landscape {TDA}

The Persistence Landscape Function

Description

The function landscape 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 colnames: "dimension", "Birth", "Death".

dimension

the dimension of the topological features under consideration. The default value is 1 (loops).

KK

a vector: the order of the landscape function. The default value is 1. (First Landscape function).

tseq

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

Value

The function landscape returns a numeric matrix with the number of row as the length of tseq and the number of column as the length of KK. The value at ith row and jth column represents the value of the KK[j]-th landscape function evaluated at tseq[i].

Author(s)

Fabrizio Lecci

References

Bubenik P (2012). "Statistical topology using persistence landscapes." arXiv:1207.6437.

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

silhouette

Examples

Diag <- matrix(c(0, 0, 10, 1, 0, 3, 1, 3, 8), ncol = 3, byrow = TRUE)
DiagLim <- 10
colnames(Diag) <- c("dimension", "Birth", "Death")

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

par(mfrow = c(1,2))
plot.diagram(Diag)
plot(tseq, Land, type = "l", xlab = "t", ylab = "landscape", asp = 1)