Unconstrained parametrizations for variance-covariance matrices

Pinheiro, José; Bates, Douglas M.

doi:10.1007/bf00140873

Cited by 425 publications

(310 citation statements)

References 12 publications

Supporting

Mentioning

310

Contrasting

Order By: Relevance

“…To investigate this situation, it is convenient to describe the orientation of a set of eigenvectors in terms of the angles by which they deviate from the axes [the so-called ''Givens angles,'' (e.g., Pinheiro and Bates 1996)]. It is easy to see that for q ¼ 2 a single angle describes the orientation of the first eigenvector relative to the x-axis and, simultaneously, the orientation of the second eigenvector relative to the y-axis.…”

Section: Bias Due To Reduced-rank Estimationmentioning

confidence: 99%

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Meyer

Kirkpatrick

2008

Genetics

View full text Add to dashboard Cite

Eigenvalues and eigenvectors of covariance matrices are important statistics for multivariate problems in many applications, including quantitative genetics. Estimates of these quantities are subject to different types of bias. This article reviews and extends the existing theory on these biases, considering a balanced one-way classification and restricted maximum-likelihood estimation. Biases are due to the spread of sample roots and arise from ignoring selected principal components when imposing constraints on the parameter space, to ensure positive semidefinite estimates or to estimate covariance matrices of chosen, reduced rank. In addition, it is shown that reduced-rank estimators that consider only the leading eigenvalues and -vectors of the ''between-group'' covariance matrix may be biased due to selecting the wrong subset of principal components. In a genetic context, with groups representing families, this bias is inverse proportional to the degree of genetic relationship among family members, but is independent of sample size. Theoretical results are supplemented by a simulation study, demonstrating close agreement between predicted and observed bias for large samples. It is emphasized that the rank of the genetic covariance matrix should be chosen sufficiently large to accommodate all important genetic principal components, even though, paradoxically, this may require including a number of components with negligible eigenvalues. A strategy for rank selection in practical analyses is outlined.

show abstract

Section: Bias Due To Reduced-rank Estimationmentioning

confidence: 99%

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Meyer

Kirkpatrick

2008

Genetics

View full text Add to dashboard Cite

show abstract

“…A reparametrization can be used to remove constraints on the parameter space. For instance, instead of estimating the unique elements of a covariance matrix K, we can estimate the elements of its Cholesky factor, taking logarithmic values of the diagonals (Meyer & Smith 1996;Pinheiro & Bates 1996). This not only allows use of unconstrained maximization procedures, but can also improve rates of convergence in an iterative estimation scheme (Groeneveld 1994).…”

Section: Estimation Of Covariance Functions: II 'Directly'mentioning

confidence: 99%

Up hill, down dale: quantitative genetics of curvaceous traits

Meyer

Kirkpatrick

2005

Phil. Trans. R. Soc. B

100

107

View full text Add to dashboard Cite

'Repeated' measurements for a trait and individual, taken along some continuous scale such as time, can be thought of as representing points on a curve, where both means and covariances along the trajectory can change, gradually and continually. Such traits are commonly referred to as 'functionvalued' (FV) traits. This review shows that standard quantitative genetic concepts extend readily to FV traits, with individual statistics, such as estimated breeding values and selection response, replaced by corresponding curves, modelled by respective functions. Covariance functions are introduced as the FV equivalent to matrices of covariances.Considering the class of functions represented by a regression on the continuous covariable, FV traits can be analysed within the linear mixed model framework commonly employed in quantitative genetics, giving rise to the so-called random regression model. Estimation of covariance functions, either indirectly from estimated covariances or directly from the data using restricted maximum likelihood or Bayesian analysis, is considered. It is shown that direct estimation of the leading principal components of covariance functions is feasible and advantageous. Extensions to multidimensional analyses are discussed.

show abstract

“…The Cholesky decomposition provides an alternative parametrization of positive †.´/ different from the exponential in (4.3) [21]. Real data typically depend on more than one external factor (for instance, on a patient's age, sex, and ethnicity in medical studies, and on time of the day, time of the year, latitude, and elevation in weather prediction).…”

Section: Continuous and Multiple Factorsmentioning

confidence: 99%

Explanation of Variability and Removal of Confounding Factors from Data through Optimal Transport

Tabak

Trigila

2017

Comm Pure Appl Math

View full text Add to dashboard Cite

A methodology based on the theory of optimal transport is developed to attribute variability in data sets to known and unknown factors and to remove such attributable components of the variability from the data. Denoting by x the quantities of interest and by´the explanatory factors, the procedure transforms x into filtered variables y through a´-dependent map, so that the conditional probability distributions .xj´/ are pushed forward into a target distribution .y/, independent of´. Among all maps and target distributions that achieve this goal, the procedure selects the one that minimally distorts the original data: the barycenter of the .xj´/. Connections are found to unsupervised learning and to fundamental problems in statistics such as conditional density estimation and sampling. Particularly simple instances of the methodology are shown to be equivalent to k-means and principal component analysis. An application is shown to a time series of ground temperature hourly data across the United States.

show abstract

Unconstrained parametrizations for variance-covariance matrices

Cited by 425 publications

References 12 publications

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Up hill, down dale: quantitative genetics of curvaceous traits

Explanation of Variability and Removal of Confounding Factors from Data through Optimal Transport

Contact Info

Product

Resources

About