Topological data analysis in investment decisions

<p style='text-indent:20px;'>In this paper, we investigate the use of the Euler characteristic for the topological data analysis, particularly over higher dimensional parameter spaces. The Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, primarily in the context of random fields. The goal of this paper, is to present the extension of using the Euler characteristic in higher dimensional parameter spaces. The topological data analysis of higher dimensional parameter spaces using stronger invariants such as homology, has and continues to be the subject of intense research. However, as important theoretical and computational obstacles remain, the use of the Euler characteristic represents an important intermediary step toward multi-parameter topological data analysis. We show the usefulness of the techniques using generated examples as well as a real world dataset of detecting diabetic retinopathy in retinal images.</p>

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.

Persistence landscapes (PL) have emerged as a transformative tool in topological data analysis (TDA), a rapidly advancing field in the current era of data-driven research. In this study, we propose a machine learning model that leverages persistence landscapes to analyse and classify time series data. We apply this model to the problem of Alzheimer's disease detection, using electroencephalogram (EEG) time series data to differentiate between a group of Alzheimer's patients and a control group of unaffected individuals. Our results demonstrate the potential of PL-based models in supporting effective classification in biomedical applications.

• 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.

The microstructure of porous electrodes of solid oxide cells (SOCs) significantly impacts their electrochemical performance. Therefore, various attempts have been made to analyze their complex porous microstructure in three dimensions. The obtained microstructures have been quantitatively evaluated using structural metrics, such as volume fraction, surface area density, and tortuosity factor of the constituent phases, as well as double- and triple-phase boundaries. These structural characteristics are useful to correlate the electrode performance with their microstructure, thereby determining optimal electrode design. However, these intuitive metrics often fails to explain the electrode performance. In fact, structural changes during long term operation of SOCs are subtle so that the performance degradation of the electrodes cannot be fully detected by these intuitive characteristics. Therefore, it is necessary to find a way to extract hidden structural metrics that characterize the electrode microstructure and to clarify their correlation with electrode performance.One of the promising candidates for such structural metrics is topological information. Topology is a structural property that is invariant to successive deformation operations, such as connected structures and the number of holes in the structures. Topology-based structural analysis has many applications in material science, where macroscopic properties of materials are correlated with topological information about their internal structures (e.g., crystal structures). For example, Wang et al. [1] performed a topological analysis on the crystal structure of double-phase steel to characterize its structure and to predict its macroscopic properties, such as stress-strain curves. Application of topological data analysis is also found in the analysis of SOFC electrodes; Pawlowski et al. [2] attempted to capture the structural changes in SOFC anodes during long-term operation from a topological perspective.In addition, topological analysis is regarded as a tool for dimensional reduction of complex structural information. Therefore, the information extracted from the topological analysis can be used to evaluate conventional metrics of porous microstructures, such as surface area and triple-phase boundary. If these metrics can be accurately evaluated from a limited number of information from the topological analysis, the numerical cost is expected to be significantly reduced.Therefore, this study investigates the applicability of the topological analysis to the electrode microstructure analysis. Fig. 1 shows the schematic diagram of the structural analysis in this study. First, persistent homology analysis [3] is employed to extract the topological information of the electrodes, where the birth and death of topological features are detected during the filtration process within the structure datasets to produce the persistent diagram (PD). The persistent diagram is then discretized and concatenated to obtain the persistent image (PI). Subsequently, principal component analysis (PCA) is conducted to further reduce the dimensionality of the structural information extracted in the persistent image. The obtained values of the principal components are correlated with the conventional structural metrics of the porous electrodes, such as volume fraction, surface area density, and triple-phase boundary density.Moreover, neural networks (NNs) consisting of fully-connected layers are constructed to quantify the structural metrics from the principal components. The constructed neural networks are trained using the real electrode microstructure datasets obtained using the focused ion beam and scanning electron microstructure [4]. To validate the developed neural networks, they are compared with the convolutional neural network (CNN), which directly quantifies the microstructural metrics from the three-dimensional structures, in terms of quantification accuracy, the number of training datasets, and required training time.The values obtained from the topological analysis followed by the principal component analysis are found to contain essential structural information in the porous electrodes. This is implied by the fact that the principal component values obtained from the real structure datasets are significantly different from those from the artificial sphere-packing structures, even though the conventional structural metrics, i.e., volume fraction and surface density, are identical between the real and artificial structures. In addition, the principal component values have sensitivity to the conventional statistical metrics, because the structures with different solid compositions form clusters in different locations in a principal component space. These suggest that the topological information will be useful not only to detect subtle structural changes in the porous electrodes undetectable in conventional structural metrics, but also to reduce the dimensionality of the complex porous electrodes without losing the information quality.Since the information extracted using the persistent homology and principal component analysis possess essential information about the electrode structures, conventional microstructural metrics are accurately quantified by the constructed neural network. It should be emphasized that the number of parameters in the neural network that need to be adjusted during the training process is significantly reduced compared with the convolutional neural network. As a result, the number of required datasets for the training of the network is significantly reduced. These clearly indicate the effectiveness of the dimensional reduction in the quantification of the electrode microstructures.

