Applications of linear and nonlinear models

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This work provides a comprehensive examination of the Grand Universe of linear and weakly nonlinear regression models across the first eight chapters, combining both algebraic and stochastic perspectives. An important lemma connects best linear uniformly unbiased estimation (BLUUE) in a Gauss-Markov model with a least squares solution (LESS) in linear systems, highlighting the interplay between stochastic models and algebraic solutions. The initial six chapters focus on underdetermined and overdetermined linear systems, including those with data defects, and explore various estimators such as MINOLESS, BLIMBE, BLUMBE, and others. 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 Bayesian design. Additional topics include continuous versus discrete networks, Grassmann-Pluecker coordinates, and Taylor-Karman criterion matrices, along with FUZZY sets. Chapter seven delves into overdetermined nonlinear systems on curved manifolds, emphasizing the von Mises-Fisher distribution for circular data. The final chapter addresses probabilistic regression within the Gauss-Markov model with random effects, leading to BLIP and VIP estimators, including Bayesian methods. Four appendices supplement the text, covering tensor algebra, sampling distributions, basic statistical concepts, and Groebner basis algebra, including the Buchberger Al