🍩 Database of Original & Non-Theoretical Uses of Topology
(found 49 matches in 0.024863s)
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Investigation of Flash Crash via Topological Data Analysis (2020)
Wonse Kim, Younng-Jin Kim, Gihyun Lee, Woong KookAbstract
Topological data analysis has been acknowledged as one of the most successful mathematical data analytic methodologies in various fields including medicine, genetics, and image analysis. In this paper, we explore the potential of this methodology in finance by applying persistence landscape and dynamic time series analysis to analyze an extreme event in the stock market, known as Flash Crash. We will provide results of our empirical investigation to confirm the effectiveness of our new method not only for the characterization of this extreme event but also for its prediction purposes. -
Wear Monitoring in Fine Blanking Processes Using Feature Based Analysis of Acoustic Emission Signals (2021)
Martin Unterberg, Herman Voigts, Ingo Felix Weiser, Andreas Feuerhack, Daniel Trauth, Thomas BergsAbstract
Tool wear during fine blanking impairs the quality of the sheared part, which is assessed in regular samples in an industrial environment. This leads to scrap production and low planning reliability due to low wear predictability. A tool condition monitoring based on acoustic emission (AE) data for the prediction of the remaining useful life of the tool would mitigate those effects. In a production series, AE signals were recorded, and the tool wear observed. The AE signals were then preprocessed using feature engineering and visualized using linear and nonlinear dimensionality reduction techniques. These visualizations preserve information about the data structure even in two dimensions and resemble the temporal dependent observed tool wear during fine blanking. -
Toward Automated Prediction of Manufacturing Productivity Based on Feature Selection Using Topological Data Analysis (2016)
Wei Guo, Ashis G. BanerjeeAbstract
In this paper, we extend the application of topological data analysis (TDA) to the field of manufacturing for the first time to the best of our knowledge. We apply a particular TDA method, known as the Mapper algorithm, on a benchmark chemical processing data set. The algorithm yields a topological network that captures the intrinsic clusters and connections among the clusters present in the high-dimensional data set, which are difficult to detect using traditional methods. We select key process variables or features that impact the final product yield by analyzing the shape of this network. We then use three prediction models to evaluate the impact of the selected features. Results show that the models achieve the same level of high prediction accuracy as with all the process variables, thereby, providing a way to carry out process monitoring and control in a more cost-effective manner. -
Topological Attention for Time Series Forecasting (2021)
Sebastian Zeng, Florian Graf, Christoph Hofer, Roland KwittAbstract
The problem of (point) forecasting univariate time series is considered. Most approaches, ranging from traditional statistical methods to recent learning-based techniques with neural networks, directly operate on raw time series observations. As an extension, we study whether local topological properties, as captured via persistent homology, can serve as a reliable signal that provides complementary information for learning to forecast. To this end, we propose topological attention, which allows attending to local topological features within a time horizon of historical data. Our approach easily integrates into existing end-to-end trainable forecasting models, such as N-BEATS, and, in combination with the latter exhibits state-of-the-art performance on the large-scale M4 benchmark dataset of 100,000 diverse time series from different domains. Ablation experiments, as well as a comparison to recent techniques in a setting where only a single time series is available for training, corroborate the beneficial nature of including local topological information through an attention mechanism. -
Analyzing Collective Motion With Machine Learning and Topology (2019)
Dhananjay Bhaskar, Angelika Manhart, Jesse Milzman, John T. Nardini, Kathleen M. Storey, Chad M. Topaz, Lori ZiegelmeierAbstract
We use topological data analysis and machine learning to study a seminal model of collective motion in biology [M. R. D’Orsogna et al., Phys. Rev. Lett. 96, 104302 (2006)]. This model describes agents interacting nonlinearly via attractive-repulsive social forces and gives rise to collective behaviors such as flocking and milling. To classify the emergent collective motion in a large library of numerical simulations and to recover model parameters from the simulation data, we apply machine learning techniques to two different types of input. First, we input time series of order parameters traditionally used in studies of collective motion. Second, we input measures based on topology that summarize the time-varying persistent homology of simulation data over multiple scales. This topological approach does not require prior knowledge of the expected patterns. For both unsupervised and supervised machine learning methods, the topological approach outperforms the one that is based on traditional order parameters. -
Identification of Key Features Using Topological Data Analysis for Accurate Prediction of Manufacturing System Outputs (2017)
Wei Guo, Ashis G. BanerjeeAbstract
Topological data analysis (TDA) has emerged as one of the most promising approaches to extract insights from high-dimensional data of varying types such as images, point clouds, and meshes, in an unsupervised manner. To the best of our knowledge, here, we provide the first successful application of TDA in the manufacturing systems domain. We apply a widely used TDA method, known as the Mapper algorithm, on two benchmark data sets for chemical process yield prediction and semiconductor wafer fault detection, respectively. The algorithm yields topological networks that capture the intrinsic clusters and connections among the clusters present in the data sets, which are difficult to detect using traditional methods. We select key process variables or features that impact the system outcomes by analyzing the network shapes. We then use predictive models to evaluate the impact of the selected features. Results show that the models achieve at least the same level of high prediction accuracy as with all the process variables, thereby, providing a way to carry out process monitoring and control in a more cost-effective manner. -
