TDA (version 1.9.1)

ripsFiltration: Rips Filtration

Description

The function ripsFiltration computes the Rips filtration built on top of a point cloud.

Usage

ripsFiltration(
    X, maxdimension, maxscale, dist = "euclidean",
    library = "GUDHI", printProgress = FALSE)

Value

The function ripsFiltration returns a list with the following elements:

cmplx

a list representing the complex. Its i-th element represents the vertices of i-th simplex.

values

a vector representing the filtration values. Its i-th element represents the filtration value of i-th simplex.

increasing

a logical variable indicating if the filtration values are in increasing order (TRUE) or in decreasing order (FALSE).

coordinates

only if dist = "euclidean": a matrix representing the coordinates of vertices. Its i-th row represents the coordinate of i-th vertex.

Arguments

X

If dist="euclidean", X is an \(n\) by \(d\) matrix of coordinates, where \(n\) is the number of points in the \(d\)-dimensional euclidean space. If dist="arbitrary", X is an \(n\) by \(n\) matrix of distances of \(n\) points.

maxdimension

integer: max dimension of the homological features to be computed. (e.g. 0 for connected components, 1 for connected components and loops, 2 for connected components, loops, voids, etc.)

maxscale

number: maximum value of the rips filtration.

dist

"euclidean" for Euclidean distance, "arbitrary" for an arbitrary distance given in input as a distance matrix.

library

a string specifying which library to compute the Rips filtration. If dist = "euclidean", the user can use either the library "GUDHI" or "Dionysus". If dist = "arbitrary", the user can use the library "Dionysus". The default value is "GUDHI" if dist = "euclidean", and "Dionysus" if dist == "arbitrary". When "GUDHI" is used for dist = "arbitrary", "Dionysus" is implicitly used.

printProgress

logical: if TRUE, a progress bar is printed. The default value is FALSE.

Author

Jisu Kim

Details

For Rips filtration based on Euclidean distance of the input point cloud, the user can decide to use either the C++ library GUDHI or Dionysus. For Rips filtration based on arbitrary distance, the user can use the C++ library Dionysus. See refereneces.

References

Maria C (2014). "GUDHI, Simplicial Complexes and Persistent Homology Packages." https://project.inria.fr/gudhi/software/ .

Morozov D (2007). "Dionysus, a C++ library for computing persistent homology". https://www.mrzv.org/software/dionysus/

Edelsbrunner H, Harer J (2010). "Computational topology: an introduction." American Mathematical Society.

See Also

ripsDiag, filtrationDiag

Examples

n <- 5
X <- cbind(cos(2*pi*seq_len(n)/n), sin(2*pi*seq_len(n)/n))
maxdimension <- 1
maxscale <- 1.5

FltRips <- ripsFiltration(X = X, maxdimension = maxdimension,
               maxscale = maxscale, dist = "euclidean", library = "GUDHI",
               printProgress = TRUE)

# plot rips filtration
lim <- rep(c(-1, 1), 2)
plot(NULL, type = "n", xlim = lim[1:2], ylim = lim[3:4],
    main = "Rips Filtration Plot")
for (idx in seq(along = FltRips[["cmplx"]])) {
  polygon(FltRips[["coordinates"]][FltRips[["cmplx"]][[idx]], , drop = FALSE],
      col = "pink", border = NA, xlim = lim[1:2], ylim = lim[3:4])
}
for (idx in seq(along = FltRips[["cmplx"]])) {
  polygon(FltRips[["coordinates"]][FltRips[["cmplx"]][[idx]], , drop = FALSE],
      col = NULL, xlim = lim[1:2], ylim = lim[3:4])
}  
points(FltRips[["coordinates"]], pch = 16)