Graphcodes handle datasets that are filtered along two real-valued scale parameters.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets, such as graphs and point clouds.

Sliding windows and persistence: an application of topological methods to signal analysis.

However, in many applications there are several different parameters one might wish to vary: for example, scale and density.

Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF can be interpreted as a higher-order extension of Sankey diagrams and reduces to a dendrogram for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures higher-order inconsistencies between clusters across scales. We demonstrate through experiments the use of MCbiF Hilbert functions as topological feature maps for downstream machine learning tasks. MCbiF feature maps outperform information-based baseline features on both regression and classification tasks on synthetic sets of non-hierarchical sequences of partitions. We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social grouping patterns across time.

Graphcodes handle datasets that are filtered along two real-valued scale parameters.

Sliding windows and persistence: an application of topological methods to signal analysis.