Statistical Inference for Persistent Homology Applied to Simulated fMRI Time Series Data (2023)
Hassan Abdallah, Adam Regalski, Mohammad Behzad Kang, Maria Berishaj, Nkechi Nnadi, Asadur Chowdury, Vaibhav A. Diwadkar, Andrew SalchAbstract
Time-series data are amongst the most widely-used in biomedical sciences, including domains such as functional Magnetic Resonance Imaging (fMRI). Structure within time series data can be captured by the tools of topological data analysis (TDA). Persistent homology is the mostly commonly used data-analytic tool in TDA, and can effectively summarize complex high-dimensional data into an interpretable 2-dimensional representation called a persistence diagram. Existing methods for statistical inference for persistent homology of data depend on an independence assumption being satisfied. While persistent homology can be computed for each time index in a time-series, time-series data often fail to satisfy the independence assumption. This paper develops a statistical test that obviates the independence assumption by implementing a multi-level block sampled Monte Carlo test with sets of persistence diagrams. Its efficacy for detecting task-dependent topological organization is then demonstrated on simulated fMRI data. This new statistical test is therefore suitable for analyzing persistent homology of fMRI data, and of non-independent data in general. -
Geometric Feature Performance Under Downsampling for EEG Classification Tasks (2021)
Bryan Bischof, Eric BunchAbstract
We experimentally investigate a collection of feature engineering pipelines for use with a CNN for classifying eyes-open or eyes-closed from electroencephalogram (EEG) time-series from the Bonn dataset. Using the Takens' embedding--a geometric representation of time-series--we construct simplicial complexes from EEG data. We then compare \$\epsilon\$-series of Betti-numbers and \$\epsilon\$-series of graph spectra (a novel construction)--two topological invariants of the latent geometry from these complexes--to raw time series of the EEG to fill in a gap in the literature for benchmarking. These methods, inspired by Topological Data Analysis, are used for feature engineering to capture local geometry of the time-series. Additionally, we test these feature pipelines' robustness to downsampling and data reduction. This paper seeks to establish clearer expectations for both time-series classification via geometric features, and how CNNs for time-series respond to data of degraded resolution. -
Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology (2018)
Mustafa Hajij, Bei Wang, Carlos Scheidegger, Paul RosenAbstract
Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in time-varying graphs. Specifically, we transform each instance of the time-varying graph into a metric space, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in time-varying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on real-world datasets and show how our method can find cyclic patterns, deviations from those patterns, and one-time events in time-varying graphs. We also examine whether a persistence-based similarity measure satisfies a set of well-established, desirable properties for graph metrics. -
Testing Topological Data Analysis for Condition Monitoring of Wind Turbines (2024)
Simone Casolo, Alexander Stasik, Zhenyou Zhang, Signe Riemer-SørensenAbstract
We present an investigation of how topological data analysis (TDA) can be applied to condition-based monitoring (CBM) of wind turbines for energy generation.TDA is a branch of data analysis focusing on extracting mean- ingful information from complex datasets by analyzing their structure in state space and computing their underlying topo- logical features. By representing data in a high-dimensional state space, TDA enables the identification of patterns, anoma- lies, and trends in the data that may not be apparent through traditional signal processing methods. For this study, wind turbine data was acquired from a wind park in Norway via standard vibration sensors at different lo- cations of the turbine’s gearbox. Both the vibration acceler- ation data and its frequency spectra were recorded at infre- quent intervals for a few seconds at high frequency and fail- ure events were labelled as either gear-tooth or ball-bearing failures. The data processing and analysis are based on a pipeline where the time series data is first split into intervals and then transformed into multi-dimensional point clouds via a time-delay embedding. The shape of the point cloud is an- alyzed with topological methods such as persistent homol- ogy to generate topology-based key health indicators based on Betti numbers, information entropy and signal persistence. Such indicators are tested for CBM and diagnosis (fault de- tection) to identify faults in wind turbines and classify them accordingly. Topological indicators are shown to be an in- teresting alternative for failure identification and diagnosis of operational failures in wind turbines. -
Dissecting Ethereum Blockchain Analytics: What We Learn From Topology and Geometry of the Ethereum Graph? (2020)
Yitao Li, Umar Islambekov, Cuneyt Akcora, Ekaterina Smirnova, Yulia R. Gel, Murat Kantarcioglu -
Topological Feature Vectors for Chatter Detection in Turning Processes (2019)
Melih C. Yesilli, Firas A. Khasawneh, Andreas Otto -
Topology-Based Signal Separation (2004)
V. Robins, N. Rooney, E. Bradley -
Improving Health Care Management Through Persistent Homology of Time-Varying Variability of Emergency Department Patient Flow (2018)
Mael Dugast, Guillaume Bouleux, Olivier Mory, Eric MarconAbstract
