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...measurement and generate large quality-related databases with the expectation that this will somehow aid in efforts to systematically identify and eliminate the major root causes of manufacturing variation. Effectively transforming the large sets of measurement data into extracted knowledge that is useful for variation reduction efforts, however, is a challenging data mining problem.
We investigate this problem following the same paradigm considered in Apley and Lee (2003), in which one views each major variation source as causing a distinct variation pattern in the data, spreading across any or all of the different variables that are measured. Each variation pattern represents the interrelated manner in which the different variables vary from part to part, due to the effects of a particular variation source. The data mining objective becomes one of unsupervised learning, in which we seek to discover the nature of any variation pattern that happens to be present in the data. We refer to this as blindly identifying the variation patterns, in the sense that we are not attempting to recognize the presence of premodeled or pre-trained patterns. Rather, we seek to identify the nature of the patterns based only on a sample of data, with no prior training or modeling required. After blindly identifying the nature of a variation pattern, the results can be graphically illustrated in order to facilitate root cause identification, which we illustrate with examples in later sections.
A number of techniques have been recently developed for blindly identifying the nature of variation patterns in large sets of manufacturing measurement data (e.g., Apley and Shi (2001), Apley and Lee (2003), and Lee and Apley (2004)). All of these approaches assume that the variation patterns can be represented using linear models, however. Although linear models are reasonably versatile, there are many situations in which nonlinear models are required to represent variation patterns, such as in the following example from autobody assembly. Figure 1 illustrates schematically the rear liftgate opening of a sports utility vehicle and shows the locations at which six cross-car (left/right) dimensional measurements on the left and right bodysides are taken (denoted [x.sub.1] through [x.sub.6]). The measurements are obtained automatically via in-process laser measurement, so that every autobody is measured (for a more detailed description of the assembly process and measurement technology, refer to Ceglarek and Shi (1996) or Apley and Shi (1998; 2001)). Although almost 200 different dimensional features were measured for each autobody, for simplicity we illustrate with only the six measurements shown in Fig. 1. The nonlinear variation pattern described in the following paragraphs primarily affected only the liftgate region of the autobody.
[FIGURE 1 OMITTED]
The assembly process is relatively complex and involves hundreds of different assembly stations and thousands of different tooling elements. When a tooling element breaks, wears, malfunctions, or is simply not designed properly, this often results in a distinct variation pattern in the dimensional measurement data. Figure 2 illustrates this with scatter plots of pairs of the six variables over a sample of 100 measured autobodies. The measurements are deviations from nominal, in units of millimeters. A positive measurement represents deviation to the right. Although the relationship between [x.sub.2] and [x.sub.3] appears linear, the scatter plots for [x.sub.2]/[x.sub.5] and [x.sub.3]/[x.sub.4] clearly illustrate that the variation pattern is nonlinear. Moreover, this nonlinear pattern appears to be approximately piecewise linear with only two segments (i.e., pieces).
[FIGURE 2 OMITTED]
One potential root cause for the variation pattern is due to a fixturing problem when locating the right bodyside in the framing station (a major assembly station in which the left and right bodysides are joined to the underbody and a set of upper cross-members). When the right bodyside deviates by only a small amount to the right, it has no affect on the left bodyside. When the right bodyside deviates by a larger amount to the right, however, it begins to interfere with the upper cross-member. The upper cross-member then interferes with the left bodyside by also pulling the left bodyside towards the right.
In situations like that depicted in Fig. 2, linear models are inadequate to represent the nonlinear relationship between the different variables. One potential method for treating nonlinear variation patterns is based on the notion of a principal curve (Hastie and Stuetzle, 1989), which is a nonlinear generalization of Principal Components Analysis (PCA). Broadly speaking, a principal curve is defined as a one-dimensional curve that passes through the middle of the distribution of higher dimensional data. Principal curve estimation is relatively robust and has been applied to a variety of nonlinear data analysis problems (Tibshirani, 1996; Chang and Ghosh, 2001; Delicado, 2001). However, most applications of principal curves have been for relatively low dimensional data, especially for two-dimensional image processing (e.g., Banfield and Raftery (1992), Kegl et al., (2000), and also Chang and Ghosh (2001)). For the high dimensional data often encountered in manufacturing, principal curve estimation becomes inefficient. Because of this, we use a common data preprocessing step that involves linear PCA to first reduce the dimensionality of the problem. This not only reduces the computational complexity, but also improves the estimation accuracy by filtering out a substantial portion of the noise (i.e., random variations that are not due to any systematic pattern). One advantage of the principal curve approach is that it lends itself well to visualizing the blindly identified variation patterns. Effective visualization of a variation pattern is crucial for identifying the root cause of the variation. In a later section, we illustrate the visualization approach with an example in which the data are extremely high dimensional, representing point cloud data from laser-scanned stamped panels.
The remainder of the paper is organized as follows. Section 2 discusses the model we use to represent nonlinear manufacturing variation patterns. Section 3 provides some background on linear PCA and nonlinear principal curves. Although PCA is a common first step for reducing dimensionality in data mining applications, including principal curve estimation (e.g., Delicado and Huerta (2003)), the amount of information that is lost ranges from nothing at all to perhaps a non-negligible amount. In Section 4, we argue that for our application virtually no information is lost. In Section 5, we describe the algorithm for principal curve estimation with the PCA preprocessing step. Examples illustrating the approach are provided in Sections 5 and 6. Section 7 includes a Monte Carlo analysis demonstrating the performance improvement when one uses linear PCA to first reduce the dimensionality...
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