View source: R/bootstrapDiagram.R
bootstrapDiagram | R Documentation |
The function bootstrapDiagram
computes a (1-alpha)
confidence set for the Persistence Diagram of a filtration of sublevel sets (or superlevel sets) of a function evaluated over a grid of points. The function returns the (1-alpha
) quantile of B
bottleneck distances (or Wasserstein distances), computed in B
iterations of the bootstrap algorithm.
bootstrapDiagram( X, FUN, lim, by, maxdimension = length(lim) / 2 - 1, sublevel = TRUE, library = "GUDHI", B = 30, alpha = 0.05, distance = "bottleneck", dimension = min(1, maxdimension), p = 1, parallel = FALSE, printProgress = FALSE, weight = NULL, ...)
X |
an n by d matrix of coordinates, used by the function |
FUN |
a function whose inputs are 1) an n by d matrix of coordinates |
lim |
a 2 by d matrix, where each column specifies the range of each dimension of the grid, over which the function |
by |
either a number or a vector of length d specifying space between points of the grid in each dimension. If a number is given, then same space is used in each dimension. |
maxdimension |
a number that indicates the maximum dimension to compute persistent homology to. The default value is d - 1, which is (dimension of embedding space - 1). |
sublevel |
a logical variable indicating if the Persistence Diagram should be computed for sublevel sets ( |
library |
a string specifying which library to compute the persistence diagram. The user can choose either the library |
B |
the number of bootstrap iterations. The default value is |
alpha |
The function |
distance |
a string specifying the distance to be used for persistence diagrams: either |
dimension |
|
p |
if |
parallel |
logical: if |
printProgress |
if |
weight |
either NULL, a number, or a vector of length n. If it is NULL, weight is not used. If it is a number, then same weight is applied to each points of |
... |
additional parameters for the function |
The function bootstrapDiagram
uses gridDiag
to compute the persistence diagram of the input function using the entire sample. Then the bootstrap algorithm, for B
times, computes the bottleneck distance between the original persistence diagram and the one computed using a subsample. Finally the (1-alpha
) quantile of these B
values is returned. See (Chazal, Fasy, Lecci, Michel, Rinaldo, and Wasserman, 2014) for discussion of the method.
The function bootstrapDiagram
returns the (1-alpha
) quantile of the values computed by the bootstrap algorithm.
The function bootstrapDiagram
uses the C++ library Dionysus for the computation of bottleneck and Wasserstein distances. See references.
Jisu Kim and Fabrizio Lecci
Chazal F, Fasy BT, Lecci F, Michel B, Rinaldo A, Wasserman L (2014). "Robust Topological Inference: Distance-To-a-Measure and Kernel Distance." Technical Report.
Wasserman L (2004), "All of statistics: a concise course in statistical inference." Springer.
Morozov D (2007). "Dionysus, a C++ library for computing persistent homology." https://www.mrzv.org/software/dionysus/
bottleneck
, bootstrapBand
,
distFct
, kde
, kernelDist
, dtm
,
summary.diagram
, plot.diagram
## confidence set for the Kernel Density Diagram # input data n <- 400 XX <- circleUnif(n) ## Ranges of the grid Xlim <- c(-1.8, 1.8) Ylim <- c(-1.6, 1.6) lim <- cbind(Xlim, Ylim) by <- 0.05 h <- .3 #bandwidth for the function kde #Kernel Density Diagram of the superlevel sets Diag <- gridDiag(XX, kde, lim = lim, by = by, sublevel = FALSE, printProgress = TRUE, h = h) # confidence set B <- 10 ## the number of bootstrap iterations should be higher! ## this is just an example alpha <- 0.05 cc <- bootstrapDiagram(XX, kde, lim = lim, by = by, sublevel = FALSE, B = B, alpha = alpha, dimension = 1, printProgress = TRUE, h = h) plot(Diag[["diagram"]], band = 2 * cc)
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