Linear and Nonlinear Models Fixed effects, random effects, and total
Here we present a nearly complete treatment of the Grand Universe of linear and weakly nonlinear regression models within the first 8 chapters. Our point of view is both an algebraic view as well as a stochastic one. For example, there is an equivalent le
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Overdetermined System of Nonlinear Equations on Curved Manifolds The spherical problem of algebraic regression – inconsistent system of directional observational equations Here we review a special issue of distributions on manifolds, in particular the spherical problem of algebraic regression or analyse the inconsistent of directional observational equations. The first section introduces loss functions on longitudinal data (ˆ D 1) and (p D 2) on the circle or on the sphere as a differential manifold of dimension ˆ D 1 and p D 2. Section 6.2 introduces the minimal distance mapping on S 1 and S 2 and constructs the related normal equations. Section 6.3 reviews the transformation from the circular normal distribution to an oblique normal distribution including a historical note to von Mises analyzing data on a circle, namely atomic weights. We conclude with note on the “angular metric.” As a case study in section four we analysze 3D angular observations with two different theodolites, namely Theodolite I and Theodolite II. The main practical result is the set of data from (tan ^n ; tan ˆn ), its solution (^n ; ˆn ) is very different from the Least Squares Solution. Finally we discuss the von Mises-Fisher distribution for dimension 1, 2, 3 and 4, namely Relativity and Earth Sciences (for instance phase observations for the Global Positioning System (GPS)), Biology, Meteorology, Psychology, Image Analysis and Astronomy. A detailed reference list stays at the end. Read only Sect. 7-2 for a careful treatment(HAPS).
“Least squares regression is not appropriate when the response variable is circular, and can lead to erroneous results. The reason for this is that the squared difference is not an appropriate measure of distance on the circle.” – U. Lund (1999)
A typical example of a nonlinear model is the inconsistent system of nonlinear observational equations generated by directional measurements (angular observations, longitudinal data). Here the observation space Y as well as the parameter space X is the hypersphere Sp RpC1 : the von Mises circle S1 ; p D 2 the Fisher sphere S2 in general the Langevin sphere Sp . For instance, assume repeated measurements of horizontal directions to one target which are distributed as polar coordinates on a unit circle clustered around a central direction. Alternatively, assume repeated measurements of horizontal and vertical directions to one target which are similarly distributed as spherical coordinates (longitude, latitude) on a unit sphere clustered around a central direction. By means of a properly chosen loss function we aim at a determination of the central direction. Let us connect all points E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models, Springer Geophysics, DOI 10.1007/978-3-642-22241-2 7, © Springer-Verlag Berlin Heidelberg 2012
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7 Overdetermined System of Nonlinear Equations on Curved Manifolds
Lemma 7.2 minimum geodesic distance: Definition 7.1 minimum geodesic distance: Definition 7.4 minimum geodesic distance:
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Lemma 7.3 mini
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