Excessive admissions at the Emergency Department (ED) is a phenomenon very closely linked to the propagation of viruses. It is a cause of overcrowding for EDs and a public health problem. The aim of this work is to give EDs’ leaders more time for decision making during this period. Based on the admissions time series associated with specific clinical diagnoses, we will first perform a Detrended Fluctuation Analysis (DFA) to obtain the corresponding variability time series. Next, we will embed this time series on a manifold to obtain a point cloud representation and use Topological Data Analysis (TDA) through persistent homology technic to propose two early realtime indicators. One is the early indicator of abnormal arrivals at the ED whereas the second gives the information on the time index of the maximum number of arrivals. The performance of the detectors is parameter dependent and it can evolve each year. That is why we also propose to solve a bi-objective optimization problem to track the variations of this parameter. -
Evasion Paths in Mobile Sensor Networks (2015)
Henry Adams, Gunnar CarlssonAbstract
Suppose that ball-shaped sensors wander in a bounded domain. A sensor does not know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In ‘Coordinate-free coverage in sensor networks with controlled boundaries via homology', Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path. -
Induction Motor Eccentricity Fault Detection and Quantification Using Topological Data Analysis (2024)
Bingnan Wang, Chungwei Lin, Hiroshi Inoue, Makoto KanemaruAbstract
In this paper, we propose a topological data analysis (TDA) method for the processing of induction motor stator current data, and apply it to the detection and quantification of eccentricity faults. Traditionally, physics-based models and involved signal processing techniques are required to identify and extract the subtle frequency components in current data related to a particular fault. We show that TDA offers an alternative way to extract fault related features, and effectively distinguish data from different fault conditions. We will introduce TDA method and the procedure of extracting topological features from time-domain data, and apply it to induction motor current data measured under different eccentricity fault conditions. We show that while the raw time-domain data are very challenging to distinguish, the extracted topological features from these data are distinct and highly associated with eccentricity fault level. With TDA processed data, we can effectively train machine learning models to predict fault levels with good accuracy, even for new data from eccentricity levels that are not seen in the training data. The proposed method is model-free, and only requires a small segment of time-domain data to make prediction. These advantages make it attractive for a wide range of data-driven fault detection applications. -
Topological Data Analysis of C. Elegans Locomotion and Behavior (2021)
Ashleigh Thomas, Kathleen Bates, Alex Elchesen, Iryna Hartsock, Hang Lu, Peter BubenikAbstract
Video of nematodes/roundworms was analyzed using persistent homology to study locomotion and behavior. In each frame, an organism's body posture was represented by a high-dimensional vector. By concatenating points in fixed-duration segments of this time series, we created a sliding window embedding (sometimes called a time delay embedding) where each point corresponds to a sequence of postures of an organism. Persistent homology on the points in this time series detected behaviors and comparisons of these persistent homology computations detected variation in their corresponding behaviors. We used average persistence landscapes and machine learning techniques to study changes in locomotion and behavior in varying environments. -
A Simplified Algorithm for Identifying Abnormal Changes in Dynamic Networks (2022)
Bouchaib Azamir, Driss Bennis, Bertrand MichelAbstract
Topological data analysis has recently been applied to the study of dynamic networks. In this context, an algorithm was introduced and helps, among other things, to detect early warning signals of abnormal changes in the dynamic network under study. However, the complexity of this algorithm increases significantly once the database studied grows. In this paper, we propose a simplification of the algorithm without affecting its performance. We give various applications and simulations of the new algorithm on some weighted networks. The obtained results show clearly the efficiency of the introduced approach. Moreover, in some cases, the proposed algorithm makes it possible to highlight local information and sometimes early warning signals of local abnormal changes. -
Practical Joint Human-Machine Exploration of Industrial Time Series Using the Matrix Profile (2023)
Felix Nilsson, Mohamed-Rafik Bouguelia, Thorsteinn RögnvaldssonAbstract
Technological advancements and widespread adaptation of new technology in industry have made industrial time series data more available than ever before. With this development grows the need for versatile methods for mining industrial time series data. This paper introduces a practical approach for joint human-machine exploration of industrial time series data using the Matrix Profile, and presents some challenges involved. The approach is demonstrated on three real-life industrial data sets to show how it enables the user to quickly extract semantic information, detect cycles, find deviating patterns, and gain a deeper understanding of the time series. A benchmark test is also presented on ECG (electrocardiogram) data, showing that the approach works well in comparison to previously suggested methods for extracting relevant time series motifs. -
A Novel Method of Extracting Topological Features From Word Embeddings (2020)
Shafie Gholizadeh, Armin Seyeditabari, Wlodek ZadroznyAbstract
In recent years, topological data analysis has been utilized for a wide range of problems to deal with high dimensional noisy data. While text representations are often high dimensional and noisy, there are only a few work on the application of topological data analysis in natural language processing. In this paper, we introduce a novel algorithm to extract topological features from word embedding representation of text that can be used for text classification. Working on word embeddings, topological data analysis can interpret the embedding high-dimensional space and discover the relations among different embedding dimensions. We will use persistent homology, the most commonly tool from topological data analysis, for our experiment. Examining our topological algorithm on long textual documents, we will show our defined topological features may outperform conventional text mining features. -
