🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.001003s)

  1. Robust Crossings Detection in Noisy Signals Using Topological Signal Processing (2024)

    Sunia Tanweer, Firas A. Khasawneh, Elizabeth Munch
    Abstract This article explores a novel method of bracketing zero-crossings for both 1-D functions and discretely sampled time series by the application of 0-D persistent homology from algebraic topology. We introduce an algorithm and demonstrate its capability of detecting crossing in noisy signals across various sampling frequencies. Compared to other software-based methods for crossing-detection in signals, our approach is typically faster, shows a higher accuracy, and has the unique ability to identify all roots within the provided interval instead of detecting only one out of all. We also discuss different options for mathematically estimating the persistence threshold— a parameter which impacts and controls the correct bracketing of roots. Finally, we explore the potential of extending our algorithm to higher dimensions.

  2. Topological Detection of Phenomenological Bifurcations With Unreliable Kernel Density Estimates (2024)

    Sunia Tanweer, Firas A. Khasawneh
    Abstract Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations requires reliable kernel density estimates computed from an ensemble of system realizations. However, in several real world signals such as Big Data, only a single system realization is available—making it impossible to estimate a reliable kernel density. This study presents an approach for detecting P-type bifurcations using unreliable density estimates. The approach creates an ensemble of objects from Topological Data Analysis (TDA) called persistence diagrams from the system’s sole realization and statistically analyzes the resulting set. We compare several methods for replicating the original persistence diagram including Gibbs point process modelling, Pairwise Interaction Point Modelling, and subsampling. We show that for the purpose of predicting a bifurcation, the simple method of subsampling exceeds the other two methods of point process modelling in performance.

  3. A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems (2024)

    Sunia Tanweer, Firas A. Khasawneh, Elizabeth Munch, Joshua R. Tempelman
    Abstract Changes in the parameters of dynamical systems can cause the state of the system to shift between different qualitative regimes. These shifts, known as bifurcations, are critical to study as they can indicate when the system is about to undergo harmful changes in its behavior. In stochastic dynamical systems, there is particular interest in P-type (phenomenological) bifurcations, which can include transitions from a monostable state to multi-stable states, the appearance of stochastic limit cycles and other features in the probability density function (PDF) of the system’s state. Current practices are limited to systems with small state spaces, cannot detect all possible behaviors of the PDFs and mandate human intervention for visually identifying the change in the PDF. In contrast, this study presents a new approach based on Topological Data Analysis that uses superlevel persistence to mathematically quantify P-type bifurcations in stochastic systems through a “homological bifurcation plot”—which shows the changing ranks of 0th and 1st homology groups, through Betti vectors. Using these plots, we demonstrate the successful detection of P-bifurcations on the stochastic Duffing, Raleigh-Vander Pol and Quintic Oscillators given their analytical PDFs, and elaborate on how to generate an estimated homological bifurcation plot given a kernel density estimate (KDE) of these systems by employing a tool for finding topological consistency between PDFs and KDEs.