Multivariate statistical shape model

In Chapter 6, a multivariate model that combines Kernel Density Estimation and the Point Distribution Model (the KDE-PDM) was introduced to provide a more accurate statistical representation of the entire shape of sheet metal stamping and assemblies. Major advancements of this approach over univariate approaches are that the model is capable of representing correlated variation modes, dealing with high dimensional data sets, and capturing non-normal distributions. The ability of this multivariate approach to capture correlated variation modes is important as geometric covariance is an inherent characteristic of sheet metal assembly variation. Through capturing correlated variation patterns more information can be extracted about the type or nature of variation present, allowing for more detailed assessments about the quality of the assembly. High dimensional data sets are made to be more manageable through the dimensional reduction technique PCA, which allows for the representation of a large set of MP's with a smaller set of factors. Non-normally distributed data is accounted for through the kernel density estimation technique which is capable of estimating any type of distribution, allowing for a more accurate inferences to be drawn from a more accurate statistical representation of the underlying data set. Areas for possible implementation of the multivariate shape model (or KDE-PDM) include data mining and diagnosis, interpreting quality, and tolerance zones.

Data mining and diagnosis

Multivariate tools have been applied for the purposes of fault diagnosis from process monitoring data. Rather than just highlighting a sample as being out-of-control as in SPC, multivariate techniques can uncover the type of variation pattern present which can often be matched with a root cause. The majority of methods involve a supervised approach to process diagnosis, where fault vectors are pre-determined from theoreti- cal assembly functions, process data is decomposed using a method such as PCA to identify physical fault patterns (Yang, 1996), and the theoretical and physically oc- curring patterns compared to identify the particular fault mode present (Ceglarek and Shi, 1996). It is not always possible or practical to identify all possible theoretical fault modes. Apley et al. aim to use unsupervised approaches to reveal unknown fault patterns in process data (2001; 2007). The statistical shape model proposed adds an- other dimension to multivariate analysis which enables further avenues for unsupervised pattern identification: the kernel (or mixture model) density estimate. Probabilistic representations can be used to reveal underlying structures in a data set. For example, probabilistic thresholds could be applied to identify clusters within a data set. In many cases it is possible that the formation of such shape clusters will be a result of process factors. Probabilistic thresholding could therefore provide targets for quality engineers

8.2. A FRAMEWORK FOR QUALITY ASSESSMENT 129 in their quality control efforts. Furthermore, corresponding physical shapes for each cluster could be used to aid in process diagnosis through an ability to visualise the three dimensional correlated variation patterns. Figure 8.1 looks at an artificial data set presented in the form of the KDE-PDM to illustrate the data mining concepts. It can be seen that the probabilistic representation highlights an underlying structure in the data.

(a) (b)

Figure 8.1: Figure (a) shows an example measurement data set in the reduced 3D PCA space. Figure (b) shows a three level density contour plot at the 7.5%, 15% and 75% levels within the same plot space (ie, 7.5% of samples will fall within the 7.5% contour level and so on). Note how the underlying data structure consisting of two main clusters is highlighted by the density estimate.

Interpreting quality

The multivariate statistical shape model has the potential to provide a more intu- itive interpretation of assembly quality through the ability to visualise the probabilistic representation of the data set in parallel with corresponding physical shapes. This im- proved interpretation of assembly variation could assist the investigation of both virtual and physical process data, and for the selection of processes for dimensional control. There are three key approaches that could be taken for improving the dimensional qual- ity of processes within the context of the multivariate statistical shape model: reducing the spread of data, and shifting the process towards the nominal and/or away from less desirable variation patterns.

1. Reducing variation or spread of the data set by pushing the process in towards the mean: An example procedural approach to achieving this goal could be to firstly reduce the number of clusters in the data set (ideally down to one cluster), and secondly to reduce the spread of each remaining cluster.

