I am just beginning to learn about topological data analysis and understand the basics. With respect to constructing a persistence diagram, I understand level sets etc. My question is regarding how components, or betti 0, join. It was my understanding that if elements of a level set are close enough to each other, that they will join to form a component, with the oldest element surviving. How do we define what is close enough?
In particular if we are looking at a n by n grid of values, how do we decide which elements are neighbours with each other? I've heard of the cross or square neighbourhood but I think there is something more to this.
I have ran into an issue when looking in the TDA package in R. The documentation states that the grid is triangulated and considers the simplices of the function values. What does this mean in terms of which grid elements are neighbours with each other? Using a cross or square neighbourhood does not seem to work here. Am I missing something, is there a piece of theory I need to know?
Any help would be appreciated.
Sample code (R):
library('TDA')
n=36
fdata = rnorm(n, mean = 5, sd = 1)
fdata = array(fdata, c(sqrt(n),sqrt(n))) Diag=gridDiag(FUNvalues = fdata)
Diag$diagram