Unconstrained parametrizations for variance-covariance matrices

By J. Pinheiro, Douglas M. Bates. 1996

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Citations in the corpus

Taking the Human Out of the Loop: A Review of Bayesian Optimization (2016, methodology): “There are many ways to parameterize the task covariance function [123].”
Bayesian Gaussian processes for sequential prediction, optimisation and quadrature (2010, background): “Fortunately, other parameterisations exist [Pinheiro and Bates, 1996].”
Gaussian Processes for Prediction Technical Report PARG-0701 (2007, background): “Fortunately, other parameterisations exist [Pinheiro and Bates, 1996].”
Covariance Estimation: The GLM and Regularization Perspectives (2011, background): “Statistical interpretation of the entries of C is difficult in its present form (Pinheiro and Bates, 1996).”
Estimating Size and Composition of Biological Communities by Modeling the Occurrence of Species (2005, methodology): “Furthermore, to ensure the positive-definiteness of estimates of , we use a log-Cholesky parameterization of (Pinheiro and Bates 1996), which implies the following one-toone transformation of variables: γ1 = logσu, γ2 = σvρuv, and γ3 = logσv + .5 log(1 − ρ2uv).”
Warmer winters reduce the advance of tree spring phenology induced by warmer springs in the Alps (2018, methodology): “Mixed effects models were performed in R (version 3.1.1; R Core team, (2015)) using the library nlme (Lindstrom & Bates 1990; Pinheiro & Bates 1996) with observation sites as random effect.”
Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models (2002, methodology): “In particular, using the Cholesky parameterization (Pinheiro & Bates 1996), we can rewrite the log-linear portion of the model as = >A@ 68H \BP "]C^EV [`_+a b 6c Ged where G d G d are i.i.d. with mean Z and identity variance-covariance matrix, and [ _fa b c is the transpose of [ a b 6c , the upper…”
Gaussian processes for time-series modelling (2013, methodology): “If the number of time series is not too large, we can arbitrarily represent the covariance matrix over the labels, using the spherical decomposition [6].”
Parsimonious Covariance Matrix Estimation for Longitudinal Data (2002, background): “A number of authors have considered the problem of ef - ciently estimating a covariance matrix for both longitudinal data and more general covariance matrices (see Pinheiro and Bates 1996 for a partial summary).”
mvmorph: an r package for fitting multivariate evolutionary models to morphometric data (2015, methodology): “Generally, one of the eﬃcient ways to solve this issue is to use an unconstrained optimization of the likelihood with a parameterization that enforces the matrices to a particular structure (Pinheiro & Bates 1996).”