Persistent homology records how the homology of a filtration evolves as its parameter increases. A filtration is a nested family , typically the sublevel sets of a function or a Vietoris–Rips complex grown by radius. Each -dimensional cycle is born at one value and dies at another, when it becomes a boundary or merges with an older class. The multiset of (birth, death) pairs forms the persistence diagram, equivalently the barcode. Points far from the diagonal are robust features, whereas points near it are noise. Algebraically the filtration is a persistence module, a functor from the poset to , and the structure theorem for finitely generated modules over splits it into interval modules, which are exactly the bars. It is the standard multiscale invariant of topological data analysis (Edelsbrunner–Letscher–Zomorodian; Zomorodian–Carlsson).