Persistence Landscapes Research Articles

Topological Approach for Detection of Structural Breakpoints in Philippine Stock Price Data Surrounding the COVID-19 Pandemic

Topological data analysis (TDA) is a novel but rigorous framework used to identify the underlying shape and structure of a huge data set. In this study, we demonstrated how TDA can be useful in analyzing and predicting stock market movement. The study collected time-series data from the Philippine Stock Exchange of companies that were heavily affected by lockdown measures implemented during the COVID-19 pandemic and those that thrived. The study found that during market crashes, persistence diagrams revealed a presence of persistence weakening phenomenon. Moreover, by analyzing the stock price movement given an interval of time, the persistence landscape graphs produced high points or bumps during a market drop, which corresponds to structural breakpoints in the time series. Indeed, persistent homology can be used to predict early warning signs of probable market crashes.

Analyzing and interpreting convolutional neural networks using latent space topology

The development of explainability methods for Convolutional Neural Networks (CNNs), under the growing framework of explainable Artificial Intelligence (xAI) for image understanding, is crucial due to neural networks success in contrast with their black box nature. However, usual methods focus on image visualizations and are inadequate to analyze the encoded contextual information (that captures the spatial dependencies of pixels considering their neighbors), as well as to explain the evolution of learning across layers without degrading the information. To address the latter, this paper presents a novel explanatory method based on the study of the latent representations of CNNs through their topology, and supported by Topological Data Analysis (TDA). For each activation layer after a convolution, the pixel values of the activation maps along the channels are considered latent space points. The persistent homology of this data is summarized via persistence landscapes, called Latent Landscapes. This provides a global view of how contextual information is being encoded, its variety and evolution, and allows for statistical analysis. The applicability and effectiveness of our approach is demonstrated by experiments conducted with CNNs trained on distinct datasets: (1) two U-Net segmentation models on RGB and pseudo-multiband images (generated by considering vegetation indices) from the agricultural benchmark CRBD were evaluated, in order to explain the difference in performance; and (2) a VGG-16 classification network on CIFAR-10 (RGB) was analyzed, showing how the information evolves within the network. Moreover, comparisons with state-of-the-art methods (Grad-CAM and occlusion) prove the consistency and validity of our proposal. It offers novel insights into the decision making process and helps to compare how models learn.

Investigation of Indian stock markets using topological data analysis and geometry-inspired network measures

Financial markets are a quintessential example of complex systems wherein strategic connections arising through the trading (buying and selling) of different stocks gives rise to dynamic correlation patterns in stock prices. Network science based approaches have been employed for more than two decades to investigate the stock market as an inferential network, with nodes representing stocks and edges as pairwise cross-correlations between stock prices. However, a recent focus has been the characterization of the higher-order structure, beyond pairwise correlation, of such networks. To this end, geometry-inspired measures (such as discrete Ricci curvatures) and topological data analysis (TDA) based methods (such as persistent homology) have become attractive tools for characterizing the higher-order structure of networks representing the financial systems. In this study, our goal is to perform a comparative analysis of both these approaches, especially by assessing the fragility and systemic risk in the Indian stock markets, which is known for its high volatility and risk. To achieve this goal, we analyze the time series of daily log-returns of stocks comprising the National Stock Exchange (NSE) and the Bombay Stock Exchange (BSE). Specifically, our aim is to monitor the changes in standard network measures, edge-centric discrete Ricci curvatures, and persistent homology based topological measures computed from cross-correlation matrices of stocks. In this study, the edge-centric discrete Ricci curvatures have been employed for the first time in the analysis of the Indian stock markets. The Indian stock markets are known to be less diverse in comparison to the US market, and hence provides us an interesting example. Our results point that, among the persistent homology based topological measures, persistent entropy is simple and more robust than L1-norm and L2-norm of persistence landscape. In a broader comparison between network analysis and TDA, we highlight that the network analysis is sensitive to the way of constructing the networks (threshold or minimum spanning tree), as well as the threshold values used to construct the correlation-based threshold networks. On the other hand, the persistent homology is a more robust approach and is able to capture the higher-order interactions and eliminate noisy data in financial systems, since it does not take into account a single value of threshold but rather a range of values.

