[[TDA|Topological Data Analysis]] ([[TDA]]) is a field of data analysis that combines ideas from topology, algebraic topology, and computational geometry to study datasets. It aims to extract meaningful information about the underlying structure and shape of the data. TDA assumes that the data is not randomly distributed but has an inherent geometric structure. By applying techniques from topology, TDA can capture and analyze this structure. It focuses on understanding the shape, connectivity, and relationships between different data points. The main goal of TDA is to provide a higher-level understanding of complex datasets that may not be easily captured by traditional statistical or machine learning techniques. It can uncover patterns, clusters, and topological features in the data that might be missed by other methods. One of the key tools in TDA is persistent homology. Homology is a mathematical concept that captures the presence of holes or voids in a dataset at different scales. Persistent homology extends this idea to identify significant topological features that persist across multiple scales. This allows for a more robust characterization of the data's shape. TDA also utilizes techniques such as simplicial complexes, which represent higher-dimensional spaces through their building blocks called simplices (e.g., points, lines, triangles). By constructing these complexes from data points and analyzing their properties, TDA can reveal important topological information. Applications of TDA span various domains such as biology, neuroscience, social sciences, computer vision, and materials science. For example, TDA has been used to analyze brain connectivity networks to understand neurological disorders or to study protein structures for drug discovery. ## What is filtration in TDA [[Filtration]] in topological data analysis ([[TDA]]) refers to the process of gradually introducing and ordering the simplicial complexes or other topological structures based on a parameter, typically called the filtration parameter. In TDA, data is often represented as a simplicial complex, which is a mathematical structure composed of vertices, edges, triangles, and higher-dimensional simplices. The filtration parameter determines which simplices are included in the complex at each step of the filtration. The filtration process starts with an empty complex and gradually adds simplices based on their relevance or significance to the data. The filtration parameter could represent various attributes such as distance, density, density gradient, or any other characteristic of the data that is relevant to the analysis. By varying the filtration parameter, TDA captures different levels of connectivity and structure in the data. This allows for a multi-scale analysis where topological features at different resolutions can be studied. The resulting sequence of complexes obtained through filtration can be visualized using a persistence diagram or barcode to identify persistent topological features or homology classes. Overall, filtration in TDA is a fundamental concept that enables the exploration and analysis of topological properties underlying complex datasets by gradually building up their simplicial representations based on a chosen parameter. # Conclusion In summary, Topological Data Analysis is an emerging field that aims to extract meaningful structure and shape information from complex datasets using tools from topology. It provides a unique perspective for understanding high-dimensional data and has potential applications in diverse scientific disciplines. # References ```dataview Table title as Title, authors as Authors where contains(subject, "TDA") or contains(subject, "Topological Data Analysis") or contains(subject, "TDL") sort title, authors, modified ```