Multiple Weighted Regression Analysis of the Curvature of a 3D Brane in a 4D Bulk Space under a Homogenous Vector Field

by Stefan von Weber and Alexander von Eye .

Abstract: Weighted multiple regression analysis is shown to be a suitable tool for finding the normal vector of an inclined space. The outer vector product is defined only in the 3-dimensional space. The use of that vector product in a space with more than three dimensions requires an extension to higher dimensions. Both methods, regression analysis and generalization of the outer vector product, use the cross-product of matrices, and are mathematically similar. However, using weighted multiple regression analysis comes with the advantage that one can perform two tasks at the same time: (1) finding the normal vector of the locally fitted hyperplane, and (2) averaging over distributed loads. Application of multiple weighted regression is suggested when modelling stochastically distorted surfaces, to obtain local fit of a plane to a curved surface, in particular in higher dimensional spaces. The example from mechanics presented in this paper belongs to this class of models.

Key Words: Multiple weighted regression, normal vector, curved surfaces, elastic membrane

Authors:
Stefan von Weber, webers@hs-furtwangen.de
Alexander von Eye, voneye@msu.edu

Editor: Graf, R. G., rgraf@sunstroke.sdsu.edu

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