Detecting Stochasticity in Discrete Signals via Nonparametric Excursion Theorem

We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number $N_\varepsilon$ of excursions of magnitude at least $\varepsilon$ with the quadratic variation $[X]_T$ of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent volatility, but fails sharply for deterministic systems -- thereby providing a theoretically-certfied method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio $K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$ is then summarized by a log-log slope deviation measuring the $\varepsilon^{-2}$ law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.

Unsupervised Learning of Density Estimates with Topological Optimization

Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a crucial hyperparameter: the kernel bandwidth. The choice of bandwidth is critical as it controls the bias-variance trade-off by over- or under-smoothing the topological features. Topological data analysis provides methods to mathematically quantify topological characteristics, such as connected components, loops, voids et cetera, even in high dimensions where visualization of density estimates is impossible. In this paper, we propose an unsupervised learning approach using a topology-based loss function for the automated and unsupervised selection of the optimal bandwidth and benchmark it against classical techniques -- demonstrating its potential across different dimensions.

Dynamical System Parameter Path Optimization using Persistent Homology

Nonlinear dynamical systems are complex and typically only simple systems can be analytically studied. In applications, these systems are usually defined with a set of tunable parameters and as the parameters are varied the system response undergoes significant topological changes or bifurcations. In a high dimensional parameter space, it is difficult to determine which direction to vary the system parameters to achieve a desired system response or state. In this paper, we introduce a new approach for optimally navigating a dynamical system parameter space that is rooted in topological data analysis. Specifically we use the differentiability of persistence diagrams to define a topological language for intuitively promoting or deterring different topological features in the state space response of a dynamical system and use gradient descent to optimally move from one point in the parameter space to another. The end result is a path in this space that guides the system to a set of parameters that yield the desired topological features defined by the loss function. We show a number of examples by applying the methods to different dynamical systems and scenarios to demonstrate how to promote different features and how to choose the hyperparameters to achieve different outcomes.

Hall Effect Thruster Forecasting using a Topological Approach for Data Assimilation

Hall Effect Thrusters (HETs) are electric thrusters that eject heavy ionized gas particles from the spacecraft to generate thrust. Although traditionally they were used for station keeping, recently They have been used for interplanetary space missions due to their high delta-V potential and their operational longevity in contrast to other thrusters, e.g., chemical. However, the operation of HETs involves complex processes such as ionization of gases, strong magnetic fields, and complicated solar panel power supply interactions. Therefore, their operation is extremely difficult to model thus necessitating Data Assimilation (DA) approaches for estimating and predicting their operational states. Because HET's operating environment is often noisy with non-Gaussian sources, this significantly limits applicable DA tools. We describe a topological approach for data assimilation that bypasses these limitations that does not depend on the noise model, and utilize it to forecast spatiotemporal plume field states of HETs. Our approach is a generalization of the Topological Approach for Data Assimilation (TADA) method that allows including different forecast functions. We show how TADA can be combined with the Long Short-Term Memory network for accurate forecasting. We then apply our approach to high-fidelity Hall Effect Thruster (HET) simulation data from the Air Force Research Laboratory (AFRL) rocket propulsion division where we demonstrate the forecast resiliency of TADA on noise contaminated, high-dimensional data.

Topological Detection of Phenomenological Bifurcations with Unreliable Kernel Densities

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.

Robust Zero-crossings Detection in Noisy Signals using Topological Signal Processing

We explore a novel application of zero-dimensional persistent homology from Topological Data Analysis (TDA) for bracketing zero-crossings of both one-dimensional continuous functions, and uniformly sampled time series. We present an algorithm and show its robustness in the presence of noise for a range of sampling frequencies. In comparison to state-of-the-art software-based methods for finding zeros of a time series, our method generally converges faster, provides higher accuracy, and is capable of finding all the roots in a given interval instead of converging only to one of them. We also present and compare options for automatically setting the persistence threshold parameter that influences the accurate bracketing of the roots.