Topological Data Analysis: Concepts, Computation, and Applications in Chemical Engineering (2021)
Alexander D. Smith, Paweł Dłotko, Victor M. ZavalaAbstract
A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g., convolutional neural nets, Fourier series). Topological Data Analysis (TDA) is a field of mathematics that analyzes data from a fundamentally different perspective. TDA represents datasets as geometric objects and provides dimensionality reduction techniques that project such objects onto low-dimensional descriptors. The key properties of these descriptors (also known as topological features) are that they provide multiscale information and that they are stable under perturbations (e.g., noise, translation, and rotation). In this work, we review the key mathematical concepts and methods of TDA and present different applications in chemical engineering. -
Cybersecurity Challenges in Downstream Steel Production Processes (2022)
Joaquín Ordieres-Meré, Andreas Wolff, Antonia Pacios-Álvarez, Antonio Bello-GarcíaAbstract
The goal of this paper is to explore proposals coming from different EU-RFCS research funded projects, in such a way that cybersecurity inside the steel industry can be increased from the Operational Technology area, with the current level of adopted Information Technology solutions. The dissemination project Control In Steel has reviewed different projects with different strategies, including ideas to be developed inside the Auto Surveillance project. An advanced control process strategy is considered and cloud based solutions are the main analysed alternatives. The different steps in the model lifecycle are considered where different cloud configurations provide different solutions. Advanced techniques such as UMAP projection are proposed to be used as detectors for anomalous behaviour in the continuous development / continuous implementation strategy, suitable for integration in processing workflows -
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 self-excited vibrations in turning. Each time series is generated using the Euler-Maruyama 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. -
An Industry Case of Large-Scale Demand Forecasting of Hierarchical Components (2019)
Rodrigo Rivera-Castro, Ivan Nazarov, Yuke Xiang, Ivan Maksimov, Aleksandr Pletnev, Evgeny BurnaevAbstract
Demand forecasting of hierarchical components is essential in manufacturing. However, its discussion in the machine-learning literature has been limited, and judgemental forecasts remain pervasive in the industry. Demand planners require easy-to-understand tools capable of delivering state-of-the-art results. This work presents an industry case of demand forecasting at one of the largest manufacturers of electronics in the world. It seeks to support practitioners with five contributions: (1) A benchmark of fourteen demand forecast methods applied to a relevant data set, (2) A data transformation technique yielding comparable results with state of the art, (3) An alternative to ARIMA based on matrix factorization, (4) A model selection technique based on topological data analysis for time series and (5) A novel data set. Organizations seeking to up-skill existing personnel and increase forecast accuracy will find value in this work. -
A Probabilistic Topological Approach to Feature Identification Using a Stochastic Robotic Swarm (2018)
Ragesh K. Ramachandran, Sean Wilson, Spring BermanAbstract
This paper presents a novel automated approach to quantifying the topological features of an unknown environment using a swarm of robots with local sensing and limited or no access to global position information. The robots randomly explore the environment and record a time series of their estimated position and the covariance matrix associated with this estimate. After the robots’ deployment, a point cloud indicating the free space of the environment is extracted from their aggregated data. Tools from topological data analysis, in particular the concept of persistent homology, are applied to a subset of the point cloud to construct barcode diagrams, which are used to determine the numbers of different types of features in the domain. We demonstrate that our approach can correctly identify the number of topological features in simulations with zero to four features and in multi-robot experiments with one to three features. -
Topological Analysis of Population Activity in Visual Cortex (2008)
Gurjeet Singh, Facundo Memoli, Tigran Ishkhanov, Guillermo Sapiro, Gunnar Carlsson, Dario L. RingachAbstract
Information in the cortex is thought to be represented by the joint activity of neurons. Here we describe how fundamental questions about neural representation can be cast in terms of the topological structure of population activity. A new method, based on the concept of persistent homology, is introduced and applied to the study of population activity in primary visual cortex (V1). We found that the topological structure of activity patterns when the cortex is spontaneously active is similar to those evoked by natural image stimulation and consistent with the topology of a two sphere. We discuss how this structure could emerge from the functional organization of orientation and spatial frequency maps and their mutual relationship. Our findings extend prior results on the relationship between spontaneous and evoked activity in V1 and illustrates how computational topology can help tackle elementary questions about the representation of information in the nervous system. -
Topological Data Analysis of Biological Aggregation Models (2015)
Chad M. Topaz, Lori Ziegelmeier, Tom HalversonAbstract
We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters. -
Extremal Event Graphs: A (Stable) Tool for Analyzing Noisy Time Series Data (2022)
Robin Belton, Bree Cummins, Brittany Terese Fasy, Tomáš GedeonAbstract
Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise \$L_\\infty\\$ distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison. -
Topological Data Analysis for Electric Motor Eccentricity Fault Detection (2022)
