Topological Data Analysis Methods Research Articles

Characterizing emerging features in cell dynamics using topological data analysis methods.

Filament-motor interactions inside cells play essential roles in many developmental as well as other biological processes. For instance, actin-myosin interactions drive the emergence or closure of ring channel structures during wound healing or dorsal closure. These dynamic protein interactions and the resulting protein organization lead to rich time-series data generated by using fluorescence imaging experiments or by simulating realistic stochastic models. We propose methods based on topological data analysis to track topological features through time in cell biology data consisting of point clouds or binary images. The framework proposed here is based on computing the persistent homology of the data at each time point and on connecting topological features through time using established distance metrics between topological summaries. The methods retain aspects of monomer identity when analyzing significant features in filamentous structure data, and capture the overall closure dynamics when assessing the organization of multiple ring structures through time. Using applications of these techniques to experimental data, we show that the proposed methods can describe features of the emergent dynamics and quantitatively distinguish between control and perturbation experiments.

Quantification of Fracture Networks and Tracer Analysis by Topological Data Analysis

Development of geological resources requires understanding fluid flow in the underground. It is necessary to characterize fracture networks in the underground that works as flow paths. However, the fracture network structures are complex and difficult to analyze. Therefore, we quantified the fracture structures using persistent homology, which is one of the topological data analysis methods. In this study, we verified whether it is possible to estimate the number of flow paths from the tracer response. The results showed that it is possible to estimate the number of flow paths using tracer responses, but the method still needs to be improved to apply to complex fracture networks.

Quantification of Fracture Networks and Tracer Analysis by Topological Data Analysis

Development of geological resources requires understanding fluid flow in the underground. It is necessary to characterize fracture networks in the underground that works as flow paths. However, the fracture network structures are complex and difficult to analyze. Therefore, we quantified the fracture structures using persistent homology, which is one of the topological data analysis methods. In this study, we verified whether it is possible to estimate the number of flow paths from the tracer response. The results showed that it is possible to estimate the number of flow paths using tracer responses, but the method still needs to be improved to apply to complex fracture networks.

Ensemble mapper

Mapper is a popular topological data analysis method to analyse structure of complex high‐dimensional data sets. As the Mapper algorithm can be applied to clustering and feature selection with visualization, it is used in various fields such as biology and chemistry. However, some resolution parameters have to be chosen by the user before applying the Mapper algorithm, and the results are sensitive to the selection. In this paper, we focus on the selection of two resolution parameters, the number of intervals and the overlapping percentage. We propose a new resolution parameter selection method in Mapper based on the ensemble technique. We generate multiple Mapper results under various parameter values and apply the fuzzy clustering ensemble method to combine the results. To evaluate Mapper algorithms including the proposed one, three real data sets are considered. The results demonstrate the superiority of the proposed ensemble Mapper method.

Time series classification via topological data analysis

In this paper, we develop topological data analysis methods for classification tasks on univariate time series. As an application, we perform binary and ternary classification tasks on two public datasets that consist of physiological signals collected under stress and non-stress conditions. We accomplish our goal by using persistent homology to engineer stable topological features after we use a time delay embedding of the signals and perform a subwindowing instead of using windows of fixed length. The combination of methods we use can be applied to any univariate time series and allows us to reduce noise and use long window sizes without incurring an extra computational cost. We then use machine learning models on the features we algorithmically engineered to obtain higher accuracies with fewer features.

A Survey of Topological Machine Learning Methods.

The last decade saw an enormous boost in the field of computational topology: methods and concepts from algebraic and differential topology, formerly confined to the realm of pure mathematics, have demonstrated their utility in numerous areas such as computational biology personalised medicine, and time-dependent data analysis, to name a few. The newly-emerging domain comprising topology-based techniques is often referred to as topological data analysis (TDA). Next to their applications in the aforementioned areas, TDA methods have also proven to be effective in supporting, enhancing, and augmenting both classical machine learning and deep learning models. In this paper, we review the state of the art of a nascent field we refer to as “topological machine learning,” i.e., the successful symbiosis of topology-based methods and machine learning algorithms, such as deep neural networks. We identify common threads, current applications, and future challenges.

Global Patterns of Recombination across Human Viruses.