TopoGCL: Topological Graph Contrastive Learning

Graph contrastive learning (GCL) has recently emerged as a new concept which allows for capitalizing on the strengths of graph neural networks (GNNs) to learn rich representations in a wide variety of applications which involve abundant unlabeled information. However, existing GCL approaches largely tend to overlook the important latent information on higher-order graph substructures. We address this limitation by introducing the concepts of topological invariance and extended persistence on graphs to GCL. In particular, we propose a new contrastive mode which targets topological representations of the two augmented views from the same graph, yielded by extracting latent shape properties of the graph at multiple resolutions. Along with the extended topological layer, we introduce a new extended persistence summary, namely, extended persistence landscapes (EPL) and derive its theoretical stability guarantees. Our extensive numerical results on biological, chemical, and social interaction graphs show that the new Topological Graph Contrastive Learning (TopoGCL) model delivers significant performance gains in unsupervised graph classification for 8 out of 12 considered datasets and also exhibits robustness under noisy scenarios.

Topo-Geometric Analysis of Variability in Point Clouds using Persistence Landscapes.

Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects known as persistence diagrams. To better aid statistical analysis, a functional representation of the diagrams, known as persistence landscapes, enable use of functional data analysis and machine learning tools. Topological and geometric variabilities inherent in point clouds are confounded in both persistence diagrams and landscapes, and it is important to distinguish topological signal from noise to draw reliable conclusions on the structure of the point clouds when using persistence homology. We develop a framework for decomposing variability in persistence diagrams into topological signal and topological noise through alignment of persistence landscapes using an elastic Riemannian metric. Aligned landscapes (amplitude) isolate the topological signal. Reparameterizations used for landscape alignment (phase) are linked to a resolution parameter used to generate persistence diagrams, and capture topological noise in the form of geometric, global scaling and sampling variabilities. We illustrate the importance of decoupling topological signal and topological noise in persistence diagrams (landscapes) using several simulated examples. We also demonstrate that our approach provides novel insights in two real data studies.

Identifying Irregular Rainfall Patterns Using Persistent Homology

Efficient investigation tools are required to elucidate the changes in climatic change caused by various climate processes, variables, and socioeconomic development activities around the world. In this study, we track the changes of daily rainfall at three flood-prone sites in Terengganu between 2012 to 2017. In recent years, topological data analysis (TDA) has been applied in many fields of data analytics to rank, classify, and cluster time series datasets. In this work, we employ Persistent Homology to quantify and identify topological patterns from a rainfall data. A sliding window (SW) approach is used for each 1D rainfall dataset to embed in higher dimensions before computing its Persistence Diagrams (PD). The topological information obtained from PD, namely connected components (H0) is then retrieved and vectorized in the form of Persistence Curves (Persistence Landscape (PL), Persistence lifetime Curve (PLC), and Persistence Lifetime Entropy (PLE)) to identify unusual rainfall patterns. We employ various types of L1-norms from these Persistence vectors to identify anomalies in rainfall data which can be used as an early warning flood system. The irregular pattern of Persistence lifetime and Persistence entropy mismatch the actual flood events suggesting that the irregular points may not be as closely related to flood risk. However, PL analysis of the irregular points shows match of about 59% to the flood events.It is expected that other determining factors, for example, land use, cloud cover, and wind information, which can be obtained via satellite gridded data may increase the predictability of flood events thus promotes an effective flood risk management strategies.

Topological Data Analysis Captures Task-Driven fMRI Profiles in Individual Participants: A Classification Pipeline Based on Persistence.

BOLD-based fMRI is the most widely used method for studying brain function. The BOLD signal while valuable, is beset with unique vulnerabilities. The most notable of these is the modest signal to noise ratio, and the relatively low temporal and spatial resolution. However, the high dimensional complexity of the BOLD signal also presents unique opportunities for functional discovery. Topological Data Analyses (TDA), a branch of mathematics optimized to search for specific classes of structure within high dimensional data may provide particularly valuable applications. In this investigation, we acquired fMRI data in the anterior cingulate cortex (ACC) using a basic motor control paradigm. Then, for each participant and each of three task conditions, fMRI signals in the ACC were summarized using two methods: a) TDA based methods of persistent homology and persistence landscapes and b) non-TDA based methods using a standard vectorization scheme. Finally, using machine learning (with support vector classifiers), classification accuracy of TDA and non-TDA vectorized data was tested across participants. In each participant, TDA-based classification out-performed the non-TDA based counterpart, suggesting that our TDA analytic pipeline better characterized task- and condition-induced structure in fMRI data in the ACC. Our results emphasize the value of TDA in characterizing task- and condition-induced structure in regional fMRI signals. In addition to providing our analytical tools for other users to emulate, we also discuss the unique role that TDA-based methods can play in the study of individual differences in the structure of functional brain signals in the healthy and the clinical brain.