Bingnan Wang, Chungwei Lin, Hiroshi Inoue, Makoto KanemaruAbstract
In this paper, we develop topological data analysis (TDA) method for motor current signature analysis (MCSA), and apply it to induction motor eccentricity fault detection. We introduce TDA and present the procedure of extracting topological features from time-domain data that will be represented using persistence diagrams and vectorized Betti sequences. The procedure is applied to induction machine phase current signal analysis, and shown to be highly effective in differentiating signals from different eccentricity levels. With TDA, we are able to use a simple regression model that can predict the fault levels with reasonable accuracy, even for the data of eccentricity levels that are not seen in the training data. The proposed method is model-free, and only requires a small segment of time-domain data to make prediction. These advantages make it attractive for a wide range of fault detection applications. -
Topological Analysis Reveals State Transitions in Human Gut and Marine Bacterial Communities (2020)
William K. Chang, David VanInsberghe, Libusha KellyAbstract
Microbiome dynamics influence the health and functioning of human physiology and the environment and are driven in part by interactions between large numbers of microbial taxa, making large-scale prediction and modeling a challenge. Here, using topological data analysis, we identify states and dynamical features relevant to macroscopic processes. We show that gut disease processes and marine geochemical events are associated with transitions between community states, defined as topological features of the data density. We find a reproducible two-state succession during recovery from cholera in the gut microbiomes of multiple patients, evidence of dynamic stability in the gut microbiome of a healthy human after experiencing diarrhea during travel, and periodic state transitions in a marine Prochlorococcus community driven by water column cycling. Our approach bridges small-scale fluctuations in microbiome composition and large-scale changes in phenotype without details of underlying mechanisms, and provides an assessment of microbiome stability and its relation to human and environmental health. -
(Quasi)Periodicity Quantification in Video Data, Using Topology (2018)
Christopher J. Tralie, Jose A. PereaAbstract
This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of periodic modes with noncommensurate periods), in a way which does not require segmentation, training, object tracking, or 1-dimensional surrogate signals. Our methodology operates directly on video data. The approach combines ideas from nonlinear time series analysis (delay embeddings) and computational topology (persistent homology) by translating the problem of finding recurrent dynamics in video data into the problem of determining the circularity or toroidality of an associated geometric space. Through extensive testing, we show the robustness of our scores with respect to several noise models/levels; we show that our periodicity score is superior to other methods when compared to human-generated periodicity rankings; and furthermore, we show that our quasiperiodicity score clearly indicates the presence of biphonation in videos of vibrating vocal folds, which has never before been accomplished quantitatively end to end. -
Severe Slugging Flow Identification From Topological Indicators (2022)
Simone CasoloAbstract
In this work, topological data analysis is used to identify the onset of severe slug flow in offshore petroleum production systems. Severe slugging is a multiphase flow regime known to be very inefficient and potentially harmful to process equipment and it is characterized by large oscillations in the production fluid pressure. Time series from pressure sensors in subsea oil wells are processed by means of Takens embedding to produce point clouds of data. Embedded sensor data is then analyzed using persistent homology to obtain topological indicators capable of revealing the occurrence of severe slugging in a condition-based monitoring approach. A large dataset of well events consisting of both real and simulated data is used to demonstrate the possibilty of authomatizing severe slugging detection from live data via topological data analysis. Methods based on persistence diagrams are shown to accurately identify severe slugging and to classify different flow regimes from pressure signals of producing wells with supervised machine learning. -
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 two-state square wave with randomly varying in-between-pulse 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. -
A Functional Data-Driven Approach to Monitor and Analyze Equipment Degradation in Multiproduct Batch Processes (2023)
Joel Sansana, Ricardo Rendall, Mark N. Joswiak, Ivan Castillo, Gloria Miller, Leo H. Chiang, Marco S. ReisAbstract
Equipment degradation is ubiquitous in the Chemical Process Industry (CPI), causing significant losses in efficiency, controllability, and plant economy, as well as an increased environmental fingerprint and additional operational safety risks. The case of fouling in heat exchangers, in particular, is well-known and pervasive but still hard to cope with, given the complexity of the underlying mechanisms and the difficulty of assessing its extension in real-time. This problem becomes even more complex in batch processes producing different products, where multiple recipes are used, bringing additional variability and new challenges to the analysis. In this work, we propose a functional data-driven approach for streamlining the analysis and monitoring of the progression of fouling taking place in heat exchangers in multiproduct batch processes. With the approach developed and presented in this paper, process analysis can be efficiently conducted by integrating historical data with engineering knowledge. Furthermore, a surrogate measure of fouling extension in heat exchangers is proposed, that can be readily implemented as an equipment health indicator (EHI) leading to a safer operation of the heat exchanger. -
Topological Early Warning Signals: Quantifying Varying Routes to Extinction in a Spatially Distributed Population Model (2022)
Laura S. Storch, Sarah L. DayAbstract
