ON THE APPLICATION OF TOPOLOGICAL DATA ANALYSIS: A Z24 BRIDGE CASE STUDY

BackgroundAttention-deficit/hyperactivity disorder (ADHD) is currently diagnosed by a diagnostic interview, mainly based on subjective reports from parents or teachers. It is necessary to develop methods that rely on objectively measureable neurobiological data to assess brain-behavior relationship in patients with ADHD. We investigated the application of a topological data analysis tool, Mapper, to analyze the brain functional connectivity data from ADHD patients.MethodsTo quantify the disease severity using the neuroimaging data, the decomposition of individual functional networks into normal and disease components by the healthy state model (HSM) was performed, and the magnitude of the disease component (MDC) was computed. Topological data analysis using Mapper was performed to distinguish children with ADHD (n = 196) from typically developing controls (TDC) (n = 214).ResultsIn the topological data analysis, the partial clustering results of patients with ADHD and normal subjects were shown in a chain-like graph. In the correlation analysis, the MDC showed a significant increase with lower intelligence scores in TDC. We also found that the rates of comorbidity in ADHD significantly increased when the deviation of the functional connectivity from HSM was large. In addition, a significant correlation between ADHD symptom severity and MDC was found in part of the dataset.ConclusionsThe application of HSM and topological data analysis methods in assessing the brain functional connectivity seem to be promising tools to quantify ADHD symptom severity and to reveal the hidden relationship between clinical phenotypic variables and brain connectivity.

Abstract: Neurodegenerative diseases, such as Alzheimer's and Parkinson's, are characterized by progressive neuronal deterioration, leading to cognitive and motor impairments. Early detection is crucial for effective intervention and management. This study proposes a novel framework utilizing Topological Data Analysis (TDA), particularly persistent homology, to identify early biomarkers from neuroimaging and physiological data. This study explores the application of TDA techniques—specifically persistent homology and topological invariants—for the early detection of neurodegenerative diseases such as Alzheimer's. By analyzing functional brain connectivity, gait dynamics, and molecular structures associated with disease progression, the research demonstrates that TDA can capture subtle, yet critical, topological signatures indicative of early neurodegenerative changes. Layer-wise topological complexity in neural networks and persistent homology in clinical datasets reveal distinct structural degradation patterns. The integration of TDA with machine learning models enhances diagnostic accuracy, interpretability, and generalization, offering a promising direction for non-invasive, data-driven diagnostics in neurology.By capturing the intrinsic topological features of brain networks, the framework aims to distinguish between healthy and early-stage neurodegenerative conditions, offering a robust tool for early diagnosis.Topological Data Analysis (TDA) offers a robust mathematical framework for extracting shape-based features from high-dimensional and noisy biomedical data. Keywords: Topological Data Analysis, Persistent Homology, Neurodegenerative Disease, Alzheimer's, Brain Connectivity, Early Diagnosis, Betti Numbers, Gait Analysis, Functional Networks, Machine Learning, Biomedical Data

Topological Data Analysis (TDA) and its mainstay computational device, persistent homology (PH), has established a strong track record of providing researchers across the data-driven sciences with new insights and methodologies by characterizing low-dimensional geometric structures in high-dimensional data. When combined with machine learning (ML) methods, PH is valued as a discriminating-feature extraction tool. This work highlights many of the recent successes at the intersection of TDA and ML, introduces some of the foundational mathematics underpinning TDA, and summarizes the efforts to strengthen the bridge between TDA and ML. Thus, this document is a launching point for experimentalists and theoreticians to consider what can be learned from the shape of their data.

