Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis

Permutation Entropy (PE) is a powerful tool for quantifying the predictability of a sequence which includes measuring the regularity of a time series. Despite its successful application in a variety of scientific domains, PE requires a judicious choice of the delay parameter $\tau$. While another parameter of interest in PE is the motif dimension $n$, Typically $n$ is selected between $4$ and $8$ with $5$ or $6$ giving optimal results for the majority of systems. Therefore, in this work we focus solely on choosing the delay parameter. Selecting $\tau$ is often accomplished using trial and error guided by the expertise of domain scientists. However, in this paper, we show that persistent homology, the flag ship tool from Topological Data Analysis (TDA) toolset, provides an approach for the automatic selection of $\tau$. We evaluate the successful identification of a suitable $\tau$ from our TDA-based approach by comparing our results to a variety of examples in published literature

Topological Signal Processing using the Weighted Ordinal Partition Network

One of the most important problems arising in time series analysis is that of bifurcation, or change point detection. That is, given a collection of time series over a varying parameter, when has the structure of the underlying dynamical system changed? For this task, we turn to the field of topological data analysis (TDA), which encodes information about the shape and structure of data. The idea of utilizing tools from TDA for signal processing tasks, known as topological signal processing (TSP), has gained much attention in recent years, largely through a standard pipeline that computes the persistent homology of the point cloud generated by the Takens' embedding. However, this procedure is limited by computation time since the simplicial complex generated in this case is large, but also has a great deal of redundant data. For this reason, we turn to a more recent method for encoding the structure of the attractor, which constructs an ordinal partition network (OPN) representing information about when the dynamical system has passed between certain regions of state space. The result is a weighted graph whose structure encodes information about the underlying attractor. Our previous work began to find ways to package the information of the OPN in a manner that is amenable to TDA; however, that work only used the network structure and did nothing to encode the additional weighting information. In this paper, we take the next step: building a pipeline to analyze the weighted OPN with TDA and showing that this framework provides more resilience to noise or perturbations in the system and improves the accuracy of the dynamic state detection

Effects of Correlated Noise on the Performance of Persistence Based Dynamic State Detection Methods

The ability to characterize the state of dynamic systems has been a pertinent task in the time series analysis community. Traditional measures such as Lyapunov exponents are often times difficult to recover from noisy data, especially if the dimensionality of the system is not known. More recent binary and network based testing methods have delivered promising results for unknown deterministic systems, however noise injected into a periodic signal leads to false positives. Recently, we showed the advantage of using persistent homology as a tool for achieving dynamic state detection for systems with no known model and showed its robustness to white Gaussian noise. In this work, we explore the robustness of the persistence based methods to the influence of colored noise and show that colored noise processes of the form $1/f^{\alpha}$ lead to false positive diagnostic at lower signal to noise ratios for $\alpha<0$

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 CGSSN and 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

Separating Persistent Homology of Noise from Time Series Data Using Topological Signal Processing

We introduce a novel method for separating significant features in the sublevel set persistence diagram based on a statistics analysis of the sublevel set persistence of a noise distribution. Specifically, the statistical analysis of the sublevel set persistence of additive noise distributions are leveraged to provide a noise cutoff or confidence interval in the sublevel set persistence diagram. This analysis is done for several common noise models including Gaussian, uniform, exponential and Rayleigh distributions. We then develop a framework implementing this statistical analysis of sublevel set persistence for signals contaminated by an additive noise distribution to separate the sublevel sets associated to noise and signal. This method is computationally efficient, does not require any signal pre-filtering, is widely applicable, and has open-source software available. We demonstrate the functionality of the method with both numerically simulated examples and an experimental data set. Additionally, we provide an efficient $O(nlog(n))$ algorithm for calculating the zero-dimensional sublevel set persistence homology

Formal Concept Lattice Representations and Algorithms for Hypergraphs

There is increasing focus on analyzing data represented as hypergraphs, which are better able to express complex relationships amongst entities than are graphs. Much of the critical information about hypergraph structure is available only in the intersection relationships of the hyperedges, and so forming the "intersection complex" of a hypergraph is quite valuable. This identifies a valuable isomorphism between the intersection complex and the "concept lattice" formed from taking the hypergraph's incidence matrix as a "formal context": hypergraphs also generalize graphs in that their incidence matrices are arbitrary Boolean matrices. This isomorphism allows connecting discrete algorithms for lattices and hypergraphs, in particular s-walks or s-paths on hypergraphs can be mapped to order theoretical operations on the concept lattice. We give new algorithms for formal concept lattices and hypergraph s-walks on concept lattices. We apply this to a large real-world dataset and find deep lattices implying high interconnectivity and complex geometry of hyperedges