Understanding and predicting critical transitions in spatially explicit ecological systems is particularly challenging due to their complex spatial and temporal dynamics and high dimensionality. Here, we explore changes in population distribution patterns during a critical transition (an extinction event) using computational topology. Computational topology allows us to quantify certain features of a population distribution pattern, such as the level of fragmentation. We create population distribution patterns via a simple coupled patch model with Ricker map growth and nearest neighbors dispersal on a two dimensional lattice. We observe two dominant paths to extinction within the explored parameter space that depend critically on the dispersal rate d and the rate of parameter drift, Δϵ. These paths to extinction are easily topologically distinguishable, so categorization can be automated. We use this population model as a theoretical proof-of-concept for the methodology, and argue that computational topology is a powerful tool for analyzing dynamical changes in systems with noisy data that are coarsely resolved in space and/or time. In addition, computational topology can provide early warning signals for chaotic dynamical systems where traditional statistical early warning signals would fail. For these reasons, we envision this work as a helpful addition to the critical transitions prediction toolbox. -
Optimizing Porosity Detection in Wire Laser Metal Deposition Processes Through Data-Driven AI Classification Techniques (2023)
Meritxell Gomez-Omella, Jon Flores, Basilio Sierra, Susana Ferreiro, Nicolas Hascoët, Francisco ChinestaAbstract
Additive manufacturing (AM) is an attractive solution for many companies that produce geometrically complex parts. This process consists of depositing material layer by layer following a sliced CAD geometry. It brings several benefits to manufacturing capabilities, such as design freedom, reduced material waste, and short-run customization. However, one of the current challenges faced by users of the process, mainly in wire laser metal deposition (wLMD), is to avoid defects in the manufactured part, especially the porosity. This defect is caused by extreme conditions and metallurgical transformations of the process. And not only does it directly affect the mechanical performance of the parts, especially the fatigue properties, but it also means an increase in costs due to the inspection tasks to which the manufactured parts must be subjected. This work compares three operational solution approaches, product-centric, based on signal-based feature extraction and Topological Data Analysis together with statistical and Machine Learning (ML) techniques, for the early detection and prediction of porosity failure in a wLMD process. The different forecasting and validation strategies demonstrate the variety of conclusions that can be drawn with different objectives in the analysis of the monitored data in AM problems. -
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. -
Dynamic State Analysis of a Driven Magnetic Pendulum Using Ordinal Partition Networks and Topological Data Analysis (2020)
Audun Myers, Firas A. KhasawnehAbstract
Abstract. The use of complex networks for time series analysis has recently shown to be useful as a tool for detecting dynamic state changes for a wide variety of applications. In this work, we implement the commonly used ordinal partition network to transform a time series into a network for detecting these state changes for the simple magnetic pendulum. The time series that we used are obtained experimentally from a base-excited magnetic pendulum apparatus, and numerically from the corresponding governing equations. The magnetic pendulum provides a relatively simple, non-linear example demonstrating transitions from periodic to chaotic motion with the variation of system parameters. For our method, we implement persistent homology, a shape measuring tool from Topological Data Analysis (TDA), to summarize the shape of the resulting ordinal partition networks as a tool for detecting state changes. We show that this network analysis tool provides a clear distinction between periodic and chaotic time series. Another contribution of this work is the successful application of the networks-TDA pipeline, for the first time, to signals from non-autonomous nonlinear systems. This opens the door for our approach to be used as an automatic design tool for studying the effect of design parameters on the resulting system response. Other uses of this approach include fault detection from sensor signals in a wide variety of engineering operations. -
Homological Scaffolds of Brain Functional Networks (2014)
G. Petri, P. Expert, F. Turkheimer, R. Carhart-Harris, D. Nutt, P. J. Hellyer, F. VaccarinoAbstract
Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci.10, 186–198. (doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and metrics concerned with node and link properties, e.g. degree-distribution, node centrality and modularity. Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. This can be done by focusing on the features of a set of topological objects—homological cycles—associated with the weighted functional network. We leverage the detected topological information to define the homological scaffolds, a new set of objects designed to represent compactly the homological features of the correlation network and simultaneously make their homological properties amenable to networks theoretical methods. As a proof of principle, we apply these tools to compare resting-state functional brain activity in 15 healthy volunteers after intravenous infusion of placebo and psilocybin—the main psychoactive component of magic mushrooms. The results show that the homological structure of the brain's functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo. -
Nonlinear Dynamic Approaches to Identify Atrial Fibrillation Progression Based on Topological Methods (2019)
Bahareh Safarbali, Seyed Mohammad Reza Hashemi GolpayeganiAbstract
