Scientific data has been growing in both size and complexity across the modern physical, engineering, life and social sciences. Spatial structure, for example, is a hallmark of many of the most important real-world complex systems, but its analysis is fraught with statistical challenges. Topological data analysis can provide a powerful computational window on complex systems. Here we present a framework to extend and interpret persistent homology summaries to analyse spatial data across multiple scales. We introduce hyperTDA, a topological pipeline that unifies local (e.g. geodesic) and global (e.g. Euclidean) metrics without losing spatial information, even in the presence of noise. Homology generators offer an elegant and flexible description of spatial structures and can capture the information computed by persistent homology in an interpretable way. Here the information computed by persistent homology is transformed into a weighted hypergraph, where hyperedges correspond to homology generators. We consider different choices of generators (e.g. matroid or minimal) and find that centrality and community detection are robust to either choice. We compare hyperTDA to existing geometric measures and validate its robustness to noise. We demonstrate the power of computing higher-order topological structures on spatial curves arising frequently in ecology, biophysics, and biology, but also in high-dimensional financial datasets. We find that hyperTDA can select between synthetic trajectories from the landmark 2020 AnDi challenge and quantifies movements of different animal species, even when data is limited.
翻译:例如,空间结构是许多最重要的真实世界复杂系统的标志,但其分析充满了统计挑战。 地形数据分析可以提供一个强大的计算窗口,在复杂的系统中提供强大的计算窗口。 在这里,我们提出了一个框架,用于扩展和解释持续的同质摘要,以分析多种尺度的空间数据。 我们引入了超高TDA, 这是一种将本地(例如大地测量)和全球(例如欧洲大陆)指标统一起来的地形管道,即使有噪音,也不丧失空间信息。 同性恋生成器对空间结构作了优雅和灵活的描述,并能够以可解释的方式捕捉由持久性同质学计算的信息。 这里,由持久性同质学计算的信息被转化成一个加权的超强度图, 高端与同质生成器相匹配。 我们考虑不同发电机的选择(例如,基质或最小), 发现核心和社区检测在两种选择中都很强大。 我们将超高端TDA与现有的几何计量尺度和确认其稳健性与噪声。 我们经常在高空的生物学结构中发现高端数据结构结构中, 高端数据在生物物理结构中发现高端结构结构结构结构结构结构结构结构中, 。