Persistence Landscape — persistence

Computes the persistence landscape (Bubenik 2015) from a persistence diagram. Each (birth, death) pair contributes a tent function $$\Lambda_{(b,d)}(t) = \max(0, \min(t - b, d - t)).$$ The $k$-th landscape function $\lambda^{(k)}(t)$ is the $k$-th largest of $\{\Lambda_{(b_i,d_i)}(t)\}_i$ at each $t$. Landscapes are stable under bottleneck distance and form a Banach-space embedding of persistence diagrams.

Essential classes are excluded: a tent function is undefined for an infinitely-lived class on a finite grid, so pairs with death = Inf (VR mode) and pairs with death = 0 but birth > 0 (the clique-mode encoding of an essential class) are dropped before the landscape is built. When no finite pair remains in the requested dimension, every landscape function is zero on the grid.

Usage

persistence_landscape(ph, k_max = 5L, dimension = 1L, t_grid = NULL)

# S3 method for class 'persistence_landscape'
print(x, ...)

# S3 method for class 'persistence_landscape'
plot(x, ...)

Arguments

ph: A persistent_homology object or a data.frame with columns dimension, birth, death.
k_max: Maximum landscape index to compute (default 5). Must be a single positive integer.
dimension: Integer scalar – which homology dimension to compute the landscape for. Default 1.
t_grid: Numeric vector of evaluation points. NULL (default) uses an even grid of 200 points covering the union of pair intervals.
x: For the print() and plot() methods: an object of class persistence_landscape.
...: In plot.persistence_landscape() and print.persistence_landscape(): Ignored.

Value

A persistence_landscape object with:

landscape: Data frame: k, t, value.
dimension: Integer scalar.
k_max: Integer scalar.
t_grid: Numeric vector.

In print.persistence_landscape(): The input, invisibly.

In plot.persistence_landscape(): A ggplot.

References

Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research 16, 77-102.

Examples

mat <- matrix(c(0, .6, .5, .6, 0, .4, .5, .4, 0), 3, 3)
rownames(mat) <- colnames(mat) <- c("A","B","C")
ph <- persistent_homology(mat, n_steps = 5)
pl <- persistence_landscape(ph, k_max = 3, dimension = 0)