Importance attribution in neural networks by means of persistence landscapes of time series

This article describes a method to analyze time series with a neural network using a matrix of area-normalized persistence landscapes obtained with topological data analysis. The network’s architecture includes a gating layer that is able to identify the most relevant landscape levels for a classification task, thus working as an importance attribution system. Next, a matching is performed between the selected landscape levels and the corresponding critical points of the original time series. This matching enables reconstruction of a simplified shape of the time series that gives insight into the grounds of the classification decision. As a use case, this technique is tested in the article with input data from a dataset of electrocardiographic signals. The classification accuracy obtained using only a selection of landscape levels from data was 94.00%pm 0.13 averaged after five runs of a neural network, while the original signals achieved 98.41% pm 0.09 and landscape-reduced signals yielded 97.04% pm 0.14.

Formation of features based on computational topology methods

The use of traditional methods of algebraic topology to obtain information about the shape of an object is associated with the problem of forming a small amount of information, namely, Betti numbers and Euler characteristics. The central tool for topological data analysis is the persistent homology method, which summarizes the geometric and topological information in the data using persistent diagrams and barcodes. Based on persistent homology methods, topological data can be analyzed to obtain information about the shape of an object. The construction of persistent barcodes and persistent diagrams in computational topology does not allow one to construct a Hilbert space with a scalar product. The possibility of applying the methods of topological data analysis is based on mapping persistent diagrams into a Hilbert space; one of the ways of such mapping is a method of constructing a persistence landscape. It has an advantage of being reversible, so it does not lose any information and has persistence properties. The paper considers mathematical models and functions for representing persistence landscape objects based on the persistent homology method. Methods for converting persistent barcodes and persistent diagrams into persistence landscape functions are considered. Associated with persistence landscape functions is a persistence landscape kernel that forms a mapping into a Hilbert space with a dot product. A formula is proposed for determining a distance between the persistence landscapes, which allows the distance between images of objects to be found. The persistence landscape functions map persistent diagrams into a Hilbert space. Examples of determining the distance between images based on the construction of persistence landscape functions for these images are given. Representations of topological characteristics in various models of computational topology are considered. Results for one-parameter persistence modules are extended onto multi-parameter persistence modules.

0257 Topological Data Analysis of Electroencephalogram Signals for Pediatric Obstructive Sleep Apnea

Abstract Introduction Obstructive sleep apnea (OSA) is difficult to diagnose and leads to significant complications, including poor rest and heart disease. Identification of OSA in patients typically requires a polysomnogram, which includes acquisition of electroencephalogram (EEG) signals in order to assist with identification of sleep stages. Recent advancements in signal processing technology, including topological data analysis (TDA), provide a novel new way to analyze EEG signals to assist with identification of OSA before disordered breathing occurs. Methods We leverage EEG signals from polysomnogram studies included in the Nationwide Children’s Hospital Sleep DataBank (NCHSDB) dataset for this study. This dataset includes 3,984 pediatric patients, of which we leverage 2,881 containing usable data. We analyze each sleep state (awake before sleeping, awake in between sleep, awake after study, N1, N2, N3, and REM sleep) individually. We first break the data into 30-second chunks, then compute a distance matrix based on the multivariate time series obtained as part of the polysomnogram for the delta, theta, alpha, beta, and gamma bands individually. We then apply TDA to the distance matrices, leading to a set of persistence landscapes for the 0th homology group. We then group persistence landscapes for OSA positive patients and OSA negative patients, and perform a permutation test for each frequency band, leading to a p-value indicating significance. Results Based on our permutation test, we find significant (p < 0.05) results in all bands for the N1 (p=0.0 for all bands), N2 (p=0.0 for all bands), N3 (p=0.0 for all bands), and REM sleep (p=0.023 for delta band, p=0.0 for all other bands) cases. In the cases when the patient is awake, we find significant differences in the delta (p=0.022), theta (p=0.007), and beta (p=0.002) bands in the awake before study case, the delta (p=0.003), theta (p=0.003), and beta (p=0.0) bands for the inter-sleep awake case, and no significant differences in the awake after study case. Conclusion Our results indicate that a statistically significant difference between OSA positive and OSA negative patients can be identified when patients are awake through topological data analysis. Support (if any)