Topological Data Analysis is a set of mathematical methods for capturing the structure of data forms. TDA has been used to cope with noisy signals and time series. TDA is used to discover the key characteristics of data, which are shapes. One of the basic techniques for TDA is persistent homology which is used to do a study of time series and signal processing. Human activity analysis is a well-researched issue in the field of vision, with large research on the topic. The produced time- series data is typically high-dimensional, intrinsic, non-linear, and non-stationary, leading to pattern changes. The data acquired by a person’s numerous sensors, as well as their placement and orientation, often produces noisy and complicated data, resulting in poor model performance. The implementation method must be data-driven, independent of sensor location and orientation, and employ a coordinate and parameter-free strategy to identify the variable time series data. We need technology that can represent a low-dimensional manifold compactly, identify emergent behaviors, and globalize the system. The persistent homology having property being co- ordinate free. Sensor placement has little effect on time series factorization using persistent homology. This paper gives an overview of topological approaches as well as the problems of human activity analysis, as well as recommendations for the future.

Asthma is a complex respiratory disease characterised by spatial ventilation heterogeneity (VH) in the lung. Multiple breath washout (MBW) and forced oscillation technique (FOT) testing has been shown to quantify VH at the mouth. Methods to provide fresh insight into the mechanisms driving VH are urgently needed to support treatment development. Computed tomography (CT) with parametric response map (PRM) analysis provides spatial and functional lung data. Integrated data sets of imaging features and VH markers present the problem of extracting meaningful information from high dimensional objects; topological data analysis (TDA) has demonstrated recent success in solving this problem. In this thesis topological methods are applied to extract information about lung anatomy and function from imaging data to study association with VH in innately high dimensional integrated data sets. It was sought to describe the spatial distributions of voxel based metrics, visualise the lungs using TDA methods, and combine PRM with simulated airway tree models to improve computational modelling of FOT. Data was obtained from a clinical trial involving 52 subjects, 41 asthmatic (11 healthy). An algorithm was developed to quantify directional Hounsfield unit change (ΔHU) gradients. TDA was applied to integrated data sets and voxel sets with measures of airway tree depth. FOT modelling was augmented with ΔHU about simulated acinar airways.Inferior-to-superior ΔHU gradient best differentiated VH markers. Visualisations of the lungs illustrated gradient reversal and increased heterogeneity in ΔHU in asthmatics. FOT modelling using ΔHU parameterisation showed increased correlation between clinically measured and simulated FOT small airway resistance (R5-R20). In this thesis additional insight is presented into mechanisms affecting VH markers derived from MBW and FOT. Topological methods to analyse imaging data were developed and implemented, including workflows to visualise the lungs and lung function. Linking simulated airway tree geometry with ΔHU showed improvement in FOT modelling.

Single-cell technologies have revolutionized our ability to interrogate biological systems at unprecedented resolution, revealing complex cellular heterogeneity and dynamic processes that underlie development, disease, and immune responses. However, the high dimensionality and nonlinear structure of single-cell data present substantial analytical challenges. Topological data analysis offers a powerful mathematical framework for capturing the intrinsic shape of data, providing novel insights that complement and extend traditional statistical and machine learning methods. By leveraging tools such as persistent homology and the Mapper algorithm, TDA enables the detection of subtle, multiscale patterns – including rare cell populations, transitional states, and branching trajectories – that are often obscured by conventional approaches. In this review, we explore the theoretical foundations of topological data analysis and examine its emerging applications across single-cell transcriptomics, proteomics, and spatial biology. We highlight how this approach can unveil previously unrecognized biological phenomena, from alternative differentiation paths to complex tissue architectures, and discuss the growing ecosystem of computational tools that support its use. As single-cell datasets become increasingly large and multimodal, topological data analysis stands out as a uniquely robust and interpretable approach, with the potential to deepen our understanding of cellular identity and function in health and disease. TDA is specially suited for fields such as systems immunology since it can capture the complex, nonlinear structures inherent in high-dimensional immune data helping to identify distinct immune cell states, differentiation pathways, and dynamic responses to infection or therapy. This topological perspective complements traditional statistical approaches, providing a robust, scale-invariant framework for uncovering hidden organization within the immune system’s complexity.

Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.

Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach that can deal with various data modalities of varying complexities. One of the most commonly used tools in TDA is persistent homology (PH), which can extract topological properties from data at various scales. The aim of this article is to introduce TDA concepts to a statistical audience and provide an approach to analyzing multivariate time series data. The application's focus will be on multivariate brain signals and brain connectivity networks. Finally, this paper concludes with an overview of some open problems and potential application of TDA to modeling directionality in a brain network, as well as the casting of TDA in the context of mixed effect models to capture variations in the topological properties of data collected from multiple subjects.

Persistent homology is a fundamental tool in topological data analysis used for the most diverse applications. Information captured by persistent homology is commonly visualized using scatter plots representations. Despite being widely adopted, such a visualization technique limits user understanding and is prone to misinterpretation. This article proposes a new approach for the efficient computation of persistence cycles, a geometric representation of the features captured by persistent homology. We illustrate the importance of rendering persistence cycles when analyzing scalar fields, and we discuss the advantages that our approach provides compared to other techniques in topology-based visualization. We provide an efficient implementation of our approach based on discrete Morse theory, as a new module for the Topology Toolkit. We show that our implementation has comparable performance with respect to state-of-the-art toolboxes while providing a better framework for visually analyzing persistent homology information.

The natural pseudo-distance \(d_G\) associated with a group G of self-homeomorphisms of a topological space X is a pseudo-metric developed to compare real-valued functions defined on X, when the equivalence between functions is expressed by the group G. In this paper, we illustrate \(d_G\), its role in topological data analysis, its main properties and its link with persistent homology.

Topological data analysis is a family of recent mathematical techniques seeking to understand the ‘shape’ of data, and has been used to understand the structure of the descriptor space produced from a standard chemical informatics software from the point of view of solubility. We have used the mapper algorithm, a TDA method that creates low-dimensional representations of data, to create a network visualization of the solubility space. While descriptors with clear chemical implications are prominent features in this space, reflecting their importance to the chemical properties, an unexpected and interesting correlation between chlorine content and rings and their implication for solubility prediction is revealed. A parallel representation of the chemical space was generated using persistent homology applied to molecular graphs. Links between this chemical space and the descriptor space were shown to be in agreement with chemical heuristics. The use of persistent homology on molecular graphs, extended by the use of norms on the associated persistence landscapes allow the conversion of discrete shape descriptors to continuous ones, and a perspective of the application of these descriptors to quantitative structure property relations is presented.

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.

In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel ridge regression, and t-distributed stochastic neighbor embedding. We then give a brief overview of some of the topological notions used in topological data analysis, such as barcodes, persistent homology, and Wasserstein distance. Theoretically, when these methods are applied on a data set, they can be interpreted differently. From EEG data embedded into a manifold of high dimension, we discuss these methods and we compare them across persistent homologies of dimensions 0, 1, and 2, that is, across connected components, tunnels and holes, shells around voids, or cavities. We find that from three dimension clouds of points, it is not clear how distinct from each other the methods are, but Wasserstein and Bottleneck distances, topological tests of hypothesis, and various methods show that the methods qualitatively and significantly differ across homologies. We can infer from this analysis that topological persistent homologies do change dramatically at seizure, a finding already obtained in previous analyses. This suggests that looking at changes in homology landscapes could be a predictor of seizure.

In this work, we present a combination of dimension reduction techniques and persistent homology for detection of epileptic events in electroencephalograms for a special kind of epilepsy called petit-mal epilepsy. Persistent homology, one of the main methods in topological data analysis, extracts information about the structures appearing in a given data set. Since during an epileptic seizure of the above type the electrical brain activity is more synchronized, we take the resulting structure in the EEG signal as classification feature that is analyzed topologically by means of persistent homology. As preprocessing step, the dimension of the data is reduced by two alternative techniques, principal component analysis and dynamical component analysis, and their performance is compared. Our results show that in comparison to principal component analysis, dynamical component analysis captures the dynamics of the system when projected onto a low-dimensional subspace. Furthermore, the results prove that persistent homology is well-suited for the detection of petit-mal epileptic seizures by means of their inherent structure.

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.