Viral recombination is a major evolutionary mechanism driving adaptation processes, such as the ability of host-switching. Understanding global patterns of recombination could help to identify underlying mechanisms and to evaluate the potential risks of rapid adaptation. Conventional approaches (e.g., those based on linkage disequilibrium) are computationally demanding or even intractable when sequence alignments include hundreds of sequences, common in viral data sets. We present a comprehensive analysis of recombination across 30 genomic alignments from viruses infecting humans. In order to scale the analysis and avoid the computational limitations of conventional approaches, we apply newly developed topological data analysis methods able to infer recombination rates for large data sets. We show that viruses, such as ZEBOV and MARV, consistently displayed low levels of recombination, whereas high levels of recombination were observed in Sarbecoviruses, HBV, HEV, Rhinovirus A, and HIV. We observe that recombination is more common in positive single-stranded RNA viruses than in negatively single-stranded RNA ones. Interestingly, the comparison across multiple viruses suggests an inverse correlation between genome length and recombination rate. Positional analyses of recombination breakpoints along viral genomes, combined with our approach, detected at least 39 nonuniform patterns of recombination (i.e., cold or hotspots) in 18 viral groups. Among these, noteworthy hotspots are found in MERS-CoV and Sarbecoviruses (at spike, Nucleocapsid and ORF8). In summary, we have developed a fast pipeline to measure recombination that, combined with other approaches, has allowed us to find both common and lineage-specific patterns of recombination among viruses with potential relevance in viral adaptation.

Semantic segmentation of microscopic neuroanatomical data by combining topological priors with encoder-decoder deep networks.

Understanding of neuronal circuitry at cellular resolution within the brain has relied on neuron tracing methods which involve careful observation and interpretation by experienced neuroscientists. With recent developments in imaging and digitization, this approach is no longer feasible with the large scale (terabyte to petabyte range) images. Machine learning based techniques, using deep networks, provide an efficient alternative to the problem. However, these methods rely on very large volumes of annotated images for training and have error rates that are too high for scientific data analysis, and thus requires a significant volume of human-in-the-loop proofreading. Here we introduce a hybrid architecture combining prior structure in the form of topological data analysis methods, based on discrete Morse theory, with the best-in-class deep-net architectures for the neuronal connectivity analysis. We show significant performance gains using our hybrid architecture on detection of topological structure (e.g. connectivity of neuronal processes and local intensity maxima on axons corresponding to synaptic swellings) with precision/recall close to 90% compared with human observers. We have adapted our architecture to a high performance pipeline capable of semantic segmentation of light microscopic whole-brain image data into a hierarchy of neuronal compartments. We expect that the hybrid architecture incorporating discrete Morse techniques into deep nets will generalize to other data domains.

Topological Data Analysis of Customer Relationship

In the modern society, if manufacturers want to make huge profits, they must not only control the production cost, but also classify customers and increase the loyalty of key customers. Thus, existing data analysis methods, represented by Time Series Analysis and Topological Data Analysis (TDA), have emerged at this point of time to explore customer relationships. Based on the latest research of time series analysis and TDA, application of analysis is explored in practice. The relationship between topology and datasets will be given by displaying the process of TDA. In the end, there will be a discussion about the efficiency of different approaches, the improvement of tools in this paper and their extended application. This work is intended to provide a brief introduction of TDA and other methods to help readers understand concepts easily and provide suggestions for future studies.

Uncovering Dynamic Brain Reconfiguration in MEG Working Memory n-Back Task Using Topological Data Analysis.

The increasing availability of high temporal resolution neuroimaging data has increased the efforts to understand the dynamics of neural functions. Until recently, there are few studies on generative models supporting classification and prediction of neural systems compared to the description of the architecture. However, the requirement of collapsing data spatially and temporally in the state-of-the art methods to analyze functional magnetic resonance imaging (fMRI), electroencephalogram (EEG) and magnetoencephalography (MEG) data cause loss of important information. In this study, we addressed this issue using a topological data analysis (TDA) method, called Mapper, which visualizes evolving patterns of brain activity as a mathematical graph. Accordingly, we analyzed preprocessed MEG data of 83 subjects from Human Connectome Project (HCP) collected during working memory n-back task. We examined variation in the dynamics of the brain states with the Mapper graphs, and to determine how this variation relates to measures such as response time and performance. The application of the Mapper method to MEG data detected a novel neuroimaging marker that explained the performance of the participants along with the ground truth of response time. In addition, TDA enabled us to distinguish two task-positive brain activations during 0-back and 2-back tasks, which is hard to detect with the other pipelines that require collapsing the data in the spatial and temporal domain. Further, the Mapper graphs of the individuals also revealed one large group in the middle of the stimulus detecting the high engagement in the brain with fine temporal resolution, which could contribute to increase spatiotemporal resolution by merging different imaging modalities. Hence, our work provides another evidence to the effectiveness of the TDA methods for extracting subtle dynamic properties of high temporal resolution MEG data without the temporal and spatial collapse.

Fast Estimation of Recombination Rates Using Topological Data Analysis.

