“Functional data analysis” and “Profile monitoring” are new and effective instruments that are spreading in industrial ambit to improve the analysis of process and post-production data.
Marco Grasso (MUSP Laboratory)
Bianca Maria Colosimo (Engineering Department, Milan Polytechnics)
In industrial ambit, signals and data that may be represented by a curve, by a surface or by hyper-surfaces in spaces with multiple dimensions, are more and more often analysed with techniques not completely effective and/or efficient.
In general, the variables considered in production ambit are “simple random” (for instance, the diameter of a pin obtained by turning, etc.), that’s to say “polluted” owing to uncertainties and disturbances that occur at level of process, materials, resources etc. The measured values, then, are distributed around a “mean” value and show certain dispersion; the task of a Quality Control department consists in granting that the statistical distribution of random variables is stable in time. When, as it often happens, the quality characteristics of a process are represented by more variables, that’s to say a curve in the space or a time profile, as in the cases represented in the proposed figures , the different failure modalities or the process anomalies result in form variations as to a characteristic model. In these cases, a common analysis practice consists in calculating one or more indexes that synthesize some profile properties (for instance the average, the peak-peak width, the maximum or minimum value etc.) to apply then the traditional instruments to those synthetic indexes. This, however, limits the analysis since it controls only some features and gives up great part of the information content coupled with the profile shape.
To widen the analysis spectrum, in particular in the ambit of problems concerning monitoring and quality control, some new important study instruments are available, such as the “Functional Data Analysis” and the “Profile Monitoring”, although not very diffused on a large scale, yet, and prevailingly in the industrial sector. Nevertheless the ever-increasing availability of great quantities of data and especially of new formats (profiles, surfaces, hyper-surfaces) drives towards a constant use of techniques of this kind, also because understanding the nature of data and knowing the suitable techniques for representing their information content correctly is the first step towards the unending improvement of processes and technologies.
Functional Data Analysis
The “Functional Data Analysis” system, recently devised, performs the statistical study of data sets that can be interpreted as realizations of random functions .
That means that it is possible to choose a suitable model for describing the type of profile analysed and then to estimate the parameters that provide the best fitting of the starting profile. The density profile shown in fig. 2, for instance, can be interpreted as function of a space coordinate (the panel thickness), while the profiles in fig. 1 and 3 as time functions. In general, the profile nature depends on the applicative case: for instance, it may have polynomial form or it can be described in terms of Fourier series expansion. In the case of more complex patterns, you can adopt either spline models or wavelet models, or other.
In the cases in which the profile is affected by noise, then in almost all industrial applications, it is possible to use mixed models in which shape features on different scales are modelled with different formalisms.
If the approximation offered by the model is adequate to the problem under examination, the functional transformation allows switching from a domain depending on synthetic indexes to a domain where it is possible to reason in terms of profile shape.
What above said can be applied also to space data, time data and spatio-temporal data and is extended to the more complex case of surfaces and hyper-surfaces, that’s to say surfaces defined in multi-dimension spaces.
The target of the Statistical Process Control (SPC) consists in granting the stability of a process and in improving its performances through the reduction of the variability of the features of interest. It concerns both the “post process” measuring in metrological room (or directly on machine board), and “in process” ones, which are variable depending on time and/or frequency. Measures are more and more often detected by instruments with enhanced sensitivity and precision (cameras, lasers, probes etc.), and monitored “on line” and in real time. In all these cases, we need new monitoring and/or adaptation systems of the instruments already existing in SPC ambit, and the Profile Monitoring  represents an efficient and up-to-date solution.
More precisely, Profile Monitoring techniques allow estimating the parameters of a model that correctly approximates the data pattern under normal operation conditions of the system and/or of the process, and defining the control limits not only on those parameters but also on the model “residues”, in order to recognize as fast as possible the presence of eventual anomalies, failures or process instabilities.
A concrete applicative example of Profile Monitoring is schematically illustrated in fig. 4 : the statistical process control of roundness profiles is obtained by measuring sections of cylinders machined by turning with a coordinate measuring machine. Each profile is the set of the various dimensions acquired in a full round angle. The common industrial practice provides for calculating a synthetic index for each section (difference between the maximum and minimum deviation as to the centre of the reference circle) and monitoring the process on the grounds of this single index. This, however, does not permit to evaluate the information contained in the measured profile. Thanks to the Profile Monitoring approach, instead, it is possible to identify a representative model in the normal process operation and to estimate its parameters. In this way it is possible to recognize and to evaluate in faster and more reliable way “any” modification and to operate consequently.
 W.H.Woodall, D.J.Spitzner, D.C.Montgomery, S.Gupta, Using Control Charts to monitor Process and Product Quality Profiles, Journal of Quality Technology, Vol.36, N°3, pp.309-320, 2004
 J.D.Williams, W.H.Woodall, J.B.Birch, Statistical Monitoring of Nonlinear Product and Processes Quality Profiles, Quality and Reliability Engineering International, Vol.23, pp.925-941, 2007
 B.M.Colosimo, Q.Semeraro, M.Pacella, Statistical process control for geometric specifications: on the monitoring of roundness profiles, Journal of quality technology, Vol. 40, No 1, pp. 1-18, 2008.
 B.M.Colosimo, M.Pacella, On the Use of Principal Component Analysis to Identify Systematic Patterns in Roundness Profiles, Quality and Reliability Engineering International, Vol. 23, Issue 6, pp. 707-725, 2007
 J.O.Ramsay, B.W.Silverman, Functional Data Analysis, Springer Series in Statistics, 2005
 R. Noorossana, A. Saghaei, A. Amiri, Statistical Analysis of Profile Monitoring, Wiley Series in Probability and Statistics, John Wiley & Sons, 2011