130 CHAPTER 8. CONCLUSION

• As seen in section 6.4.5, it is likely that the formation of clusters in a data set is an indicator of mean shifts such as operator error or process setup differences. Through targeting these clusters, it is likely that a great deal of the problematic variation in a process will be eliminated. Figures 8.2 (a) and (b) show a graphical representation of the process of eliminating separate clusters using the probabilistic representation of the multivariate shape model.

• Finally, once the number of clusters have been reduced, the remaining av- enue of dimensional improvement is to reduce the spread of the remaining clusters, which could for example be achieved by identifying a more optimal clamping sequence for dimensional control. Figures 8.2 (c) and (d) graphi- cally demonstrate this example procedural approach.

(a) (b)

(c) (d)

Figure 8.2: Figures (a) and (b) show how the elimination of major variation sources such as process setup errors could reduce the amount of clusters. Figures (c) and (d) show how the process could be pushed towards a mean by a method such as identifying a more robust clamping sequence. A three level density contour plot is shown at the 7.5%, 15% and 75% levels in all plots.

8.2. A FRAMEWORK FOR QUALITY ASSESSMENT 131 2. Moving the mean shape closer to the nominal specification: Once the spread of the data has been controlled the next stage could be to shift the mean of the data set towards the nominal design specification as indicated in Figure 8.3.

Figure 8.3: Moving the process towards its nominal specifications could for example be achieved by clamp shimming (Yi et al., 2005). A three level density contour plot is shown at the 7.5%, 15% and 75% levels.

3. Moving the process from undesirable variation patterns to desirable variation patterns: Some manufactured shapes will be more or less desirable according to the amount and type of variation present. The desirability of a manufactured shape will often be a subjective issue that manufacturers will need to decide upon, however, the probabilistic and physical representations made available by the multivariate shape model should provide much needed assistance for this purpose. Section 8.2.2 will proceed to discuss the local shape descriptors that delve into the concept of the desirability of correlated variation patterns in relation to quality perceptions.

Dimensional quality of a population of assemblies or stampings according to the global statistical shape model can be seen to depend on the spread of data including the number of clusters, distance between the population mean and nominal specifica- tion, and the type of global variation patterns present. The process for dimensional control utilising the global shape model differs from a univariate approach in that correlated variation patterns, rather than just individual MP's, can be targeted with process control measures. Coupling the multivariate global shape model with shape visualisation capabilities would further enhance the diagnostic toolset by allowing pro- cess engineering to see what variation patterns are present, which could lead to a more intuitive understanding of the process and how to manipulate it for dimensional control.

132 CHAPTER 8. CONCLUSION

New tolerancing approach

The multivariate statistical shape model provides an advanced probabilistic represen- tation of auto-body assembly processes. In practice tolerances are often based on a univariate probabilistic representation where 99.7% or more of process data should fall within the specified tolerance zone. This concept could be extended to the multivariate shape model. Here, probabilistic thresholds could be set such that all regions above this threshold account for 99.7% or more of the training process data. Test process samples that fall above the set probability threshold would be within the tolerance re- gion, and test samples that fall below the threshold will be outside the tolerance region. This concept is illustrated in Figure 8.4 using the same example data set as in section 8.2.1. This tolerancing approach would allow for the monitoring of a single proba- bilistic model, rather than a series of independent models as is currently the practice, which would be very advantageous given high dimensional data sets made available by OCMM's. In general, it provides a more informative and implementable approach for statistical modelling and tolerancing of exible assemblies given high dimensional measurement data. Again, when coupled with the visualisation of corresponding phys- ical shape variation patterns the approach has the potential to enable a more intuitive diagnosis of out-of-control process cases.

Figure 8.4: Example tolerance zone within a multivariate statistical shape model. A three level probability contour at the 10%, 40%, and 99.7% levels is indicated. Here, 99.7% of process samples should fall within the indicated contours. For a new toler- ancing approach, samples falling within the 99.7% contour could be deemed within- tolerance, and samples falling outside this zone out-of-tolerance.

8.2. A FRAMEWORK FOR QUALITY ASSESSMENT 133