PERCEPT: A New Online Change-Point Detection Method using Topological Data Analysis

Continuation of point clouds via persistence diagrams

In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.

Detecting structural changes in time-series data is crucial in many applications. However, the changes in the data may appear as global structural changes that cannot be detected by conventional methods. In recent years, Topological Data Analysis (TDA) has been used to detect such global structural changes. In TDA, information on connected components or holes of data is encoded into a two-dimensional plot called a persistence diagram (PD), which can be used to detect global changes in time-series. However, only a few studies on TDA conducted change detection assuming probabilistic structure on PDs. In this paper, we introduce probability structures into PD, with which we conduct change detection on the basis of the minimum description length principle. We propose the following two methods: (1) A parametric method: We employ the Gaussian mixture model for PD modeling and then detect global changes by tracking the changes in the optimal number of mixture components. (2) A non-parametric method: We employ kernel densities for PD modeling and then detect changes by tracking changes in their global complexity. These methods not only improve the detection accuracy of global structural changes but also provide the explainability of global changes. We showcase the effectiveness of the proposed methods using synthetic data and real-world financial time-series data.

• A novel foundation for topological pattern recognition of point cloud data, through persistent homology of graph approximations. • An extension of persistent homology to point cloud data of metric trees, including theoretical and experimental verification. • The first charting of cell trajectory data sets that explains some of the difficulties this field is confronted with. The rising field of Topological Data Analysis (TDA) provides a new approach to learning from data through persistence diagrams , which are topological signatures that quantify topological properties of data in a comparable manner. For point clouds, these diagrams are often derived from the Vietoris-Rips filtration—based on the metric equipped on the data—which allows one to deduce topological patterns such as components and cycles of the underlying space. In metric trees these diagrams often fail to capture other crucial topological properties, such as the present leaves and multifurcations. Prior methods and results for persistent homology attempting to overcome this issue mainly target Rips graphs, which are often unfavorable in case of non-uniform density across our point cloud. We therefore introduce a new theoretical foundation for learning a wider variety of topological patterns through any given graph . Given particular powerful functions defining persistence diagrams to summarize topological patterns, including the normalized centrality or eccentricity , we prove a new stability result, explicitly bounding the bottleneck distance between the true and empirical diagrams for metric trees. This bound is tight if the metric distortion obtained through the graph and its maximal edge-weight are small. Through a case study of gene expression data, we demonstrate that our newly introduced diagrams provide novel quality measures and insights into cell trajectory inference.

Topological Data Analysis (TDA) is used to extract topological features such as rings from point clouds. Recent works have identified that existing methods, which construct persistence diagrams in TDA, are not robust to noise and varied densities in a point cloud. This causes these methods to obtain incorrect topological features. We analyze the necessary properties of an approach that can address these two issues, and propose a new filter function for TDA based on a new data-dependent kernel that possesses these properties. Our empirical evaluation reveals that (i) the proposed kernel provides a better mean for UMAP dimensionality reduction (ii) the proposed filter function can significantly improve the performance of Topological Point Cloud Clustering (iii) the proposed filter function is a more effective way of constructing Persistence Diagram for t-SNE visualization and SVM classification than three existing methods of TDA, In addition, we explore the proposed filter’s performance on a more complex deformation named Riemannian stretching. Our proposed filter equipped with Sample Fermat distance outperforms all the other filters when noise and Riemannian stretching coexist. Code is available at https://github.com/IsolationKernel/Codes/tree/main/Lambda-kernel.

Topological data analysis is an approach to study the shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions. Multiple data sets can be compared by the similarity of their diagrams to understand their behaviors relative to each other. The bottleneck and Wasserstein distances are often used as a tool to indicate the similarity. In this paper, we introduce the cosine similarity as a new indicator for the similarity between persistence diagrams and investigate its properties. Furthermore, it leads to the new notion of orthogonality between persistence diagrams. It turns out that the orthogonality refers to perfect dissimilarity between persistence diagrams under the cosine similarity. Through data demonstration, the cosine similarity is shown to be more accurate than the standard distances to measure the similarity between persistence diagrams.