Automated Surface Texture Analysis via Discrete Cosine Transform and Discrete Wavelet Transform

Surface roughness and texture are critical to the functional performance of engineering components. The ability to analyze roughness and texture effectively and efficiently is much needed to ensure surface quality in many surface generation processes, such as machining, surface mechanical treatment, etc. Discrete Wavelet Transform (DWT) and Discrete Cosine Transform (DCT) are two commonly used signal decomposition tools for surface roughness and texture analysis. Both methods require selecting a threshold to decompose a given surface into its three main components: form, waviness, and roughness. However, although DWT and DCT are part of the ISO surface finish standards, there exists no systematic guidance on how to compute these thresholds, and they are often manually selected on case by case basis. This makes utilizing these methods for studying surfaces dependent on the user's judgment and limits their automation potential. Therefore, we present two automatic threshold selection algorithms based on information theory and signal energy. We use machine learning to validate the success of our algorithms both using simulated surfaces as well as digital microscopy images of machined surfaces. Specifically, we generate feature vectors for each surface area or profile and apply supervised classification. Comparing our results with the heuristic threshold selection approach shows good agreement with mean accuracies as high as 95\%. We also compare our results with Gaussian filtering (GF) and show that while GF results for areas can yield slightly higher accuracies, our results outperform GF for surface profiles. We further show that our automatic threshold selection has significant advantages in terms of computational time as evidenced by decreasing the number of mode computations by an order of magnitude compared to the heuristic thresholding for DCT.

Transfer Learning for Autonomous Chatter Detection in Machining

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. These three challenges can be grouped under the umbrella of transfer learning. 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), 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 the transfer learning potential of each approach by training and testing both within and across the turning and milling data sets. 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.

Data-driven and Automatic Surface Texture Analysis Using Persistent Homology

Surface roughness plays an important role in analyzing engineering surfaces. It quantifies the surface topography and can be used to determine whether the resulting surface finish is acceptable or not. Nevertheless, while several existing tools and standards are available for computing surface roughness, these methods rely heavily on user input thus slowing down the analysis and increasing manufacturing costs. Therefore, fast and automatic determination of the roughness level is essential to avoid costs resulting from surfaces with unacceptable finish, and user-intensive analysis. In this study, we propose a Topological Data Analysis (TDA) based approach to classify the roughness level of synthetic surfaces using both their areal images and profiles. We utilize persistent homology from TDA to generate persistence diagrams that encapsulate information on the shape of the surface. We then obtain feature matrices for each surface or profile using Carlsson coordinates, persistence images, and template functions. We compare our results to two widely used methods in the literature: Fast Fourier Transform (FFT) and Gaussian filtering. The results show that our approach yields mean accuracies as high as 97%. We also show that, in contrast to existing surface analysis tools, our TDA-based approach is fully automatable and provides adaptive feature extraction.

On Transfer Learning of Traditional Frequency and Time Domain Features in Turning

There has been an increasing interest in leveraging machine learning tools for chatter prediction and diagnosis in discrete manufacturing processes. Some of the most common features for studying chatter include traditional signal processing tools such as Fast Fourier Transform (FFT), Power Spectral Density (PSD), and the Auto-correlation Function (ACF). In this study, we use these tools in a supervised learning setting to identify chatter in accelerometer signals obtained from a turning experiment. The experiment is performed using four different tool overhang lengths with varying cutting speed and the depth of cut. We then examine the resulting signals and tag them as either chatter or chatter-free. The tagged signals are then used to train a classifier. The classification methods include the most common algorithms: Support Vector Machine (SVM), Logistic Regression (LR), Random Forest (RF), and Gradient Boost (GB). Our results show that features extracted from the Fourier spectrum are the most informative when training a classifier and testing on data from the same cutting configuration yielding accuracy as high as %96. However, the accuracy drops significantly when training and testing on two different configurations with different structural eigenfrequencies. Thus, we conclude that while these traditional features can be highly tuned to a certain process, their transfer learning ability is limited. We also compare our results against two other methods with rising popularity in the literature: Wavelet Packet Transform (WPT) and Ensemble Empirical Mode Decomposition (EEMD). The latter two methods, especially EEMD, show better transfer learning capabilities for our dataset.