Persistent Betti numbers
In the presence of noise one has to take into account robustness of topology. The measurement for this robustness is called persistence. Persistent Betti numbers are ranks of the persistent homology groups.
Below is an example of how we determine the persistent Betti numbers of a point cloud computed with JPlex. To be precise, these are relative Betti numbers, i.e., the edges are identified (see quotient spaces).
In the above barcodes, there is one feature in dimension 0 that has a much longer lifespan than other dimension 0 features. In dimension 1, no single feature persists significantly.
In the above barcodes, we see that in dimension 0, one feature persists longer that other dimension 0 features. In dimension 1, no single feature has a particularly long lifespan compared to other dimension 1 features. In dimension 2, one features has a significant lifespan.
By analyzing the barcodes of relative Betti numbers, we determine the dimension of our data set. In this example, the persistent Betti numbers are: $$B_0 = 1, B_1 = 0, B_2 = 0, …$$ Thus the data set constitutes a single part, and has no other topological features. Next, the persistent relative Betti numbers are: $$B_0 = 1, B_1 = 0, B_2 = 1, B_3 = 0, …$$ It follows that behind the data set is a two-dimensional plane.