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This chapter aims to discuss a method of quantifying the ‘shape’ of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of data, from the homology of a simplicial complex, calculated over a range of values. The required background the...
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This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of data, from the homology of a simplicial complex, calculated over a range of values. The required background theor...
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... The fatigue of structures under dynamic loading is difficult to monitor with traditional methods. Non-Destructive Testing and Evaluation (NDT/NDE) aim to inspect materials and/or structures for their present condition without altering their serviceability [77,78]. On the other hand, Structural Health Monitoring (SHM) characterizes and evaluates the condition of materials and/or structures for the performance of an in-service structure [79,80]. ...
Structural Health Monitoring (SHM) is critical for ensuring the longevity and safety of civil engineering structures. Traditional SHM techniques often struggle with the high-dimensional, complex data generated by modern sensor technologies. Topological Data Analysis (TDA) offers a robust alternative by capturing the underlying topological features of this data, enabling more accurate and efficient monitoring. This paper explores the integration of TDA into SHM, discussing its theoretical foundations, practical benefits, and challenges. We review recent advancements, including deep learning enhancements and real-world applications, highlighting how TDA can improve damage detection and predictive maintenance. Future research directions are proposed to further the adoption of TDA in SHM, emphasizing the need for real-time, industrial-grade solutions.
... The fatigue of structures under dynamic loading is difficult to monitor with traditional methods. Non-Destructive Testing and Evaluation (NDT/NDE) aim to inspect materials and/or structures for their present condition without altering their serviceability [77,78]. On the other hand, Structural Health Monitoring (SHM) characterizes and evaluates the condition of materials and/or structures for the performance of an in-service structure [79,80]. ...
Structural Health Monitoring (SHM) is critical for ensuring the longevity and safety of civil engineering structures. Traditional SHM techniques often struggle with the high-dimensional, complex data generated by modern sensor technologies. Topological Data Analysis (TDA) offers a robust alternative by capturing the underlying topological features of this data, enabling more accurate and efficient monitoring. This paper explores the integration of TDA into SHM, discussing its theoretical foundations, practical benefits, and challenges. We review recent advancements, including deep learning enhancements and real-world applications, highlighting how TDA can improve damage detection and predictive maintenance. Future research directions are proposed to further the adoption of TDA in SHM, emphasizing the need for real-time, industrial-grade solutions.
... The fatigue of structures under dynamic loading is difficult to monitor with traditional methods. Non-Destructive Testing and Evaluation (NDT/NDE) aim to inspect materials and/or structures for their present condition without altering their serviceability [77,78]. On the other hand, Structural Health Monitoring (SHM) characterizes and evaluates the condition of materials and/or structures for the performance of an in-service structure [79,80]. ...
Structural Health Monitoring (SHM) is critical for ensuring the longevity and safety of civil engineering structures. Traditional SHM techniques often struggle with the high-dimensional, complex data generated by modern sensor technologies. Topological Data Analysis (TDA) offers a robust alternative by capturing the underlying topological features of this data, enabling more accurate and efficient monitoring. This paper explores the integration of TDA into SHM, discussing its theoretical foundations, practical benefits, and challenges. We review recent advancements, including deep learning enhancements and real-world applications, highlighting how TDA can improve damage detection and predictive maintenance. Future research directions are proposed to further the adoption of TDA in SHM, emphasizing the need for real-time, industrial-grade solutions.
... The PH and TDA tools are widely known because Euler used them to solve the Seven Bridges of Königsberg problem. These analysis techniques allow one to reduce the problems into their main features, which turn the increasingly used high-dimensional data into easy-to-see and easy-to-understand problems [6,7]. ...
Persistent Homology (PH) analysis is a powerful tool for understanding many relevant topological features from a given dataset. PH allows finding clusters, noise, and relevant connections in the dataset. Therefore, it can provide a better view of the problem and a way of perceiving if a given dataset is equal to another, if a given sample is relevant, and how the samples occupy the feature space. However, PH involves reducing the problem to its simplicial complex space, which is computationally expensive and implementing PH in such Resource-Scarce Embedded Systems (RSES) is an essential add-on for them. However, due to its complexity, implementing PH in such tiny devices is considerably complicated due to the lack of memory and processing power. The following paper shows the implementation of 0-Dimensional Persistent Homology Analysis in a set of well-known RSES, using a technique that reduces the memory footprint and processing power needs of the 0-Dimensional PH algorithm. The results are positive and show that RSES can be equipped with this real-time data analysis tool.
Geometric realization of simplicial complexes makes them a unique representation of complex systems. The existence of local continuous spaces at the simplices level with global discrete connectivity between simplices makes the analysis of dynamical systems on simplicial complexes a challenging problem. In this work, we provide some examples of complex systems in which this representation would be a more appropriate model of real-world phenomena. Here, we generalize the concept of metaplexes to embrace that of geometric simplicial complexes, which also includes the definition of dynamical systems on them. A metaplex is formed by regions of a continuous space of any dimension interconnected by sinks and sources that works controlled by discrete (graph) operators. The definition of simplicial metaplexes given here allows the description of the diffusion dynamics of this system in a way that solves the existing problems with previous models. We make a detailed analysis of the generalities and possible extensions of this model beyond simplicial complexes, e.g., from polytopal and cell complexes to manifold complexes, and apply it to a real-world simplicial complex representing the visual cortex of a macaque.
As more and more devices are being deployed across networks to gather data and use them to perform intelligent tasks, it is vital to have a tool to perform real-time data analysis. Data are the backbone of Machine Learning models, the core of intelligent systems. Therefore, verifying whether the data being gathered are similar to those used for model building is essential. One fantastic tool for the performance of data analysis is the 0-Dimensional Persistent Diagrams, which can be computed in a Resource-Scarce Embedded System (RSES), a set of memory and processing-constrained devices that are used in many IoT applications because they are cost-effective and reliable. However, it is challenging to compare Persistent Diagrams, and Persistent Landscapes are used because they allow Persistent Diagrams to be passed to a space where the mean concept is well-defined. The following work shows how one can perform a Persistent Landscape analysis in an RSES. It also shows that the distance between two Persistent Landscapes makes it possible to verify whether two devices collect the same data. The main contribution of this work is the implementation of Persistent Landscape analysis in an RSES, which is not provided in the literature. Moreover, it shows that devices can now verify, in real-time, whether they can trust the data being collected to perform the intelligent task they were designed to, which is essential in any system to avoid bugs or errors.
