FisherRaoDistance

This is a repository for calculating the Fisher-Rao Distance between densities and performing hypothesis testing.

Overview

In addition to the fact that the Fisher-Rao Distance is a proper distace metric, key advantages of using the Fisher-Rao Distance for comparing sets of data are that it can be used:

• parametrically or non-parametrically
• for scalar or multi-dimensional data
• in various domains (i.e. Rⁿ or Sⁿ ) by changing the form of the density

In this current implementation, we use a nonparametric Gaussian KDE and LOOCV for calculating its bandwidth parameter.

Example 1: Framework Usage

For this example we generate and compare three data sets generated from a Normal Distribution in R¹⁰ and mapped to R.

using FisherRaoDistance
using KernelDensityEstimatePlotting #currently not working in Atom; needs Atom dev update

The density estimation requires sets of points. These points can either be the original data or can be the result of some sort of dimension reduction. This example calculates pairwise distances and uses those in a classicalMDS setting

Generate the Points and Perform Dimension Reduction

points1 = randn(10, 500)
points2 = randn(10, 500)
points3 = randn(10, 500) + ones(1, 500) * 1.5


#below calculates the pairwise Euclidean distance matrix between the points
#and uses those for classical mds
Points = [points1, points2, points3]
lowdimpoints = get_low_dim_points(Points, 1)

Estimate the Densities

pdf1 = kde!(lowdimpoints[:,:,1])
pdf2 = kde!(lowdimpoints[:,:,2])
pdf3 = kde!(lowdimpoints[:,:,3])

#plot pane is not currently working in Atom-- waiting for Atom dev fix
plot([pdf1; pdf2; pdf3], c = ["red"; "green"; "blue"])

Estimate the Fisher-Rao Distances

dfr1_2 = fisherraodistance(
    pdf1,
    pdf2,
    lowdimpoints[:, :, 1],
    lowdimpoints[:, :, 2],
)

returns: 0.089

dfr1_3 = fisherraodistance(
    pdf1,
    pdf3,
    lowdimpoints[:, :, 1],
    lowdimpoints[:, :, 3],
)

returns: 0.495

Estimate the Fisher-Rao p-values

p1_2 = fisherraotest(pdf1, pdf2, n1, n2, dfr1_2)

returns: 0.62

p1_3 = fisherraotest(pdf1, pdf3, n1, n3, dfr1_3)

returns: 0.00

Example 2: Basic Function Usage from KernelDensityEstimate.jl

data = rand(100) #generate some random data
pdf = kde!(data) #estimate a pdf from given data
data_likelihoods = evaluateDualTree(pdf,data) #get data likelihoods
sample_points, ind = sample(pdf,100) #sample 100 points from the pdf

References

Henning, Wade. "A Framework for Comparing Shape Distributions", 2014, Ph.D. Thesis, Florida State University.

Henning, Wade; Srivastava, Anuj. "A Two-Sample Test for Statistical Comparisons of Shape Populations", 2016, IEEE Winter Conference on Applications in Computer Vision

Srivastava, Anuj; Jermyn, Ian; Joshi, Shantau. "Riemannian analysis of probability density functions with applications in vision", 2007, IEEE Computer Vision and Pattern Recognition

Sudderth, Erik B.; Ihler, Alexander; et al. "Nonparametric belief propagation.", 2010, Communications of the ACM