Complex PCA as an Exploratory Tool for 3-Dimensional Profiles

During the past few years, an increasing number of approaches and applications of profile monitoring have been proposed in the scientific literature. As a matter of fact, very often product and/or process quality is characterized by profiles or functional data. In this type of applications, the quality outcome (dependent variable) is actually a function of one or more spatial or temporal location variables (independent variables). Up to now, profile monitoring techniques have been constrained to situations in which the dependent variable is a scalar which can be modelled as a function of one or more independent variables via linear models or data-reduction approaches as PCA. As a matter of fact, PCA has been demonstrated to be an excellent exploratory method for interpreting and modelling profiles in a scalar field of data, obtained from manufacturing processes (Colosimo and Pacella 2007). However, when the quality of products is related to geometric tolerances, very often the profile cannot be simply modelled via a scalar variable, since the profile or curve lies in a 3-dimensional space. Examples range from the simplest requirement of axial straightness to very complex curves in free-form geometric tolerance. This paper explores problems arising when 3D curves has to be monitored over time proposing possible solutions. A real case dealing with the straightness of cylindrical components machined by turning is used as reference throughout the paper. In particular, an approach is discussed in which a generalization of PCA is used to model two-dimensional vector observations (i.e., directional data). The method is based on an appropriate use of complex, rather than real numbers in the analysis. The so-called “complex PCA” is routinely used in the case of horizontal velocity components in geophysical measurements (such an approach was firstly proposed by Hardy and Walton, 1978). Our aim is to explore the use of complex PCA also for modelling 3D profiles obtained from manufacturing processes. References Colosimo B. M. and Pacella M., 2007. On the Use of Principal Component Analysis to Identify Systematic Patterns in Roundness Profiles. Quality and Reliability Engineering International, 23(6), 707-725. Hardy D. M. and Walton J. J., 1978. Principal Components of Vector Wind Measurements. Journal of Applied Meteorology, 17(8), 1153–1162.