Persistenzdiagrammen

Persistenzdiagrammen, also known as persistence diagrams, are a fundamental tool in applied topology, specifically within the field of topological data analysis (TDA). They are used to visualize and summarize the topological features of a data set, such as connected components, loops, and voids. A persistence diagram is a scatter plot where each point represents a topological feature. The x-coordinate of a point typically signifies the "birth" time or scale at which a feature appears, and the y-coordinate represents its "death" time or scale at which it disappears. The difference between these two values, y-x, is referred to as the persistence of the feature. Features with higher persistence are considered more significant or robust.

The construction of a persistence diagram begins with a filtration, which is a sequence of nested topological spaces. This filtration is often derived from a point cloud by gradually growing balls around each point or by considering a thresholded distance function. As the filtration progresses, topological features emerge and disappear. A persistence diagram captures these events. The diagonal line y=x in the diagram represents features that are born and die at the same scale, which are generally considered noise. Points far from the diagonal indicate persistent topological structures. These diagrams provide a compact and informative summary of the shape of the data, invariant under certain transformations, making them valuable for tasks like shape recognition, material science, and image analysis.