Accurate estimation of recombination rates is critical for studying the origins and maintenance of genetic diversity. Because the inference of recombination rates under a full evolutionary model is computationally expensive, we developed an alternative approach using topological data analysis (TDA) on genome sequences. We find that this method can analyze datasets larger than what can be handled by any existing recombination inference software, and has accuracy comparable to commonly used model-based methods with significantly less processing time. Previous TDA methods used information contained solely in the first Betti number ([Formula: see text]) of a set of genomes, which aims to capture the number of loops that can be detected within a genealogy. These explorations have proven difficult to connect to the theory of the underlying biological process of recombination, and, consequently, have unpredictable behavior under perturbations of the data. We introduce a new topological feature, which we call ψ, with a natural connection to coalescent models, and present novel arguments relating [Formula: see text] to population genetic models. Using simulations, we show that ψ and [Formula: see text] are differentially affected by missing data, and package our approach as TREE (Topological Recombination Estimator). TREE's efficiency and accuracy make it well suited as a first-pass estimator of recombination rate heterogeneity or hotspots throughout the genome. Our work empirically and theoretically justifies the use of topological statistics as summaries of genome sequences and describes a new, unintuitive relationship between topological features of the distribution of sequence data and the footprint of recombination on genomes.

Construction a Neural-Net Model of Network Traffic Using the Topologic Analysis of Its Time Series Complexity

The dynamics of data traffic intensity is examined using traffic measurements at the interface switch input. The wish to prevent failures of trunk line equipment and take the full advantage of network resources makes it necessary to be able to predict the network usage. The research tackles the problem of building a predicting neural-net model of the time sequence of network traffic. Topological data analysis methods are used for data preprocessing. Nonlinear dynamics algorithms are used to choose the neural net architecture. Topological data analysis methods allow the computation of time sequence invariants. The probability function for random field maxima cannot be described analytically. However, computational topology algorithms make it possible to approximate this function using the expected value of Euler’s characteristic defined over a set of peaks. The expected values of Euler’s characteristic are found by constructing persistence diagrams and computing barcode lengths. A solution of the problem with the help of R-based libraries is given. The computation of Euler’s characteristics allows us to divide the whole data set into several uniform subsets. Predicting neural-net models are built for each of such subsets. Whitney and Takens theorems are used for determining the architecture of the sought-for neural net model. According to these theorems, the associative properties of a mathematical model depend on how accurate the dimensionality of the dynamic system is defined. The sub-problem is solved using nonlinear dynamics algorithms and calculating the correlation integral. The goal of the research is to provide ways to secure the effective transmission of data packets.

Topological Signature of 19th Century Novelists: Persistent Homology in Text Mining

Topological Data Analysis (TDA) refers to a collection of methods that find the structure of shapes in data. Although recently, TDA methods have been used in many areas of data mining, it has not been widely applied to text mining tasks. In most text processing algorithms, the order in which different entities appear or co-appear is being lost. Assuming these lost orders are informative features of the data, TDA may play a significant role in the resulted gap on text processing state of the art. Once provided, the topology of different entities through a textual document may reveal some additive information regarding the document that is not reflected in any other features from conventional text processing methods. In this paper, we introduce a novel approach that hires TDA in text processing in order to capture and use the topology of different same-type entities in textual documents. First, we will show how to extract some topological signatures in the text using persistent homology-i.e., a TDA tool that captures topological signature of data cloud. Then we will show how to utilize these signatures for text classification.

Identify and monitor clinical variation using machine intelligence: a pilot in colorectal surgery.

Standardized clinical pathways are useful tool to reduce variation in clinical management and may improve quality of care. However the evidence supporting a specific clinical pathway for a patient or patient population is often imperfect limiting adoption and efficacy of clinical pathway. Machine intelligence can potentially identify clinical variation and may provide useful insights to create and optimize clinical pathways. In this quality improvement project we analyzed the inpatient care of 1786 patients undergoing colorectal surgery from 2015 to 2016 across multiple Ohio hospitals in the Cleveland Clinic System. Data from four information subsystems was loaded in the Clinical Variation Management (CVM) application (Ayasdi, Inc., Menlo Park, CA). The CVM application uses machine intelligence and topological data analysis methods to identify groups of similar patients based on the treatment received. We defined "favorable performance" as groups with lower direct variable cost, lower length of stay, and lower 30-day readmissions. The software auto-generated 9 distinct groups of patients based on similarity analysis. Overall, favorable performance was seen with ketorolac use, lower intra-operative fluid use (< 2000cc) and surgery for cancer. Multiple sub-groups were easily created and analyzed. Adherence reporting tools were easy to use enabling almost real time monitoring. Machine intelligence provided useful insights to create and monitor care pathways with several advantages over traditional analytic approaches including: (1) analysis across disparate data sets, (2) unsupervised discovery, (3) speed and auto-generation of clinical pathways, (4) ease of use by team members, and (5) adherence reporting.

