Different formulations of principal component analysis for 3D profiles and surfaces modeling

摘要

During the past few years, an increasing number of approaches and applications of profile monitoring have been proposed in the literature as the quality of product and process is very often characterized by functional data. In the context of geometric tolerances, where curves and surfaces describe the shape of manufactured item, the quality outcome (dependent variable) is a function of one or more spatial location variables (independent variables). Up to now, profile monitoring has been mainly constrained to situations in which the dependent variable is a scalar, which is modeled as a function of a single location variable via linear models or data-reduction approaches as Principal Component Analysis (PCA). When the quality of products is related to geometric tolerances (e.g., roundness or circularity, straightness, cylindricity, flatness or planarity) the geometry of the item lies in a 3-dimensional (3D) space and cannot be modeled as a scalar function of one location variable. This paper presents solutions to problems arising when 3D features (either curves or surfaces) are considered and data-reduction techniques are implemented as modeling tool. Two PCA-based approaches are presented, namely (i) the complex PCA (i.e., PCA performed on matrices of complex numbers) and the (ii) multilinear PCA (i.e., PCA performed on tensor data). These two approaches are explored as viable solutions to modeling 3D profiles and surfaces respectively, in the context of geometric tolerance monitoring.