Estimating linear covariance models with numerical nonlinear algebra

Abstract

Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.

23 pages, 2 figures, 5 tables

References in corpus (3)

• Estimation of a Covariance Matrix with Zeros
• The Chow form of a reciprocal linear space
• Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

Cited by in corpus (11)

• Pencils of Quadrics: Old and New
• Reciprocal maximum likelihood degrees of diagonal linear concentration models
• Maximum likelihood degree of the two-dimensional linear Gaussian covariance model
• Computing Maximum Likelihood Estimates for Gaussian Graphical Models with Macaulay2
• Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance
• Proof of a conjecture of Sturmfels, Timme and Zwiernik
• Likelihood Geometry of Correlation Models
• The maximum likelihood degree of sparse polynomial systems
• The Geometries of Jordan nets and Jordan webs
• Applications of intersection theory: from maximum likelihood to chromatic polynomials
• Geometry of the Gaussian graphical model of the cycle