On the automatic parameter selection for permutation entropy.

Permutation Entropy (PE) is a cost effective tool for summarizing the complexity of a time series. It has been used in many applications including damage detection, disease forecasting, detection of dynamical changes, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension n and embedding delay τ. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. Both of these methods can be time-consuming and lead to inaccurate results. In this work, we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information Function and Multi-scale Permutation Entropy (MPE) for determining τ. We then compare our methods as well as current methods in the literature for obtaining both τ and n against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter τ, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and electrocardiogram/electroencephalogram data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension n, both False Nearest Neighbors and MPE provide accurate values for n for most of the systems with a value of n=5 being suitable in most cases.

Persistent homology of coarse-grained state-space networks.

This work is dedicated to the topological analysis of complex transitional networks for dynamic state detection. Transitional networks are formed from time series data and they leverage graph theory tools to reveal information about the underlying dynamic system. However, traditional tools can fail to summarize the complex topology present in such graphs. In this work, we leverage persistent homology from topological data analysis to study the structure of these networks. We contrast dynamic state detection from time series using a coarse-grained state-space network (CGSSN) and topological data analysis (TDA) to two state of the art approaches: ordinal partition networks (OPNs) combined with TDA and the standard application of persistent homology to the time-delay embedding of the signal. We show that the CGSSN captures rich information about the dynamic state of the underlying dynamical system as evidenced by a significant improvement in dynamic state detection and noise robustness in comparison to OPNs. We also show that because the computational time of CGSSN is not linearly dependent on the signal's length, it is more computationally efficient than applying TDA to the time-delay embedding of the time series.

Low-cost double pendulum for high-quality data collection with open-source video tracking and analysis.

The double pendulum is a system that manifests fascinating non-linear behavior. This made it a popular tool in academic settings for illustrating the intricate response of a seemingly simple physical apparatus, or to validate tools for studying nonlinear phenomena. In addition, the double pendulum is also widely used in several modeling applications including robotics and human locomotion analysis. However, surprisingly, there is a lack of a thoroughly documented hardware that enables designing, building, and reliably tracking and collecting data from a double pendulum. This paper provides comprehensive documentation of a research quality bench top double pendulum. The contributions of our work include (1) providing detailed CAD drawings, part lists, and assembly instructions for building a low friction double pendulum. (2) A new tracking algorithm written in Python for tracking the position of both links of the double pendulum. This algorithm measures the angles of the links by examining each frame, and computes uncertainties in the measured angles by following several trackers on each link. Additionally, our tracking algorithm bypasses the data transmission difficulties caused by instrumenting the bottom link with physical sensors. (3) A derivation of the equations of motion of the actual physical system. (4) A description of the process (with provided Python code) for extracting the model parameters-e.g., damping-with error bounds from physical measurements.

Persistent homology of complex networks for dynamic state detection.

In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.