5 research outputs found
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 . While another
parameter of interest in PE is the motif dimension , Typically is
selected between and with or giving optimal results for the
majority of systems. Therefore, in this work we focus solely on choosing the
delay parameter. Selecting 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 . We
evaluate the successful identification of a suitable from our TDA-based
approach by comparing our results to a variety of examples in published
literature
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
algorithm for calculating the zero-dimensional sublevel set persistence
homology
ICML 2023 topological deep learning challenge. Design and results
This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two-month duration. This paper describes the design of the challenge and summarizes its main finding