EmpiricalMeasures.jl

This package implements empirical probability measures in Julia. Atention: This package is quite new and I'm still improving it's efficiency. Thus, the internal data structure might change.

Given a set of points x₁,...,xₙ ∈ ℝⁿ, the generalized empirical probability measure is a discrete finite measure defined as:

where each pᵢ is the probability of each point ("sample") xᵢ multiplying a Dirac measures, i.e

This package is compatible with Distributions.jl, and creates a new distribution called MvDiscreteNonParametric which is a multivariate version of DiscreteNonParametric.

EmpiricalMeasures.jl vs EmpiricalDistributions.jl vs EmpiricalCDFs.jl

Let's clarify the difference between EmpiricalMeasures.jl and other similar packages.

EmpiricalMeasures.jl - Implements a multivariate discrete distribution as a sum of weighted Dirac measures;
EmpiricalDistributions.jl - Provides "binned approximations" of continuous distributions;
EmpiricalCDFs.jl - Computes the cumulative density function of 1D arrays.

Thus, each package fulfills very different tasks.

Basic Use

Let's do a quick overview of how to use this package.

Declaring (Multivariate) Discrete Non-Parametric Measures

This package provides the function empiricalmeasure(support, p), which dispatches to either DiscreteNonParametric (1D case from Distributions.jl) or MvDiscreteNonParametric based on the support.

using EmpiricalMeasures
using LinearAlgebra

# 1D DiscreteNonParametric example
ν = empiricalmeasure(rand(10), normalize!(rand(10),1))

# MvDiscreteNonParametric example
μ = empiricalmeasure(rand(10,2), normalize!(rand(10),1))

Note that in the example above, the multivariate discrete measure μ was constructed using a matrix. The default behavior is to interpret the rows as samples, thus, μ is a probability measure in ℝ² with 10 "mass-points". If all points have the same probability, you can omit the p and just do

μ = empiricalmeasure(rand(10,2))

Instead of passing matrices, one can also pass an array of arrays, e.g. empiricalmeasure([[0,0],[1,1]]). One can also use MvDiscreteNonParametric to declare in multivariate discrete distributions without the possibility of dispatching for the 1D case.

using EmpiricalMeasures
using LinearAlgebra
using ArraysOfArrays

n = 4 # number of "samples"
m = 2 # dimension of each "sample"
A = nestedview(rand(n,m)')
p = normalize!(rand(n),1);

μ = MvDiscreteNonParametric(A,p)

Basic Functionalities

This package implements all the basic functionalities suggested by Distributions.jl for new multivariate distributions.

using EmpiricalMeasures

p = ([3/5, 1/5, 1/5])
A = [[1,0],[1,1],[0,1]]
μ = MvDiscreteNonParametric(A,p)

length(μ) # returns 2
size(μ) # returns (10,2)

Note that μ is of type ::MvDiscreteNonParametric which is a struct containing the support and the probability of each point.

μ.support, μ.p

([[1, 0], [1, 1], [0, 1]], [0.6, 0.2, 0.2])

The support and probabilities can also be obtained using support(μ) and probs(μ).

As in any distribution from Distributions.jl, we can easily calculate things like the mean, variance, covariance, and even sample from it:

mean(μ)
var(μ)
cov(μ)

rand(μ,10)

2×10 Matrix{Int64}:
 1  0  1  1  1  1  1  1  0  1
 1  1  1  0  0  0  0  0  1  0

Future Plans