In recent years, atrial fibrillation (AF) development from paroxysmal to persistent or permanent forms has become an important issue in cardiovascular disorders. Information about AF pattern of presentation (paroxysmal, persistent, or permanent) was useful in the management of algorithms in each category. This management is aimed at reducing symptoms and stopping severe problems associated with AF. AF classification has been based on time duration and episodes until now. In particular, complexity changes in Heart Rate Variation (HRV) may contain clinically relevant signals of imminent systemic dysregulation. A number of nonlinear methods based on phase space and topological properties can give more insight into HRV abnormalities such as fibrillation. Aiming to provide a nonlinear tool to qualitatively classify AF stages, we proposed two geometrical indices (fractal dimension and persistent homology) based on HRV phase space, which can successfully replicate the changes in AF progression. The study population includes 38 lone AF patients and 20 normal subjects, which are collected from the Physio-Bank database. “Time of Life (TOL)” is proposed as a new feature based on the initial and final Čech radius in the persistent homology diagram. A neural network was implemented to prove the effectiveness of both TOL and fractal dimension as classification features. The accuracy of classification performance was 93%. The proposed indices provide a signal representation framework useful to understand the dynamic changes in AF cardiac patterns and to classify normal and pathological rhythms. -
Persistent Homology of Time-Dependent Functional Networks Constructed From Coupled Time Series (2017)
Bernadette J. Stolz, Heather A. Harrington, Mason A. PorterAbstract
We use topological data analysis to study “functional networks” that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging data that were acquired from human subjects during a simple motor-learning task in which subjects were monitored for three days during a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process. -
Topological Data Analysis of Financial Time Series: Landscapes of Crashes (2017)
Marian Gidea, Yuri KatzAbstract
We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional time series. Using a sliding window, we extract time-dependent point cloud data sets, to which we associate a topological space. We detect transient loops that appear in this space, and we measure their persistence. This is encoded in real-valued functions referred to as a 'persistence landscapes'. We quantify the temporal changes in persistence landscapes via their \$L\textasciicircump\$-norms. We test this procedure on multidimensional time series generated by various non-linear and non-equilibrium models. We find that, in the vicinity of financial meltdowns, the \$L\textasciicircump\$-norms exhibit strong growth prior to the primary peak, which ascends during a crash. Remarkably, the average spectral density at low frequencies of the time series of \$L\textasciicircump\$-norms of the persistence landscapes demonstrates a strong rising trend for 250 trading days prior to either dotcom crash on 03/10/2000, or to the Lehman bankruptcy on 09/15/2008. Our study suggests that TDA provides a new type of econometric analysis, which goes beyond the standard statistical measures. The method can be used to detect early warning signals of imminent market crashes. We believe that this approach can be used beyond the analysis of financial time series presented here. -
Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies (2016)
Sofya Chepushtanova, Michael Kirby, Chris Peterson, Lori ZiegelmeierAbstract
The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper, we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold Gk,ï źn whose points parameterize the k-dimensional subspaces of \$\$\mathbb \R\\textasciicircumn\$\$Rn, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves. -
A Topological Approach to Selecting Models of Biological Experiments (2019)
M. Ulmer, Lori Ziegelmeier, Chad M. TopazAbstract
We use topological data analysis as a tool to analyze the fit of mathematical models to experimental data. This study is built on data obtained from motion tracking groups of aphids in [Nilsen et al., PLOS One, 2013] and two random walk models that were proposed to describe the data. One model incorporates social interactions between the insects via a functional dependence on an aphid’s distance to its nearest neighbor. The second model is a control model that ignores this dependence. We compare data from each model to data from experiment by performing statistical tests based on three different sets of measures. First, we use time series of order parameters commonly used in collective motion studies. These order parameters measure the overall polarization and angular momentum of the group, and do not rely on a priori knowledge of the models that produced the data. Second, we use order parameter time series that do rely on a priori knowledge, namely average distance to nearest neighbor and percentage of aphids moving. Third, we use computational persistent homology to calculate topological signatures of the data. Analysis of the a priori order parameters indicates that the interactive model better describes the experimental data than the control model does. The topological approach performs as well as these a priori order parameters and better than the other order parameters, suggesting the utility of the topological approach in the absence of specific knowledge of mechanisms underlying the data. -
Topological Data Analysis in Text Classification: Extracting Features With Additive Information (2020)
Shafie Gholizadeh, Ketki Savle, Armin Seyeditabari, Wlodek ZadroznyAbstract
