurement" as one that, when obtained repeatedly, captures all of the information created by the system's evolution with minimal redundancy. Finding an optimal measurement is challenging and has generally required intimate knowledge of the dynamics in the few cases where it has been done. We establish an equivalence between a perfect measurement and a variant of the information bottleneck. As a consequence, we can employ machine learning to optimize measurement processes that efficiently extract information from trajectory data. We obtain approximately optimal measurements for multiple chaotic maps and lay the necessary groundwork for efficient information extraction from general time series.

Recurrence plots bridge deterministic systems and stochastic systems topologically and measure-theoretically

We connect a common conventional value to quantify a recurrence plot with its motifs, which have recently been termed "recurrence triangles." The common practical value we focus on is DET, which is the ratio of the points forming diagonal line segments of length 2 or longer within a recurrence plot. As a topological value, we use different recurrence triangles defined previously. As a measure-theoretic value, we define the typical recurrence triangle frequency dimension, which generally fluctuates around 1 when the underlying dynamics are governed by deterministic chaos. By contrast, the dimension becomes higher than 1 for a purely stochastic system. Additionally, the typical recurrence triangle frequency dimension correlates most precisely with DET among the above quantities. Our results show that (i) the common practice of using DET could be partly theoretically supported using recurrence triangles, and (ii) the variety of recurrence triangles behaves more consistently for identifying the strength of stochasticity for the underlying dynamics. The results in this study should be useful in checking basic properties for modeling a given time series.