🍩 Database of Original & NonTheoretical Uses of Topology
(found 11 matches in 0.002533s)


A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems (2024)
Sunia Tanweer, Firas A. Khasawneh, Elizabeth Munch, Joshua R. TempelmanAbstract
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 Ptype (phenomenological) bifurcations, which can include transitions from a monostable state to multistable 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 Ptype 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 Pbifurcations on the stochastic Duffing, RaleighVander 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. 
Measuring Hidden Phenotype: Quantifying the Shape of Barley Seeds Using the Euler Characteristic Transform (2021)
Erik J. Amézquita, Michelle Y. Quigley, Tim Ophelders, Jacob B. Landis, Daniel Koenig, Elizabeth Munch, Daniel H. ChitwoodAbstract
Shape plays a fundamental role in biology. Traditional phenotypic analysis methods measure some features but fail to measure the information embedded in shape comprehensively. To extract, compare, and analyze this information embedded in a robust and concise way, we turn to Topological Data Analysis (TDA), specifically the Euler Characteristic Transform. TDA measures shape comprehensively using mathematical representations based on algebraic topology features. To study its use, we compute both traditional and topological shape descriptors to quantify the morphology of 3121 barley seeds scanned with Xray Computed Tomography (CT) technology at 127 micron resolution. The Euler Characteristic Transform measures shape by analyzing topological features of an object at thresholds across a number of directional axes. A KruskalWallis analysis of the information encoded by the topological signature reveals that the Euler Characteristic Transform picks up successfully the shape of the crease and bottom of the seeds. Moreover, while traditional shape descriptors can cluster the seeds based on their accession, topological shape descriptors can cluster them further based on their panicle. We then successfully train a support vector machine (SVM) to classify 28 different accessions of barley based exclusively on the shape of their grains. We observe that combining both traditional and topological descriptors classifies barley seeds better than using just traditional descriptors alone. This improvement suggests that TDA is thus a powerful complement to traditional morphometrics to comprehensively describe a multitude of “hidden” shape nuances which are otherwise not detected. 
Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems (2020)
Sarah Tymochko, Elizabeth Munch, Firas A. KhasawnehAbstract
Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here we present Bifurcations using ZigZag (BuZZ), a onestep method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system. 
Automatic Tree Ring Detection Using Jacobi Sets (2020)
Kayla Makela, Tim Ophelders, Michelle Quigley, Elizabeth Munch, Daniel Chitwood, Asia DowtinAbstract
Tree ring widths are an important source of climatic and historical data, but measuring these widths typically requires extensive manual work. Computer vision techniques provide promising directions towards the automation of tree ring detection, but most automated methods still require a substantial amount of user interaction to obtain high accuracy. We perform analysis on 3D Xray CT images of a crosssection of a tree trunk, known as a tree disk. We present novel automated methods for locating the pith (center) of a tree disk, and ring boundaries. Our methods use a combination of standard image processing techniques and tools from topological data analysis. We evaluate the efficacy of our method for two different CT scans by comparing its results to manually located rings and centers and show that it is better than current automatic methods in terms of correctly counting each ring and its location. Our methods have several parameters, which we optimize experimentally by minimizing edit distances to the manually obtained locations. 
Evolutionary Homology on Coupled Dynamical Systems With Applications to Protein Flexibility Analysis (2020)
Zixuan Cang, Elizabeth Munch, GuoWei WeiAbstract
While the spatial topological persistence is naturally constructed from a radiusbased ﬁltration, it has hardly been derived from a temporal ﬁltration. Most topological models are designed for the global topology of a given object as a whole. There is no method reported in the literature for the topology of an individual component in an object to the best of our knowledge. For many problems in science and engineering, the topology of an individual component is important for describing its properties. We propose evolutionary homology (EH) constructed via a time evolutionbased ﬁltration and topological persistence. Our approach couples a set of dynamical systems or chaotic oscillators by the interactions of a physical system, such as a macromolecule. The interactions are approximated by weighted graph Laplacians. Simplices, simplicial complexes, algebraic groups and topological persistence are deﬁned on the coupled trajectories of the chaotic oscillators. The resulting EH gives rise to timedependent topological invariants or evolutionary barcodes for an individual component of the physical system, revealing its topologyfunction relationship. In conjunction with Wasserstein metrics, the proposed EH is applied to protein ﬂexibility analysis, an important problem in computational biophysics. Numerical results for the Bfactor prediction of a benchmark set of 364 proteins indicate that the proposed EH outperforms all the other stateoftheart methods in the ﬁeld. 
Chatter Diagnosis in Milling Using Supervised Learning and Topological Features Vector (2019)
Melih C. Yesilli, Sarah Tymochko, Firas A. Khasawneh, Elizabeth MunchAbstract
Chatter detection has become a prominent subject of interest due to its effect on cutting tool life, surface finish and spindle of machine tool. Most of the existing methods in chatter detection literature are based on signal processing and signal decomposition. In this study, we use topological features of data simulating cutting tool vibrations, combined with four supervised machine learning algorithms to diagnose chatter in the milling process. Persistence diagrams, a method of representing topological features, are not easily used in the context of machine learning, so they must be transformed into a form that is more amenable. Specifically, we will focus on two different methods for featurizing persistence diagrams, Carlsson coordinates and template functions. In this paper, we provide classification results for simulated data from various cutting configurations, including upmilling and downmilling, in addition to the same data with some added noise. Our results show that Carlsson Coordinates and Template Functions yield accuracies as high as 96% and 95%, respectively. We also provide evidence that these topological methods are noise robust descriptors for chatter detection. 