While the strength of Topological Data Analysis has been explored in many studies on high dimensional numeric data, it is still a challenging task to apply it to text. As the primary goal in topological data analysis is to define and quantify the shapes in numeric data, defining shapes in the text is much more challenging, even though the geometries of vector spaces and conceptual spaces are clearly relevant for information retrieval and semantics. In this paper, we examine two different methods of extraction of topological features from text, using as the underlying representations of words the two most popular methods, namely word embeddings and TF-IDF vectors. To extract topological features from the word embedding space, we interpret the embedding of a text document as high dimensional time series, and we analyze the topology of the underlying graph where the vertices correspond to different embedding dimensions. For topological data analysis with the TF-IDF representations, we analyze the topology of the graph whose vertices come from the TF-IDF vectors of different blocks in the textual document. In both cases, we apply homological persistence to reveal the geometric structures under different distance resolutions. Our results show that these topological features carry some exclusive information that is not captured by conventional text mining methods. In our experiments we observe adding topological features to the conventional features in ensemble models improves the classification results (up to 5\%). On the other hand, as expected, topological features by themselves may be not sufficient for effective classification. It is an open problem to see whether TDA features from word embeddings might be sufficient, as they seem to perform within a range of few points from top results obtained with a linear support vector classifier. -
Novel Production Prediction Model of Gasoline Production Processes for Energy Saving and Economic Increasing Based on AM-GRU Integrating the UMAP Algorithm (2023)
Jintao Liu, Liangchao Chen, Wei Xu, Mingfei Feng, Yongming Han, Tao Xia, Zhiqiang GengAbstract
Gasoline, as an extremely important petroleum product, is of great significance to ensure people's living standards and maintain national energy security. In the actual gasoline industrial production environment, the point information collected by industrial devices usually has the characteristics of high dimension, high noise and time series because of the instability of manual operation and equipment operation. Therefore, it is difficult to use the traditional method to predict and optimize gasoline production. In this paper, a novel production prediction model using an attention mechanism (AM) based gated recurrent unit (GRU) (AM-GRU) integrating the uniform manifold approximation and projection (UMAP) is proposed. The data collected in the industrial plant are processed by the box plot to remove the data outside the quartile. Then, the UMAP is used to remove the strong correlation between the data, which can improve the running speed and the performance of the AM-GRU. Compared with the existing time series data prediction method, the superiority of the AM-GRU is verified based on University of California Irvine (UCI) benchmark datasets. Finally, the production prediction model of actual complex gasoline production processes for energy saving and economic increasing based on the proposed method is built. The experiment results show that compared with other time series data prediction models, the proposed model has better stability and higher accuracy with reaching 0.4171, 0.9969, 0.2538 and 0.5038 in terms of the mean squared error, the average absolute accuracy, the mean squared error and the root mean square error. Moreover, according to the optimal scheme of the raw material, the inefficiency production points can be expected to increase about 0.69 tons of the gasoline yield and between about \$645.1 and \$925.6 of economic benefits of industrial production. -
Transfer Learning for Autonomous Chatter Detection in Machining (2022)
Melih C. Yesilli, Firas A. Khasawneh, Brian P. MannAbstract
Large-amplitude chatter vibrations are one of the most important phenomena in machining processes. It is often detrimental in cutting operations causing a poor surface finish and decreased tool life. Therefore, chatter detection using machine learning has been an active research area over the last decade. Three challenges can be identified in applying machine learning for chatter detection at large in industry: an insufficient understanding of the universality of chatter features across different processes, the need for automating feature extraction, and the existence of limited data for each specific workpiece-machine tool combination, e.g., when machining one-off products. These three challenges can be grouped under the umbrella of transfer learning, which is concerned with studying how knowledge gained from one setting can be leveraged to obtain information in new settings. This paper studies automating chatter detection by evaluating transfer learning of prominent as well as novel chatter detection methods. We investigate chatter classification accuracy using a variety of features extracted from turning and milling experiments with different cutting configurations. The studied methods include Fast Fourier Transform (FFT), Power Spectral Density (PSD), the Auto-correlation Function (ACF), and decomposition based tools such as Wavelet Packet Transform (WPT) and Ensemble Empirical Mode Decomposition (EEMD). We also examine more recent approaches based on Topological Data Analysis (TDA) and similarity measures of time series based on Discrete Time Warping (DTW). We evaluate transfer learning potential of each approach by training and testing both within and across the turning and milling data sets. Four supervised classification algorithms are explored: support vector machine (SVM), logistic regression, random forest classification, and gradient boosting. In addition to accuracy, we also comment on the automation potential of feature extraction for each approach which is integral to creating autonomous manufacturing centers. Our results show that carefully chosen time-frequency features can lead to high classification accuracies albeit at the cost of requiring manual pre-processing and the tagging of an expert user. On the other hand, we found that the TDA and DTW approaches can provide accuracies and F1-scores on par with the time-frequency methods without the need for manual preprocessing via completely automatic pipelines. Further, we discovered that the DTW approach outperforms all other methods when trained using the milling data and tested on the turning data. Therefore, TDA and DTW approaches may be preferred over the time-frequency-based approaches for fully automated chatter detection schemes. DTW and TDA also can be more advantageous when pooling data from either limited workpiece-machine tool combinations, or from small data sets of one-off processes.