multipBootstrap: Multiplier Bootstrap for Persistence Landscapes and...

In TDA: Statistical Tools for Topological Data Analysis

Multiplier Bootstrap for Persistence Landscapes and Silhouettes

Description

The function multipBootstrap computes a confidence band for the average landscape (or the average silhouette) using the multiplier bootstrap.

Usage

multipBootstrap(
    Y, B = 30, alpha = 0.05, parallel = FALSE,
    printProgress = FALSE)

Arguments

Y	an N by m matrix of values of N persistence landscapes (or silhouettes) evaluated over a 1 dimensional grid of length m.
B	the number of bootstrap iterations.
alpha	multipBootstrap returns a 1-alpha confidence band for the mean landscape (or silhouette).
parallel	logical: if TRUE the bootstrap iterations are parallelized, using the library parallel.
printProgress	logical: if TRUE a progress bar is printed. The default value is FALSE.

Details

See Algorithm 1 in the reference.

Value

The function multipBootstrap returns a list with the following elements:

width	number: half of the width of the unfiorm confidence band; that is, the distance of the upper and lower limits of the band from the empirical average landscape (or silhouette).
mean	a numeric vector of length m, storing the values of the empirical average landscape (or silhouette) over a 1 dimensional grid of length m.
band	an m by 2 matrix that stores the values of the lower limit of the confidence band (first column) and upper limit of the confidence band (second column), evaluated over a 1 dimensional grid of length m.

Author(s)

Fabrizio Lecci

References

Chazal F, Fasy BT, Lecci F, Rinaldo A, Wasserman L (2014). "Stochastic Convergence of Persistence Landscapes and Silhouettes." Proceedings of the 30th Symposium of Computational Geometry (SoCG). (arXiv:1312.0308)

See Also

landscape, silhouette

Examples

nn <- 3000  #large sample size
mm <- 50    #small subsample size
NN <- 5     #we will compute NN diagrams using subsamples of size mm

XX <- circleUnif(nn)  ## large sample from the unit circle

DiagLim <- 2
maxdimension <- 1
tseq <- seq(0, DiagLim, length = 1000)

Diags <- list()  #here we will store the NN rips diagrams
                 #constructed using different subsamples of mm points
#here we'll store the landscapes
Lands <- matrix(0, nrow = NN, ncol = length(tseq))

for (i in seq_len(NN)){
  subXX <- XX[sample(seq_len(nn), mm), ]
  Diags[[i]] <- ripsDiag(subXX, maxdimension, DiagLim)
  Lands[i, ] <- landscape(Diags[[i]][["diagram"]], dimension = 1, KK = 1, tseq)
}

## now we use the NN landscapes to construct a confidence band
B <- 50
alpha <- 0.05
boot <- multipBootstrap(Lands, B, alpha)

LOWband <- boot[["band"]][, 1]
UPband <- boot[["band"]][, 2]
MeanLand <- boot[["mean"]]

plot(tseq, MeanLand, type = "l", lwd = 2, xlab = "", ylab = "",
     main = "Mean Landscape with band", ylim = c(0, 1.2))
polygon(c(tseq, rev(tseq)), c(LOWband, rev(UPband)), col = "pink")
lines(tseq, MeanLand, lwd = 1, col = 2)

TDA documentation built on Feb. 16, 2023, 6:35 p.m.