Diagnosing neuropsychiatric systemic lupus erythematosus (NPSLE) and differentiating it from systemic lupus erythematosus (SLE) without neuropsychiatric manifestations remains a substantial clinical challenge due to the absence of specific biomarkers. Topological data analysis (TDA) is a novel computational technique that enables the visualization, exploration, and analysis of complex data structures. This study aimed to identify distinct neuroimaging biomarkers in patients with NPSLE (NPSLE group) and differentiate them from patients with SLE without neuropsychiatric symptoms (non-NPSLE group) by employing TDA. We conducted a retrospective cohort study involving 30 patients with NPSLE and 30 without neuropsychiatric symptoms between 2005 and 2020. TDA was utilized to extract topological features, specifically connected components and holes, from fluid-attenuated inversion recovery (FLAIR) sequences obtained via brain magnetic resonance imaging (MRI). Summary statistics, including critical point count, persistence lifetime, centroid coordinates, perimeter, area, and filamentarity, were derived from persistence diagrams. Multiple logistic regression analyses, adjusted for age, cerebrovascular comorbidities, and 50% hemolytic unit of complement levels, demonstrated a significant association between NPSLE and the perimeter of the holes (odds ratio [OR]: 1.67, 95% confidence interval [CI]: 1.07-2.63, p = 0.025) and the area of the holes (OR: 4.42, 95% CI: 1.35-19.6, p = 0.026) of the identified topological features. Additionally, both areas under the receiver operating characteristic curve (AUC) exceeded 0.8, indicating good diagnostic accuracy. This study identified novel neuroimaging biomarkers for the diagnosis of NPSLE. The application of TDA to brain MRI features in patients with SLE proved to be a valuable diagnostic tool, particularly through the analysis of persistence diagrams.

The Vietoris–Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is of-ten extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris–Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the run-ning time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence di-agram of the Vietoris–Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guar-antees.

Cytometry experiments yield high-dimensional point cloud data that is difficult to interpret manually. Boolean gating techniques coupled with comparisons of relative abundances of cellular subsets is the current standard for cytometry data analysis. However, this approach is unable to capture more subtle topological features hidden in data, especially if those features are further masked by data transforms or significant batch effects or donor-to-donor variations in clinical data. We present that persistent homology, a mathematical structure that summarizes the topological features, can distinguish different sources of data, such as from groups of healthy donors or patients, effectively. Analysis of publicly available cytometry data describing non-naïve CD8+ T cells in COVID-19 patients and healthy controls shows that systematic structural differences exist between single cell protein expressions in COVID-19 patients and healthy controls. We identify proteins of interest by a decision-tree based classifier, sample points randomly and compute persistence diagrams from these sampled points. The resulting persistence diagrams identify regions in cytometry datasets of varying density and identify protruded structures such as ‘elbows’. We compute Wasserstein distances between these persistence diagrams for random pairs of healthy controls and COVID-19 patients and find that systematic structural differences exist between COVID-19 patients and healthy controls in the expression data for T-bet, Eomes, and Ki-67. Further analysis shows that expression of T-bet and Eomes are significantly downregulated in COVID-19 patient non-naïve CD8+ T cells compared to healthy controls. This counter-intuitive finding may indicate that canonical effector CD8+ T cells are less prevalent in COVID-19 patients than healthy controls. This method is applicable to any cytometry dataset for discovering novel insights through topological data analysis which may be difficult to ascertain otherwise with a standard gating strategy or existing bioinformatic tools.

We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.

Metal–organic frameworks (MOFs), self-assembled porous materials synthesized from metal ions and organic ligands, are promising candidates for the direct capture of CO2 from the atmosphere. In this work, we developed a regression model to predict the optimal component of the MOF that governs the amount of CO2 adsorption per volume based on experimentally observed adsorption and structure data combined with MOF adsorption sites. The structural descriptors were generated by topological data analysis with persistence diagrams, an advanced mathematical method for quantifying the rings and cavities within the MOF. This enables us to analyze direct effects and significance of the geometric structure of the MOF on the efficiency of CO2 adsorption in a novel way. The proposed approach is proved to be highly correlated with experimental data and thus offers an effective screening tool for MOFs with optimized structures.