On the Application of Topological Data Analysis and Machine Learning to Flood Incidents, and Decision Making

Simplicial degree in complex networks. Applications of topological data analysis to network science

Topological data analysis (TDA) is increasingly recognized as a promising tool in the field of neuroscience, unveiling the underlying topological patterns within brain signals. However, most TDA related methods treat brain signals as if they were static, i.e., they ignore potential non-stationarities and irregularities in the statistical properties of the signals. In this study, we develop a novel fractal dimension-based testing approach that takes into account the dynamic topological properties of brain signals. By representing EEG brain signals as a sequence of Vietoris-Rips filtrations, our approach accommodates the inherent non-stationarities and irregularities of the signals. The application of our novel fractal dimension-based testing approach in analyzing dynamic topological patterns in EEG signals during an epileptic seizure episode exposes noteworthy alterations in total persistence across 0, 1, and 2-dimensional homology. These findings imply a more intricate influence of seizures on brain signals, extending beyond mere amplitude changes.

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.

The article discusses Takens embedding for 2D and 3D visualization of 1D time series data. The paper considers the use of topological data analysis in conjunction with Takens embedding to analyze the output information of a dynamic system - a rigid body. 3D images of curves constructed from three components of the angular velocity vector of a rigid body for various values of the main moments of inertia are constructed. Three-dimensional images of the curves constructed on the basis of the Takens embedding for the component of the angular velocity of a rigid body for various values of the principal moments of inertia are constructed. Distances between 3D images of curves are determined by constructing persistent landscape functions. Using the method of topological data analysis in conjunction with Takens embedding for image comparison allows you to classify and identify images and output information of the instrumental composition.

Traditional deep learning methods for histopathological analysis suffer from a lack of interpretability, which limits their use in the clinic despite their high accuracy. This paper proposes a Topological Data Analysis (TDA) framework for interpretable colorectal cancer tissue classification. We used persistent homology to extract topological features from 5000 histological images representing eight tissue classes, combining persistence landscapes with Support Vector Machine (SVM) classification. This method achieved an overall accuracy rate of 82.70%, while providing biologically interpretable features that are directly related to tissue morphology. Topological features successfully represented cellular connectivity as well as structural patterns, enabling perfect classification of morphologically distinct tissue pairs. This research demonstrates that topological data analysis (TDA) represents a promising alternative to non-transparent methods, offering competitive efficiency while ensuring interpretability, a crucial aspect for its clinical integration in computational pathology.

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.

Unbalanced cluster solutions are affected by very different cluster sizes, with some clusters being very large while others contain almost no data. We demonstrate that this phenomenon is connected to ‘hubness’, a recently discovered general problem of machine learning in high dimensional data spaces. Hub objects have a small distance to an exceptionally large number of data points, and anti-hubs are far from all other data points. In an empirical study of K-medoids clustering we show that hubness gives rise to very unbalanced cluster sizes resulting in impaired internal and external evaluation indices. We compare three methods which reduce hubness in the distance spaces and show that with the balancing of the clusters evaluation indices improve. This is done using artificial and real data sets from diverse domains.

We present an investigation of how topological data analysis (TDA) can be applied to condition-based monitoring (CBM) of wind turbines for energy generation.TDA is a branch of data analysis focusing on extracting mean- ingful information from complex datasets by analyzing their structure in state space and computing their underlying topo- logical features. By representing data in a high-dimensional state space, TDA enables the identification of patterns, anoma- lies, and trends in the data that may not be apparent through traditional signal processing methods. For this study, wind turbine data was acquired from a wind park in Norway via standard vibration sensors at different lo- cations of the turbine’s gearbox. Both the vibration acceler- ation data and its frequency spectra were recorded at infre- quent intervals for a few seconds at high frequency and fail- ure events were labelled as either gear-tooth or ball-bearing failures. The data processing and analysis are based on a pipeline where the time series data is first split into intervals and then transformed into multi-dimensional point clouds via a time-delay embedding. The shape of the point cloud is an- alyzed with topological methods such as persistent homol- ogy to generate topology-based key health indicators based on Betti numbers, information entropy and signal persistence. Such indicators are tested for CBM and diagnosis (fault de- tection) to identify faults in wind turbines and classify them accordingly. Topological indicators are shown to be an in- teresting alternative for failure identification and diagnosis of operational failures in wind turbines.

Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. Mapper, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the codomain of the map. This cover, via a pullback operation to the domain, produces a simplicial complex connecting the data points.The resulting view of the data through a cover of the codomain offers flexibility in analyzing the data. However, it offers only a view at a fixed scale at which the cover is constructed. Inspired by the concept, we explore a notion of a tower of covers which induces a tower of simplicial complexes connected by simplicial maps, which we call multiscale mapper. We study the resulting structure, and design practical algorithms to compute its persistence diagrams efficiently. Specifically, when the domain is a simplicial complex and the map is a real-valued piecewise-linear function, the algorithm can compute the exact persistence diagram only from the 1-skeleton of the input complex. For general maps, we present a combinatorial version of the algorithm that acts only on vertex sets connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that we show to hold.