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Generalized linear models
Published in Kenneth G. Russell, Design of Experiments for Generalized Linear Models, 2018
In the regression context, the columns of X are usually of full column rank, which means that no column of X can be written as a linear combination of other columns. In this case, the p × p matrix X⊤X is nonsingular and has a proper inverse, (X⊤X)−1. Then β^ is given by β^=(X⊤X)−1X⊤Y. It can be shown that E(β^)=β, and that the covariance matrix of β^ is equal to cov(β^)=σ2(X⊤X)−1.
Efficient analysis of split-plot experimental designs using model averaging
Published in Journal of Quality Technology, 2023
Chuen Yen Hong, David Fletcher, Jiaxu Zeng, Christina M. McGraw, Christopher E. Cornwall, Vonda J. Cummings, Neill G. Barr, Emily J. Frost, Peter W. Dillingham
Model (1) can be expressed as a linear mixed model, i.e. where Y is a vector corresponding to the response variable and X is a n×p matrix corresponding to the parameter vector of fixed effects with Z is the n×q matrix corresponding to the vector b of whole-plot random effects, with and e the vector of subplot error. We assumed that and Cov( where is the n×n identity matrix. Model (2) is obtained by simply removing the term.
Metainferential duality
Published in Journal of Applied Non-Classical Logics, 2020
Bruno Da Ré, Federico Pailos, Damian Szmuc, Paula Teijeiro
In this vein, we can simply present our target substructural logics and as induced, respectively, by the p-matrix and the q-matrix – where is the three-element (strong) Kleene algebra. Given theses definitions, it is straightforward to observe that has the same valid inferences that Classical Logic, whereas has no valid inferences at all (if the language does not count with truth-constants for the truth-values or ).
A projection and contraction method for symmetric cone complementarity problem
Published in Optimization, 2019
Sanyang Liu, Xiangjing Liu, Junfeng Chen
Because NCP, SOCCP and SDCP are special cases of SCCP, the idea of extending the methods on second-order cone or semi-definite cone to symmetric cone arises spontaneously. On the other hand, in the wake of developments in Jordan algebras, many researchers began to analyze the SCCP and develop algorithms by use of the Euclidean Jordan algebras and got some achievements, see [5–10]. The interior-point algorithm was first extended for solving a symmetric cone optimization problem by Faybusovich [11]. Yoshise [12] then extended the interior-point algorithm to the monotone nonlinear SCCP. Wang [13] proposed a new class of polynomial interior point algorithms based on a parametric kernel function for solving the Cartesian P-matrix linear complementarity problem over symmetric cone. Whereafter, the smoothing Newton method was extended to the SCCP, see [14–18] and the references therein. As is known, the projection method is a class of simple and efficient method for nonlinear complementarity problem whose basic idea is to establish an equivalent relationship between complementarity problems and fixed point problems by using a projection operator. The solution of complementarity problems can be obtained by solving the projection equation. Many researchers developed the properties of the projection onto symmetric cones. Sun et al. [19] showed the projection operator is strongly semismooth. Kong [20] gave the exact expression of the Clarke generalized Jacobian of the projection onto symmetric cones, which is a generalization of the corresponding results on second-order cones and the cones of symmetric positive semidefinite matrices.