Estimating linear covariance models with numerical nonlinear algebra
Bernd Sturmfels, Sascha Timme, Piotr Zwiernik
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.
Topics & Concepts
MathematicsApplied mathematicsCovarianceCovariance functionToeplitz matrixEstimatorNonlinear systemLinear algebraGaussianCovariance matrixLikelihood functionRestricted maximum likelihoodEstimation of covariance matricesMaximaAlgebra over a fieldEstimation theoryStatisticsPure mathematicsGeometryPerformance artArt historyArtQuantum mechanicsPhysicsBayesian Modeling and Causal InferenceSoil Geostatistics and MappingStatistical Methods and Bayesian Inference