Basics

In this section, we will show the basic usage, construction and plotting of persistence intervals and diagrams. We start by importing the library.

using PersistenceDiagrams

A PersistenceDiagram is composed of persistence intervals, so we will introduce those first.

finite_interval = PersistenceInterval(1, 2)

PersistenceInterval(1.0, 2.0)

We can access, the birth death and persistence of the interval with the appropriately named functions.

birth(finite_interval), death(finite_interval), persistence(finite_interval)

(1.0, 2.0, 1.0)

To represent infinite intervals, we use the floating point value Inf.

infinite_interval = PersistenceInterval(0, Inf)

PersistenceInterval(0.0, Inf)

The finiteness of the interval can be checked with the isfinite function from Base.

isfinite(infinite_interval)

false

A persistence interval can also have a representative attached. There are no limitations on what type the representative can be, but it's intended to be used with things such as representative cycles.

interval_with_rep = PersistenceInterval(3, 7, [1, 2, 3])

PersistenceInterval(3.0, 7.0) with representative:
3-element Array{Int64,1}:
 1
 2
 3

representative(interval_with_rep)

3-element Array{Int64,1}:
 1
 2
 3

A PersistenceDiagram is essentially the same as a Vector of PersistenceInterval, but again has additional information attached. A diagram can be constructed as follows.

diagram_1 = PersistenceDiagram(0, [finite_interval, infinite_interval])

2-element 0-dimensional PersistenceDiagram:
 [1.0, 2.0)
 [0.0, ∞)

We can access the dimension of a diagram with the dim function.

dim(diagram_1)

0

Functions that work on arrays should all work on PersistenceDiagrams as well.

sort(diagram_1, by=death, rev=true)

2-element 0-dimensional PersistenceDiagram:
 [0.0, ∞)
 [1.0, 2.0)

diagram_1[1]

PersistenceInterval(1.0, 2.0)

Optionally, we can add a threshold to the diagram. The threshold is currently only used in plotting. We will show the effect of the threshold argument later.

diagram_1_thresh = PersistenceDiagram(0, [finite_interval, infinite_interval], threshold=5)

2-element 0-dimensional PersistenceDiagram:
 [1.0, 2.0)
 [0.0, ∞)

threshold(diagram_1_thresh)

5.0

threshold(diagram_1)

Inf

Alternatively, we can also construct a diagram by passing it an array of tuples.

diagram_2 = PersistenceDiagram(1, [(0, 1), (4, 7), (5, Inf)])

3-element 1-dimensional PersistenceDiagram:
 [0.0, 1.0)
 [4.0, 7.0)
 [5.0, ∞)

We plot a diagram by calling the plot function from Plots.jl.

using Plots

plot(diagram_1_thresh)

Note that the infinty line was placed at the value of the threshold.

Passing persistence=true will plot the diagram in a birth vs. persistence coordinates.

plot(diagram_1, persistence=true)

To plot the diagram as a barcode, we use the barcode function.

barcode(diagram_2)

If we want to plot multiple diagrams on the same plot, simply pass an array of PersistenceDiagrams or multiple arguments to the functions.

plot([diagram_1_thresh, diagram_2])

barcode(diagram_1_thresh, diagram_2)

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