Machining processes are most accurately described using complex dynamical systems that include nonlinearities, time delays and stochastic effects. Due to the nature of these models as well as the practical challenges which include time-varying parameters, the transition from numerical/analytical modeling of machining to the analysis of real cutting signals remains challenging. Some studies have focused on studying the time series of cutting processes using machine learning algorithms with the goal of identifying and predicting undesirable vibrations during machining referred to as chatter. These tools typically decompose the signal using Wavelet Packet Transforms (WPT) or Ensemble Empirical Mode Decomposition (EEMD). However, these methods require a significant overhead in identifying the feature vectors before a classifier can be trained. In this study, we present an alternative approach based on featurizing the time series of the cutting process using its topological features. We utilize support vector machine classifier combined with feature vectors derived from persistence diagrams, a tool from persistent homology, to encode distinguishing characteristics based on embedding the time series as a point cloud using Takens embedding. We present the results for several choices of the topological feature vectors, and we compare our results to the WPT and EEMD methods using experimental time series from a turning cutting test. Our results show that in most cases combining the TDA-based features with a simple Support Vector Machine (SVM) yields accuracies that either exceed or are within the error bounds of their WPT and EEMD counterparts.
Distinguishing between classes of time series sampled from dynamic systems is a common challenge in systems and control engineering, for example in the context of health monitoring, fault detection, and quality control. The challenge is increased when no underlying model of a system is known, measurement noise is present, and long signals need to be interpreted. In this paper we address these issues with a new non parametric classifier based on topological signatures. Our model learns classes as weighted kernel density estimates (KDEs) over persistent homology diagrams and predicts new trajectory labels using Sinkhorn divergences on the space of diagram KDEs to quantify proximity. We show that this approach accurately discriminates between states of chaotic systems that are close in parameter space, and its performance is robust to noise.
The increasing availability of sensor data at machine tools makes automatic chatter detection algorithms a trending topic in metal cutting. Two prominent and advanced methods for feature extraction via signal decomposition are Wavelet Packet Transform (WPT) and Ensemble Empirical Mode Decomposition (EEMD). We apply these two methods to time series acquired from an acceleration sensor at the tool holder of a lathe. Different turning experiments with varying dynamic behavior of the machine tool structure were performed. We compare the performance of these two methods with Support Vector Machine (SVM) classifier combined with Recursive Feature Elimination (RFE). We also show that the common WPT-based approach of choosing wavelet packets with the highest energy ratios as representative features for chatter does not always result in packets that enclose the chatter frequency, thus reducing the classification accuracy. Further, we test the transfer learning capability of each of these methods by training the classifier on one of the cutting configurations and then testing it on the other cases. It is found that when training and testing on data from the same cutting configuration both methods yield high accuracies reaching in one of the cases as high as 94% and 91%, respectively, for WPT and EEMD. However, EEMD is shown to outperform WPT in transfer learning applications with accuracy of up to 84%. Therefore, for systems where the movement of the cutting center leads to significant variations in the stiffness of the machine-tool system, we recommend using EEMD over WPT for training a classifier. This is because EEMD retains higher accuracy rates in comparison to WPT when the input data stream deviates from the data that was used to train the classifier.
This work incorporates topological and geometric features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are planar sets that summarize the shape details of given data. A new metric on persistence diagrams generates input features for the classification algorithm. The metric accounts for the similarity of persistence diagrams using a linear combination of matching costs and cardinality differences. Investigation of the stability properties of this metric provides theoretical justification for the use of the metric for comparisons of such diagrams. The crystal structure of materials are successfully classified based on noisy and sparse data retrieved from synthetic Atomic Probe Tomography experiments.
Topological Data Analysis (tda) is a recent and fast growing eld providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for non experts. 1 Introduction and motivation Topological Data Analysis (tda) is a recent eld that emerged from various works in applied (algebraic) topology and computational geometry during the rst decade of the century. Although one can trace back geometric approaches for data analysis quite far in the past, tda really started as a eld with the pioneering works of Edelsbrunner et al. (2002) and Zomorodian and Carlsson (2005) in persistent homology and was popularized in a landmark paper in 2009 Carlsson (2009). tda is mainly motivated by the idea that topology and geometry provide a powerful approach to infer robust qualitative, and sometimes quantitative, information about the structure of data-see, e.g. Chazal (2017). tda aims at providing well-founded mathematical, statistical and algorithmic methods to infer, analyze and exploit the complex topological and geometric structures underlying data that are often represented as point clouds in Euclidean or more general metric spaces. During the last few years, a considerable eort has been made to provide robust and ecient data structures and algorithms for tda that are now implemented and available and easy to use through standard libraries such as the Gudhi library (C++ and Python) Maria et al. (2014) and its R software interface Fasy et al. (2014a). Although it is still rapidly evolving, tda now provides a set of mature and ecient tools that can be used in combination or complementary to other data sciences tools. The tdapipeline. tda has recently known developments in various directions and application elds. There now exist a large variety of methods inspired by topological and geometric approaches. Providing a complete overview of all these existing approaches is beyond the scope of this introductory survey. However, most of them rely on the following basic and standard pipeline that will serve as the backbone of this paper: 1. The input is assumed to be a nite set of points coming with a notion of distance-or similarity between them. This distance can be induced by the metric in the ambient space (e.g. the Euclidean metric when the data are embedded in R d) or come as an intrinsic metric dened by a pairwise distance matrix. The denition of the metric on the data is usually given as an input or guided by the application. It is however important to notice that the choice of the metric may be critical to reveal interesting topological and geometric features of the data.