# multipBootstrap: Multiplier Bootstrap for Persistence Landscapes and... In TDA: Statistical Tools for Topological Data Analysis

 multipBootstrap R Documentation

## 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.

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)

`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.