On the use of principal component analysis to identify systematic patterns in roundness profiles

Abstract

In many industrial applications, quality of products or processes is related to profiles. With reference to mechanical components, profiles and surfaces play a relevant role, as shown by the high number of geometric specifications characterizing most of the technical drawings. In this framework, an important step consists in identifying the systematic pattern which characterizes all the profiles machined while the process is in its standard or nominal state. With reference to this aim, this paper focuses on the use of principal component analysis (PCA) for profile data (Functional PCA). Since a usual objection to PCA is that principal components (PCs) are often difficult or impossible to interpret, this paper explores what types of profile features allow one to obtain interpretable PCs. Within the paper, a real case study related to roundness profiles of mechanical components is used as reference. In particular, functional PCA is applied to the set of real profile data to derive the significant PCs and the corresponding eigenfunctions. In order to gain insight into the information behind the retained PCs, both simulations and analytical results are used. In particular, the analytical results, outlined in the literature on functional data analysis, allow one to link the eigenfunctions to specific profile features, given that profile data admit an orthogonal basis series expansion.

A comparison study of control charts for statistical monitoring of functional data

“…In the reference case study of this paper, Colosimo and Pacella (2007) showed that the number of PCs to be retained is equal to m = 4. In this last case, the PCA approach can take full advantage of both the T 2 and the Q control charts.…”

“…Different approaches can be used to select the proper set of PCs (Jackson 2003). For instance, cross-validation can be effectively used to chose the number m of significant PCs (Colosimo and Pacella 2007). When a subset of m out of the whole number of p PCs is retained (m < p), the original profile can be estimated as:…”

“…The original measurements sampled by using a Coordinate Measuring Machine were scaled by subtracting the least squares estimate of the radius and centred at the least squares estimate of the centre. A further step was eventually applied to register all the sampled profiles by minimizing the phase delay caused by the random contact angle (Colosimo and Pacella 2007). Starting from actual measurements, the general SARX model of equation (2) was fitted to data.…”

“…Note that when PCA is performed in the first case (fixed-effect model), no significant PCs are identified (Colosimo and Pacella 2007). In fact, PCA is usually performed after data centring and this first step consists in subtracting the mean pattern (described by the fixed-effect model) from each profile data.…”

“…Furthermore, in order to signal any deviation from the in-control behavior, coefficients used to describe process' signature were used to design a T 2 control chart for profile monitoring, while the estimate of the residual variance was monitored with a univariate control chart. Colosimo and Pacella (2007) explored the advantages of using a data-reduction approach for geometric product specification modelling. In particular, they showed that principal component analysis (PCA) can be an excellent exploratory method for interpreting profiles obtained from manufacturing processes.…”