Topological Data Analysis (TDA) looks for interesting topological features of a dataset, by measuring its “persistent homology”. Roughly speaking, 0th dimensional homology counts how many clusters the data is in. the 1st dimensional homology counts, roughly, how many circles or loops are in the data. The 2nd dimensional homology will detect and count bubbles, and higher dimensional homology will count higher-dimensional “bubbles”!
This project from late 2016 used simulated data to show some different possible shapes of 3D data, and how to interpret the corresponding persistent homology “barcodes”.