Applications of Linear and Nonlinear Models

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This work offers a comprehensive exploration of linear and weakly nonlinear regression models across the first eight chapters, blending algebraic and stochastic perspectives. It highlights the connection between best linear uniformly unbiased estimation (BLUUE) in Gauss-Markov models and least squares solutions (LESS) in linear systems. The initial six chapters focus on underdetermined and overdetermined linear systems, including those with datum defects, and present various estimators such as MINOLESS, BLIMBE, and Total Least Squares. A key feature is the simultaneous determination of the first and second central moments of a probability distribution through the E-D correspondence and its Bayes design. The text also contrasts continuous and discrete networks and introduces Grassmann-Pluecker coordinates and Taylor-Karman criterion matrices. Chapter seven specializes in overdetermined nonlinear equations on curved manifolds, while chapter eight addresses probabilistic regression and the Gauss-Markov model with random effects, leading to BLIP and VIP estimators, including Bayesian methods. Four appendices supplement the main content, covering tensor algebra, sampling distributions, elementary statistics, and Groebner basis algebra, including the Buchberger Algorithm and Gauss's combinatorial algorithm.