Repository for producing random sampling from unit balls in hyperbolic, Euclidian and spherical spaces, and analyzing them using Betti curves.
The code is written in the Julia language and requires Julia v.0.7.
*.jl files contain necessary functions, and the notebooks illustrate the use and show the plots comparing the three geometries.
GeometricSampling.jl -- functions for sampling and distance
AverageBettis.jl -- functions for computing and plotting average Betti curves
sampling.ipynb -- notebook illustrating sampling
betti_curves.ipynb -- notebook containing average Betti curves
GeometricSampling.jl requires the following Julia packages: LinearAlgebra
AverageBettis.jl requires the following packages: Plots, Eirene, Ripser, Statistics
To intall a package, open Julia command line promt and type ].
The prompt should change from julia> to (vX.X) pkg> where X.X is the installed version of Julia, indicating that you enterted the package manager. Use add to install packages. You can install multiple packages by listing them after add, separated by spaces.
(v0.7) pkg> add LinearAlgebra Plots Eirene Ripser Statistics
To update all packages, run
(v0.7) pkg> update
To exit the package manager and get back to Julia REPL, use backspace or ^C.
This isn't a package (yet), so to use the functions, download the file into the working directory and run:
julia> include("[filename].jl")
GeometricSampling.jl contains functions for sampling from different geometries and coomputing distances. Only sampling within a ball of radius 1 and in curvature 0, +1, -1 are implemented as of 02/08/2019.
sample_ball(d::Int, numofpts=1, curvature=0.0, radius=1.0)::Array{Array{Float64,1},1}
Uniformly sample n=numofpts points in a ball of dimension d or radius radius, in a space with constant curvature curvature.
Note: This returns an array of coordinates of the Euclidian representation of the points (i.e. points within the Poincare disk for curvature < 0, or points in the upper d+1 hemisphere for curvature > 0).
sample_hyp(d::Int, numofpts=1, curvature=-1.0, radius=1.0)::Array{Array{Float64,1},1}
sample_euc(d::Int, numofpts=1, radius=1.0)::Array{Array{Float64,1},1}
sample_sph(d::Int, numofpts=1, curvature= 1.0, radius=1.0)::Array{Array{Float64,1},1}
Aliases for sample_ball in hyperbolic/Euclidian/spherical space respectively.
sample_sphere(d::Int, numofpts::Int=1, radius=1.0)::Array{Array{Float64,1},1}
Uniformly sample n=numofpts points on the surface of a sphere in d-dimensional Euclidian space (so a (d-1)-sphere) of radius radius.
distance_matrix(pts::Array{Array{Float64,1},1}; curvature=0.0)::Array{Float64, 2}
Returns the distance matrix for the collection of points pts, i.e. the (i,j)th entry is the distance between pts[i] and pts[j], in the space with curvature curvature.
hyp_distance(pts::Array{Array{Float64,1},1}; curvature=-1.0)
euc_distance(pts::Array{Array{Float64,1},1})
sph_distance(pts::Array{Array{Float64,1},1}; curvature=1.0)
Aliases for distance_matrix in the hyperbolic, Euclidian, and spherical space respectively.
rejection_sampling(dens::Function, maxval::Float64, numofpts=1)::Array{Array{Float64,1},1}
Perform the simplest version of the rejection sampling of n=numofpts points from the density function dens, assuming that the support of the density function is [0, maxval]. The proposal distribution is taken to be the unifrom distribution on [0, dens(maxval)].
to_density(matr::Array{T,2})
to_density!(matr::Array{T,2})
Convert a symmetric n x n matrix to a density matrix, i.e. replace matr[i,j] with the the number of entries in the upper triangle of matr that are less than matr[i,j], divided by \binom{n}{2}. This ensures that the matrix entries are on [0,1] scale while preserving the Vietoris-Rips complex of the matrix.
AverageBettis.jl contains functions for computing and plotting average Betti curves.
bettis(matr::Array{Float64,2},
maxdim::Int;
mintime::Float64 = -Inf,
maxtime::Float64 = Inf
numofsteps::Int = Inf
method::Symbol = :ripser)::Array{Float64,2}
Compute Betti numbers of the Vietoris-Rips complex defined by the distance matrix matr up to dimension maxdim. Parameters mintime, maxtime, numofsteps define the filtration, where mintime and maxtime is the start and end points of the filtration, and numofsteps is the number of steps in the filtration. Parameter method determines the software to use for persistent homology (possible values method=:ripser for Ripser.jl and method=:eirene for Eirene.jl).
Returns numofsteps x maxdim matrix, where (i,j)'th entry is the jth Betti number at filtration step i. Betti_0 is discarded.
average_bettis(arrs::Array{Array{Float64,2},1})::Array{Float64,2}
Assuming arrs is array of outputs of bettis(..) of same size, find average values for Betti numbers for each point in the filtration for each dimesion.
std_bettis(arrs::Array{Array{Float64,2},1})::Array{Float64,2}
Assuming arrs is array of outputs of bettis(..) of same size, find standard deviations for Betti numbers for each point in the filtration for each dimesion.
plot_averages(xvals::Array{Float64,1},
means::Array{Float64,1},
stds::Array{Float64,1};
ribbon::Bool=true,
label::String = "",
linestyle = :solid,
color = :auto)
plot_averages!(xvals, means, stds; ribbon=true, label="", linestyle=:solid, color=:auto)
Plot average Betti curves means with standard deviations stds at filtration values given by xvals. Ribbon parameter determins whether to plot the error ribbon (of width=standard deviation) around the curve.
plot_averages(xvals::Array{Float64,1},
means::Array{Float64,1},
stds::Array{Float64,1};
ribbon::Bool=true,
label::String = "",
linestyle = :solid,
color = :auto)
plot_averages!(xvals, means, stds; ribbon=true, label="", linestyle=:solid, color=:auto)
Plot average Betti curves means with standard deviations stds at filtration values given by xvals. Ribbon parameter determins whether to plot the error ribbon (of width=standard deviation) around the curve.
plot_averages(xvals::Array{Float64,1},
file::String;
dim=1,
ribbon=true,
label="",
linestyle=:solid,
color=:auto)
plot_averages!(xvals, file::String; dim=1, ribbon=true, label="", linestyle=:solid, color=:auto)
Plot average Betti curves in dimension dim at filtration values given by xvals, given that file contains all the Betti numbers (e.g. it contains many outputs of bettis).