TDApplied (version 3.0.3)

vr_graphs: Compute Vietoris-Rips graphs of a dataset at particular epsilon radius values.

Description

Persistence diagrams computed from Rips-Vietoris filtrations contain information about distance radius scales at which topological features of a dataset exist, but the features can be challenging to visualize, analyze and interpret. In order to help solve this problem the `vr_graphs` function computes the 1-skeleton (i.e. graph) of Rips complexes at particular radii, called "Vietoris-Rips graphs" (VR graphs) in the literature.

Usage

vr_graphs(X, distance_mat = FALSE, eps, return_clusters = TRUE)

Value

A list with a `vertices` field, containing the rownames of `X`, and then a list `graphs` one (named) entry for each value in `eps`. Each entry is a list with a `graph` field, storing the (undirected) edges in the Rips-Vietoris complex in matrix format, and a `clusters` field, containing vectors of the data indices (or row names) in each connected component of the Rips graph.

Arguments

X

either a point cloud data frame/matrix, or a distance matrix.

distance_mat

a boolean representing if the input `X` is a distance matrix, default value is `FALSE`.

eps

a numeric vector of the positive scales at which to compute the Rips-Vietoris complexes, i.e. all edges at most the specified values.

return_clusters

a boolean determining if the connected components (i.e. data clusters) of the complex should be explicitly returned, default is `TRUE`.

Author

Shael Brown -

Details

This function may be used in conjunction with the igraph package to visualize the graphs (see plot_vr_graph).

References

A Zomorodian, The tidy set: A minimal simplicial set for computing homology of clique complexes in Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, SoCG ’10. (Association for Computing Machinery, New York, NY, USA), p. 257–266 (2010).

See Also

plot_vr_graph for plotting VR graphs.

Examples

if(require("TDAstats") & require("igraph"))
{
  # simulate data from the unit circle and calculate 
  # its diagram
  df <- TDAstats::circle2d[sample(1:100,25),]
  diag <- TDAstats::calculate_homology(df,
                                       dim = 1,
                                       threshold = 2)
  
  # get minimum death radius of any data cluster
  min_death_H0 <- 
  min(diag[which(diag[,1] == 0),3L])
  
  # get birth and death radius of the loop
  loop_birth <- as.numeric(diag[nrow(diag),2L])
  loop_death <- as.numeric(diag[nrow(diag),3L])

  # compute VR graphs at radii half of 
  # min_death_H0 and the mean of loop_birth and 
  # loop_death, returning clusters
  graphs <- vr_graphs(X = df,eps = 
  c(0.5*min_death_H0,(loop_birth + loop_death)/2))

  # verify that there are 25 clusters for the smaller radius
  length(graphs$graphs[[1]]$clusters)
  
}