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GMMParameterEstimation.jl Documentation

GMMParameterEstimation.jl is a package for estimating the parameters of Gaussian k mixture models using the method of moments. It works for general k with known mixing coefficients, and for k=2,3,4 for unknown mixing coefficients.

Parameter estimation

The main functionality of this package stems from

GMMParameterEstimation.estimate_parameters — Function
estimate_parameters(d::Integer, k::Integer, sample::Array{Float64}, diagonal::Bool)

Compute an estimate for the parameters of a d-dimensional Gaussian k-mixture model from a sample.

If diagonal is true, the covariance matrices are assumed to be diagonal. If w is provided it is taken as the mixing coefficients, otherwise those are computed as well. The sample should be a d x sample-size array.

estimate_parameters(d::Integer, k::Integer, sample::Array{Float64}, diagonal::Bool, w::Array{Float64})

Compute an estimate for the parameters of a d-dimensional Gaussian k-mixture model from a sample.

If diagonal is true, the covariance matrices are assumed to be diagonal. If w is provided it is taken as the mixing coefficients, otherwise those are computed as well. The sample should be a d x sample-size array.

For example, the following code snippet will generate a 3D 2-mixture, take a sample, compute the necessary moments, and then return an estimate of the parameters using the method of moments.

using GMMParameterEstimation
d = 3
k = 2
diagonal = true
num_samples = 10^4
w, true_means, true_covariances = generateGaussians(d, k, diagonal)
sample = getSample(num_samples, w, true_means, true_covariances)
first_moms, diagonal_moms, off_diagonals = sampleMoments(sample, k)
pass, (mixing_coefficients, means, covariances) = estimate_parameters(d, k, first_moms, diagonal_moms, off_diagonals, diagonal)

Generate and sample from Gaussian Mixture Models

GMMParameterEstimation.generateGaussians — Function
generateGaussians(d::Integer, k::Integer, diagonal::Bool)

Generate means and covariances for k Gaussians with dimension d.

diagonal should be true for spherical case, and false for dense covariance matrices.

GMMParameterEstimation.getSample — Function
getSample(numb::Integer, w::Vector{Float64}, means::Matrix{Float64}, covariances::Array{Float64, 3})

Generate a Gaussian mixture model sample with numb entries, mixing coefficients w, means means, and covariances covariances.

Missing docstring.

Missing docstring for sampleMoments. Check Documenter's build log for details.

Missing docstring.

Missing docstring for perfectMoments. Check Documenter's build log for details.

Build the polynomial systems

GMMParameterEstimation.build1DSystem — Function
build1DSystem(k::Integer, m::Integer)

Build the polynomial system for a mixture of 1D Gaussians where 'm' is the highest desired moment.

If a is given, use a as the mixing coefficients, otherwise leave them as unknowns.

build1DSystem(k::Integer, m::Integer, a::Union{Vector{Float64}, Vector{Variable}})

Build the polynomial system for a mixture of 1D Gaussians where 'm' is the highest desired moment.

If a is given, use a as the mixing coefficients, otherwise leave them as unknowns.

GMMParameterEstimation.selectSol — Function
selectSol(k::Integer, solution::Result, polynomial::Expression, moment::Number)

Select a k mixture solution from solution accounting for polynomial and moment.

Sort out a k mixture statistically significant solutions from solution, and return the one closest to moment when polynomial is evaluated at those values.

GMMParameterEstimation.mixedMomentSystem — Function
mixedMomentSystem(d, k, mixing, ms, vs)

Build a linear system for finding the off-diagonal covariances entries.

For a d dimensional Gaussian k-mixture model with mixing coefficients mixing, means ms, and covariances vs where the diagonal entries have been filled in and the off diagonals are variables.

Index