A Course in Combinatorial Optimization by Schrijver A.

By Schrijver A.

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In general, it is preferable to ensure that estimates of the covariance parameters in h , updated from one iteration of an optimization algorithm to the next, imply positive definiteness of D and Ri at every step of the estimation process. Unfortunately, it is difficult to meet these requirements, so software procedures set much simpler conditions that are necessary, but not sufficient, to meet positive definiteness constraints. Specifically, it is much simpler to ensure that elements on the diagonal of the estimated D and Ri matrices are greater than zero during the entire iteration process, and this method is often used by software procedures in practice.

Other covariance structures, such as the Toeplitz structure, allow more flexibility in the correlations, but at the expense of using more covariance parameters in the h R vector. In any given analysis, we try to determine the structure for the Ri matrix that seems most appropriate and parsimonious, given the observed data and knowledge about the relationships between observations on an individual subject. , males and females). Specifically, we might assume the same structures for the matrices in different groups, but with different values for the covariance parameters in the h D and h R vectors.

Although some solutions have been proposed to overcome these shortcomings, the EM algorithm is rarely used to fit LMMs, except to provide starting values for other algorithms. The N–R algorithm and its variations are the most commonly used algorithms in ML and REML estimation of LMMs. 2 for REML estimation. At every iteration, the N–R algorithm requires calculation of the vector of partial derivatives (the gradient), and the second derivative matrix with respect to the covariance parameters (the observed Hessian matrix).

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