Feature Construction Using Persistence Landscapes for Clustering Noisy IoT Time Series

With the advancement of IoT technologies, there is a large amount of data available from wireless sensor networks (WSN), particularly for studying climate change. Clustering long and noisy time series has become an important research area for analyzing this data. This paper proposes a feature-based clustering approach using topological data analysis, which is a set of methods for finding topological structure in data. Persistence diagrams and landscapes are popular topological summaries that can be used to cluster time series. This paper presents a framework for selecting an optimal number of persistence landscapes, and using them as features in an unsupervised learning algorithm. This approach reduces computational cost while maintaining accuracy. The clustering approach was demonstrated to be accurate on simulated data, based on only four, three, and three features, respectively, selected in Scenarios 1–3. On real data, consisting of multiple long temperature streams from various US locations, our optimal feature selection method achieved approximately a 13 times speed-up in computing.

Cycle Registration in Persistent Homology With Applications in Topological Bootstrap.

We propose a novel approach for comparing the persistent homology representations of two spaces (or filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes, along with suitable metrics (e.g., Wasserstein). These summaries are useful for computational purposes, but they are merely a marginal of the actual topological information that persistent homology can provide. Instead, our approach compares between two topological representations directly in the data space. We do so by defining a correspondence relation between individual persistent cycles of two different spaces, and devising a method for computing this correspondence. Our matching of cycles is based on both the persistence intervals and the spatial placement of each feature. We demonstrate our new framework in the context of topological inference, where we use statistical bootstrap methods in order to differentiate between real features and noise in point cloud data.

Homology of homologous knotted proteins.

Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here, we develop a mathematical pipeline that systematically analyses protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class. This difference is localized and interpreted geometrically with recent advancements in systematic computation of homology generators. The topological and geometric quantification we find is robust to noisy input data, which demonstrates the potential of this approach in contexts where standard knot theoretic tools fail.

Topological data analysis of noise: Uniform unimodal distributions

Random point clouds determine random simplicial complexes which can be characterized via homology. In this work we consider Vietoris–Rips complexes built on top of random point clouds in the D-dimensional Euclidian space in two setups: (i) a homogeneous Poisson point process (HPPP) and (ii) N i.i.d. points sampled from a uniform elliptical distribution. We derive expected topological summaries for such complexes in terms of the average Betti numbers (ABN) and of the newly introduced average integral persistence landscape (AIPL). The latter summary is derived from the average persistence landscape (APL) and preserves its 1-norm. In the HPPP model, we derive general analytical expressions for the density of ABN and AIPL and their 1-norms. For finite samples, we establish the functional form of decline of the 1-norm of the APL with the decrease of the sample size and/or increase of the cloud dimensionality. Furthermore, we derive analytical estimates of boundary effects in a disk, an ellipse, a D-dimensional ball, and 3-dimensional spheroids. We develop asymptotic models accounting for degeneration of a highly eccentric ellipse or a prolate spheroid to a line-segment and of an oblate spheroid to a disk, which describe the decline or growth of the 1-norm of the APL upon degeneration of an elliptic or spheroidal point cloud to a one- or two-dimensional cloud, respectively. Our results can serve as a “null model” for selection of the hyperparameters of the method D and N, as well as for the interpretation of observations in a broad range of real-world problems.

A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science

Abstract Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely, persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the sugar, fish, flower, and gravel dataset produced for the study of mesoscale organization of clouds by Rasp et al. We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and we explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that readers of this paper will leave with a better understanding of TDA and persistent homology, will be able to identify problems and datasets of their own for which persistent homology could be helpful, and will gain an understanding of the results they obtain from applying the included GitHub example code. Significance Statement Information such as the geometric structure and texture of image data can greatly support the inference of the physical state of an observed Earth system, for example, in remote sensing to determine whether wildfires are active or to identify local climate zones. Persistent homology is a branch of topological data analysis that allows one to extract such information in an interpretable way—unlike black-box methods like deep neural networks. The purpose of this paper is to explain in an intuitive manner what persistent homology is and how researchers in environmental science can use it to create interpretable models. We demonstrate the approach to identify certain cloud patterns from satellite imagery and find that the resulting model is indeed interpretable.