Gene Coexpression Network Comparison via Persistent Homology.

Persistent homology, a topological data analysis (TDA) method, is applied to microarray data sets. Although there are a few papers referring to TDA methods in microarray analysis, the usage of persistent homology in the comparison of several weighted gene coexpression networks (WGCN) was not employed before to the very best of our knowledge. We calculate the persistent homology of weighted networks constructed from 38 Arabidopsis microarray data sets to test the relevance and the success of this approach in distinguishing the stress factors. We quantify multiscale topological features of each network using persistent homology and apply a hierarchical clustering algorithm to the distance matrix whose entries are pairwise bottleneck distance between the networks. The immunoresponses to different stress factors are distinguishable by our method. The networks of similar immunoresponses are found to be close with respect to bottleneck distance indicating the similar topological features of WGCNs. This computationally efficient technique analyzing networks provides a quick test for advanced studies.

TDAstats: R pipeline for computing persistent homology in topological data analysis.

SummaryHigh-dimensional datasets are becoming more common in a variety of scientific fields. Well-known examples include next-generation sequencing in biology, patient health status in medicine, and computer vision in deep learning. Dimension reduction, using methods like principal component analysis (PCA), is a common preprocessing step for such datasets. However, while dimension reduction can save computing and human resources, it comes with the cost of significant information loss. Topological data analysis (TDA) aims to analyze the “shape” of high-dimensional datasets, without dimension reduction, by extracting features that are robust to small perturbations in data. Persistent features of a dataset can be used to describe it, and to compare it to other datasets. Visualization of persistent features can be done using topological barcodes or persistence diagrams (Figure 1). Application of TDA methods has granted greater insight into high-dimensional data (Lakshmikanth et al., 2017); one prominent example of this is its use to characterize a clinically relevant subgroup of breast cancer patients (Nicolau, Levine, & Carlsson, 2011). This is a particularly salient study as Nicolau et al. (2011) used a topological method, termed Progression Analysis of Disease, to identify a patient subgroup with 100% survival using that remains invisible to other clustering methods.

A topological approach for human movement classification and anticipation

The motion capture systems are increasingly used for biomedical purposes. In order to recognize and classify the movements, however whole-body movements using passive markers, generate a huge amount of data. Can topological data analysis methods improve the recognition of movements? Can we use the results of this analysis combined with particular types of neural networks to anticipate the continuation of a movement?

Identification of key features using topological data analysis for accurate prediction of manufacturing system outputs

Topological data analysis (TDA) has emerged as one of the most promising approaches to extract insights from high-dimensional data of varying types such as images, point clouds, and meshes, in an unsupervised manner. To the best of our knowledge, here, we provide the first successful application of TDA in the manufacturing systems domain. We apply a widely used TDA method, known as the Mapper algorithm, on two benchmark data sets for chemical process yield prediction and semiconductor wafer fault detection, respectively. The algorithm yields topological networks that capture the intrinsic clusters and connections among the clusters present in the data sets, which are difficult to detect using traditional methods. We select key process variables or features that impact the system outcomes by analyzing the network shapes. We then use predictive models to evaluate the impact of the selected features. Results show that the models achieve at least the same level of high prediction accuracy as with all the process variables, thereby, providing a way to carry out process monitoring and control in a more cost-effective manner.

A Multimodal Data Analysis Approach for Targeted Drug Discovery Involving Topological Data Analysis (TDA).

In silico drug discovery refers to a combination of computational techniques that augment our ability to discover drug compounds from compound libraries. Many such techniques exist, including virtual high-throughput screening (vHTS), high-throughput screening (HTS), and mechanisms for data storage and querying. However, presently these tools are often used independent of one another. In this chapter, we describe a new multimodal in silico technique for the hit identification and lead generation phases of traditional drug discovery. Our technique leverages the benefits of three independent methods-virtual high-throughput screening, high-throughput screening, and structural fingerprint analysis-by using a fourth technique called topological data analysis (TDA). We describe how a compound library can be independently tested with vHTS, HTS, and fingerprint analysis, and how the results can be transformed into a topological data analysis network to identify compounds from a diverse group of structural families. This process of using TDA or similar clustering methods to identify drug leads is advantageous because it provides a mechanism for choosing structurally diverse compounds while maintaining the unique advantages of already established techniques such as vHTS and HTS.