The rapid growth of data streams, propelled by the proliferation of sensors and Internet of Things (IoT) devices, presents significant challenges for real-time clustering of high-dimensional data. Traditional clustering algorithms struggle with high dimensionality, memory and time constraints, and adapting to dynamically evolving data. Existing dimensionality reduction methods often neglect feature ranking, leading to suboptimal clustering performance. To address these issues, we introduce E-Stream, a novel entropy-based clustering algorithm for high-dimensional data streams. E-Stream performs real-time feature ranking based on entropy within a sliding time window to identify the most informative features, which are then utilized with the DenStream algorithm for efficient clustering. We evaluated E-Stream using the NSL-KDD dataset, comparing it against DenStream, CluStream, and MR-Stream. The evaluation metrics included the average F-Measure, Jaccard Index, Fowlkes-Mallows Index, Purity, and Rand Index. The results show that E-Stream outperformed the baseline algorithms in both clustering accuracy and computational efficiency while effectively reducing dimensionality. E-Stream also demonstrated significantly less memory consumption and fewer computational requirements, highlighting its suitability for real-time processing of high-dimensional data streams. Despite its strengths, E-Stream requires manual parameter adjustment and assumes a consistent number of active features, which may limit its adaptability to diverse datasets. Future work will focus on developing a fully autonomous, parameter-free version of the algorithm, incorporating mechanisms to handle missing features and improving the management of evolving clusters to enhance robustness and adaptability in dynamic IoT environments.

This application paper presents a comprehensive experimental eval-uation of the suitability of Topological Data Analysis (TDA) for the quantitative comparison of turbulent flows. Specifically, our study documents the usage of the persistence diagram of the max-ima of flow enstrophy (an established vorticity indicator), for the topological representation of 180 ensemble members, generated by a coarse sampling of the parameter space of five numerical solvers. We document five main hypotheses reported by domain experts, describing their expectations regarding the variability of the flows generated by the distinct solver configurations. We contribute three evaluation protocols to assess the validation of the above hypothe-ses by two comparison measures: (i) a standard distance used in scientific imaging (the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{2}$</tex> norm) and (ii) an established topological distance between persistence diagrams (the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{2}$</tex> -Wasserstein metric). Extensive experiments on the input ensemble demonstrate the supe-riority of the topological distance (ii) to report as close to each other flows which are expected to be similar by domain experts, due to the configuration of their vortices. Overall, the insights reported by our study bring an experimental evidence of the suitability of TDA for representing and comparing turbulent flows, thereby providing to the fluid dynamics community confidence for its usage in future work. Also, our flow data and evaluation protocols provide to the TDA community an application-approved benchmark for the evaluation and design of further topological distances.

We propose to use Topological Data Analysis (TDA) to characterize the morphological development of brain organoids. We combine TDA with clustering strategies to characterize the morphology of three developmental stages of segmented brain organoid images. We calculate a linear regression of the H1 feature diagrams as well as entropy, dispersion, and average persistence of H0 and H1 features separately for each developmental stage. We also compute pairwise average Wasserstein distances between features, and within and inter clusters for each developmental stage. To explore all feature vectors from each group, we calculate K-L divergence on the t-SNE reduction of the TDA results.Early stages are characterized by a persistence diagram with a low slope and intercept, a high entropy within features, and a high Wasserstein distance between clusters near and far from the origin. The opposite is true for later stages. K-L divergences are particularly high between mid-stages and early or late stages, and punctually between some clusters. Results highlight specific morphological patterns at 14 days, corresponding to neuroepithelial formations.KeywordsBrain organoidCharacterisationTDA-clustering

This chapter describes a new approach for studying the stability of stochastic delay equations by investigating their time series using topological data analysis (TDA). The approach is illustrated utilizing two stochastic delay equations. The first model equation is the stochastic version of Hayes equation—a scalar autonomous delay equation—where the noise is an additive term. The second model equation is the stochastic version of Mathieu’s equation—a time-periodic delay equation. In the latter, noise is added via a multiplicative term in the time-periodic coefficient. The time series is generated using Euler–Maruyama method and a corresponding point cloud is obtained using the Takens’ embedding. The point cloud is then 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 that are described using constant as well as time-periodic coefficients. The presented approach can be used for signals generated from both numerical simulation and experiments. It can be used as a tool to study the stability of stochastic delay equations for which there are currently a limited number of analysis tools.