Topological variability in financial markets

<abstract><p>We investigate market crashes and downturns through the lens of persistent homology and persistence landscape norms. Using individual stock price data from Yahoo! Finance, we find that the variation in the persistence landscape norm as well as other measures of persistence exhibit a marked increase followed by a decline prior to historic incidents. We show that basic descriptions of persistent homology may be useful in addition to more sophisticated tools like the persistence landscape norm.</p></abstract>

Persistent Homology for Breast Tumor Classification Using Mammogram Scans

An important tool in the field of topological data analysis is persistent homology (PH), which is used to encode abstract representations of the homology of data at different resolutions in the form of persistence barcode (PB). Normally, one will obtain one PB from a digital image when using a sublevel-set filtration method. In this work, we built more than one PB representation of a single image based on a landmark selection method, known as local binary patterns (LBP), which encode different types of local texture from a digital image. Starting from the top-left corner of any 3-by-3 patch selected from an input image, the LBP process starts by subtracting the central pixel value from its eight neighboring pixel values. Then, each cell is assigned with 1 if the subtraction outcome is positive, and 0 otherwise, to obtain an 8-bit binary representation. This process will identify a set of landmark pixels to represent 0-simplices and use Vietoris–Rips filtration to obtain its corresponding PB. Using LBP, we can construct up to 56 PBs from a single image if we restrict to only using the binary codes that have two circular transitions between 1 and 0. The information within these 56 PBs contain detailed local and global topological and geometrical information, which can be used to design effective machine learning models. We used four different PB vectorizations, namely, persistence landscapes, persistence images, Betti curves (barcode binning), and PB statistics. We tested the effectiveness of the proposed landmark-based PH on two publicly available breast abnormality detection datasets using mammogram scans. The sensitivity and specificity of the landmark-based PH obtained was over 90% and 85%, respectively, in both datasets for the detection of abnormal breast scans. Finally, the experimental results provide new insights on using different PB vectorizations with sublevel set filtrations and landmark-based Vietoris–Rips filtration from digital mammogram scans.

Persistence Flamelets: Topological Invariants for Scale Spaces

In recent years there has been noticeable interest in the study of the “shape of data.” Among the many ways a “shape” could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dimension 1) and so on. There is a growing number of techniques, generally denoted as Topological Data Analysis, or TDA for short, aimed at estimating topological invariants of a fixed object; when we allow this object to change, however, little has been done to investigate the evolution in its topology. In this work we define the Persistence Flamelet, a multiscale version of one of the most popular tool in TDA, the Persistence Landscape. We examine its theoretical properties and we show its performance as both an exploratory and inferential tool. In addition, we provide open source implementation of the objects and methods presented in the R-package pflamelet. Supplementary materials for this article are available online.

Persistence landscapes of affine fractals

Abstract We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes, which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.

Topological Descriptors of Gait Nonlinear Dynamics Toward Freezing-of-Gait Episodes Recognition in Parkinson’s Disease

Gait complexity analysis and nonlinear dynamics descriptions were widely used in abnormal state detection in disease studies. Recent development in algebra topology brought novel tools to describe dynamical systems, which inspired us a novel insight toward gait complexity analysis. This work investigates the gait complexity variations in Parkinson’s disease patients with a topological nonlinear dynamics analysis approach toward detecting the Freezing-of-Gait episode. In this work, we use 3-dimensional acceleration data from eight subjects with FoG symptoms and extract the FoG features from the accelerometer signals through topological analysis of the reconstructed phase space. Descriptors of Betti curves, persistence landscapes, silhouette landscapes were extracted to form feature vector toward FoG detection tasks, which achieve a sensitivity of 91.93% (94.49%), specificity of 87.61% (93.10%), the accuracy of 89.76% (93.75%), and the receiver operating characteristic curve (AUC) score of 0.956 (0.977) in the generic (person-specific) FoG recognition model. We also found that the proposed approach outperforms the traditional complexity parameters in the FoG recognition task, showing that the topological descriptions are promising in gait analysis. This work is supposed to be the first topological validation work in gait complexity analysis. Our findings are also relevant to risk stratification and continuous monitoring of Parkinson’s disease, which bring a promising approach for automated FoG detection for better healthcare.