Topological Data Analysis for True Step Detection in Periodic Piecewise Constant Signals (2018)
Firas A. Khasawneh, Elizabeth MunchAbstract
This paper introduces a simple yet powerful approach based on topological data analysis for detecting true steps in a periodic, piecewise constant (PWC) signal. The signal is a twostate square wave with randomly varying inbetweenpulse spacing, subject to spurious steps at the rising or falling edges which we call digital ringing. We use persistent homology to derive mathematical guarantees for the resulting change detection which enables accurate identification and counting of the true pulses. The approach is tested using both synthetic and experimental data obtained using an engine lathe instrumented with a laser tachometer. The described algorithm enables accurate and automatic calculations of the spindle speed without any choice of parameters. The results are compared with the frequency and sequency methods of the Fourier and Walsh–Hadamard transforms, respectively. Both our approach and the Fourier analysis yield comparable results for pulses with regular spacing and digital ringing while the latter causes large errors using the Walsh–Hadamard method. Further, the described approach significantly outperforms the frequency/sequency analyses when the spacing between the peaks is varied. We discuss generalizing the approach to higher dimensional PWC signals, although using this extension remains an interesting question for future research. 
Chatter Classification in Turning Using Machine Learning and Topological Data Analysis (2018)
Firas A. Khasawneh, Elizabeth Munch, Jose A. PereaAbstract
Chatter identification and detection in machining processes has been an active area of research in the past two decades. Part of the challenge in studying chatter is that machining equations that describe its occurrence are often nonlinear delay differential equations. The majority of the available tools for chatter identification rely on defining a metric that captures the characteristics of chatter, and a threshold that signals its occurrence. The difficulty in choosing these parameters can be somewhat alleviated by utilizing machine learning techniques. However, even with a successful classification algorithm, the transferability of typical machine learning methods from one data set to another remains very limited. In this paper we combine supervised machine learning with Topological Data Analysis (TDA) to obtain a descriptor of the process which can detect chatter. The features we use are derived from the persistence diagram of an attractor reconstructed from the time series via Takens embedding. We test the approach using deterministic and stochastic turning models, where the stochasticity is introduced via the cutting coefficient term. Our results show a 97% successful classification rate on the deterministic model labeled by the stability diagram obtained using the spectral element method. The features gleaned from the deterministic model are then utilized for characterization of chatter in a stochastic turning model where there are very limited analysis methods. 
Chatter Detection in Turning Using Persistent Homology (2016)
Firas A. Khasawneh, Elizabeth MunchAbstract
This paper describes a new approach for ascertaining the stability of stochastic dynamical systems in their parameter space by examining their time series using topological data analysis (TDA). We illustrate the approach using a nonlinear delayed model that describes the tool oscillations due to selfexcited vibrations in turning. Each time series is generated using the EulerMaruyama method and a corresponding point cloud is obtained using the Takens embedding. The point cloud can then be analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data. The contributions of this paper include presenting for the first time a topological approach for investigating the stability of a class of nonlinear stochastic delay equations, and introducing a new application of TDA to machining processes. 
Applications of Persistent Homology to Time Varying Systems (2013)
Elizabeth MunchAbstract
\textlessp\textgreaterThis dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. In particular, given a point cloud, we can construct its persistence diagram. Since the diagram varies continuously as the point cloud varies continuously, we study the space of time varying persistence diagrams, called vineyards when they were introduced by CohenSteiner, Edelsbrunner, and Morozov.\textless/p\textgreater\textlessp\textgreaterWe will first show that with a good choice of metric, these vineyards are stable for small perturbations of their associated point clouds. We will also define a new mean for a set of persistence diagrams based on the work of Mileyko et al. which, unlike the previously defined mean, is continuous for geodesic vineyards. \textless/p\textgreater\textlessp\textgreaterNext, we study the sensor network problem posed by Ghrist and de Silva, and their application of persistent homology to understand when a set of sensors covers a given region. Giving each of these sensors a probability of failure over time, we show that an exact computation of the probability of failure of the whole system is NPhard, but give an algorithm which can predict failure in the case of a monitored system.\textless/p\textgreater\textlessp\textgreaterFinally, we apply these methods to an automated system which can cluster agents moving in aerial images by their behaviors. We build a data structure for storing and querying the information in realtime, and define behavior vectors which quantify behaviors of interest. This clustering by behavior can be used to find groups of interest, for which we can also quantify behaviors in order to determine whether the group is working together to achieve a common goal, and we speculate that this work can be extended to improving tracking algorithms as well as behavioral predictors.\textless/p\textgreater