landscape: The Persistence Landscape Function

Description

This function computes the landscape function corresponding to a given persistence diagram.

Usage

landscape(Diag, dimension = 1, KK = 1, 
          tseq=seq( min(Diag[,2:3]), max(Diag[,2:3]), length=500) )

Arguments

Diag

an object of class diagram or a $P$ by $3$ matrix, storing a persistence diagram with columns: "dimension", "Birth", "Death".

dimension

the dimension of the topological features under consideration. Default is 1 (loops).

the order of the landscape function. Default is 1. (First Landscape function).

tseq

a vector of values at which the landscape function is evaluated.

Value

Returns a numeric vector of the same length of tseq, with the values of the landscape function evaluated at each point of tseq.

References

Peter Bubenik, (2012), "Statistical topology using persistence landscapes", arXiv1207.6437.

Frederic Chazal, Brittany T. Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman, (2014), "Stochastic Convergence of Persistence Landscapes and Silhouettes", Proceedings of the 30th Symposium of Computational Geometry (SoCG). (arXiv:1312.0308)

Examples

Run this code

XX1 = circleUnif(30)
XX2 = circleUnif(30, r=2) +3
XX=rbind(XX1,XX2)

DiagLim=5
maxdimension=1
Diag=ripsDiag(XX,maxdimension,DiagLim, printStatus=TRUE)

#persistence landscape
tseq=seq(0,DiagLim, length=1000)
Land= landscape(Diag, dimension=1, KK=1, tseq)

plot(tseq, Land, type="l", xlab="t", ylab="landscape", asp=1)